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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2411.13951v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S1" title="In A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S2" title="In A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related Work</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S2.SS1" title="In 2 Related Work ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Publicly Available Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S2.SS2" title="In 2 Related Work ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Doubts Regarding Applicability of Deep Learning</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3" title="In A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Proposed Dataset</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.SS1" title="In 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Simulation Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.SS2" title="In 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Dataset Generation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.SS3" title="In 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Usability of the Dataset</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4" title="In A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Baseline Results on the Dataset</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.SS1" title="In 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Methodology</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.SS2" title="In 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Reproducibility and Benchmarking Considerations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.SS3" title="In 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Results and Discussion</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S5" title="In A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusion and Outlook</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\addbibresource</span> <p class="ltx_p" id="p1.2">references.bib</p> </div> <h1 class="ltx_title ltx_title_document">A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Lucas Correia <br class="ltx_break"/>Leiden University <br class="ltx_break"/>Leiden <br class="ltx_break"/>The Netherlands <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id1.1.id1">l.ferreira.correia@liacs.leidenuniv.nl</span> <br class="ltx_break"/>Jan-Christoph Goos <br class="ltx_break"/>Mercedes-Benz AG <br class="ltx_break"/>Stuttgart <br class="ltx_break"/>Germany <br class="ltx_break"/>Thomas Bäck <br class="ltx_break"/>Leiden University <br class="ltx_break"/>Leiden <br class="ltx_break"/>The Netherlands <br class="ltx_break"/>Anna V. Kononova <br class="ltx_break"/>Leiden University <br class="ltx_break"/>Leiden <br class="ltx_break"/>The Netherlands <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id2.2.id2">a.kononova@liacs.leidenuniv.nl</span> <br class="ltx_break"/> </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id3.id1">Benchmarking anomaly detection approaches for multivariate time series is challenging due to the lack of high-quality datasets. Current publicly available datasets are too small, not diverse and feature trivial anomalies, which hinders measurable progress in this research area. We propose a solution: a diverse, extensive, and non-trivial dataset generated via state-of-the-art simulation tools that reflects realistic behaviour of an automotive powertrain, including its multivariate, dynamic and variable-state properties. To cater for both unsupervised and semi-supervised anomaly detection settings, as well as time series generation and forecasting, we make different versions of the dataset available, where training and test subsets are offered in contaminated and clean versions, depending on the task. We also provide baseline results from a small selection of approaches based on deterministic and variational autoencoders, as well as a non-parametric approach. As expected, the baseline experimentation shows that the approaches trained on the semi-supervised version of the dataset outperform their unsupervised counterparts, highlighting a need for approaches more robust to contaminated training data.</p> </div> <span class="ltx_note ltx_role_footnotetext" id="footnotex1"><sup class="ltx_note_mark">0</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">0</sup><span class="ltx_note_type">footnotetext: </span>This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.</span></span></span> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">As the digitisation of industrial processes progresses, more and more data is recorded. Ensuring this data is representative of the process is important, as downstream tasks like modelling or optimisation can be negatively impacted by incomplete or contaminated data. For tasks that require system behaviour modelling, data deviating from the norm is hence undesired, and we speak of <em class="ltx_emph ltx_font_italic" id="S1.p1.1.1">anomalous</em> behaviour. Recorded data manifests itself in many forms depending on the application and domain, one form being time series. Examples of real-world time series applications are diverse, ranging from cardiology <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">moody_impact_2001</span>]</cite> and server metrics monitoring <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">su_robust_2019</span>]</cite> to water systems <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mathur_swat_2016</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">ahmed_wadi_2017</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">zhang_anomaly_2023</span>]</cite> and unmanned aerial vehicles <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">zhong_unmanned_2022</span>]</cite>. Note that, we use <em class="ltx_emph ltx_font_italic" id="S1.p1.1.2">time series</em> and <em class="ltx_emph ltx_font_italic" id="S1.p1.1.3">sequences</em> synonymously throughout this paper.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.5"><em class="ltx_emph ltx_font_italic" id="S1.p2.5.1">Time series</em> are signals that represent a property or feature of a dynamic system as a function of time, usually sampled at a fixed rate. An arbitrary time series <math alttext="\mathcal{S}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">𝒮</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝒮</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\mathcal{S}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">caligraphic_S</annotation></semantics></math> can be univariate, i.e. <math alttext="\mathcal{S}\in\mathbb{R}^{T}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mrow id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">𝒮</mi><mo id="S1.p2.2.m2.1.1.1" xref="S1.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml"><mi id="S1.p2.2.m2.1.1.3.2" xref="S1.p2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S1.p2.2.m2.1.1.3.3" xref="S1.p2.2.m2.1.1.3.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><in id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1.1"></in><ci id="S1.p2.2.m2.1.1.2.cmml" xref="S1.p2.2.m2.1.1.2">𝒮</ci><apply id="S1.p2.2.m2.1.1.3.cmml" xref="S1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.3.1.cmml" xref="S1.p2.2.m2.1.1.3">superscript</csymbol><ci id="S1.p2.2.m2.1.1.3.2.cmml" xref="S1.p2.2.m2.1.1.3.2">ℝ</ci><ci id="S1.p2.2.m2.1.1.3.3.cmml" xref="S1.p2.2.m2.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\mathcal{S}\in\mathbb{R}^{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">caligraphic_S ∈ blackboard_R start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, or multivariate, i.e. <math alttext="\mathcal{S}\in\mathbb{R}^{T\times d}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">𝒮</mi><mo id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">∈</mo><msup id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml"><mi id="S1.p2.3.m3.1.1.3.2" xref="S1.p2.3.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S1.p2.3.m3.1.1.3.3" xref="S1.p2.3.m3.1.1.3.3.cmml"><mi id="S1.p2.3.m3.1.1.3.3.2" xref="S1.p2.3.m3.1.1.3.3.2.cmml">T</mi><mo id="S1.p2.3.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p2.3.m3.1.1.3.3.1.cmml">×</mo><mi id="S1.p2.3.m3.1.1.3.3.3" xref="S1.p2.3.m3.1.1.3.3.3.cmml">d</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><in id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1.1"></in><ci id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2">𝒮</ci><apply id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p2.3.m3.1.1.3.1.cmml" xref="S1.p2.3.m3.1.1.3">superscript</csymbol><ci id="S1.p2.3.m3.1.1.3.2.cmml" xref="S1.p2.3.m3.1.1.3.2">ℝ</ci><apply id="S1.p2.3.m3.1.1.3.3.cmml" xref="S1.p2.3.m3.1.1.3.3"><times id="S1.p2.3.m3.1.1.3.3.1.cmml" xref="S1.p2.3.m3.1.1.3.3.1"></times><ci id="S1.p2.3.m3.1.1.3.3.2.cmml" xref="S1.p2.3.m3.1.1.3.3.2">𝑇</ci><ci id="S1.p2.3.m3.1.1.3.3.3.cmml" xref="S1.p2.3.m3.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">\mathcal{S}\in\mathbb{R}^{T\times d}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">caligraphic_S ∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="T" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_T</annotation></semantics></math> refers to the number of discrete time steps and <math alttext="d" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_d</annotation></semantics></math> to the number of features in the time series. More specifically, univariate time series solely possess a temporal correlation, i.e. along the time axis, whereas multivariate time series can also contain correlation along the feature axis.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.5">Detecting anomalous behaviour in time series is referred to <em class="ltx_emph ltx_font_italic" id="S1.p3.5.1">time series anomaly detection</em>, which can be split into two main areas: <em class="ltx_emph ltx_font_italic" id="S1.p3.5.2">continuous- and discrete-sequence</em> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_online_2024</span>]</cite>, where the former is the most common type present in public datasets. It is defined as detecting anomalies in a process that runs for a continuous time period without breaks. Typically, the test subset <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><msup id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1">superscript</csymbol><ci id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2">𝒟</ci><ci id="S1.p3.1.m1.1.1.3a.cmml" xref="S1.p3.1.m1.1.1.3"><mtext id="S1.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="S1.p3.1.m1.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math> in the dataset <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">caligraphic_D</annotation></semantics></math> in a continuous-sequence problem consists of a singular multivariate time series composed of multiple nominal and anomalous sub-sequences, i.e. <math alttext="\mathcal{D}^{\text{test}}=\{\mathcal{S}_{1}\}" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><msup id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.1.1.3.2" xref="S1.p3.3.m3.1.1.3.2.cmml">𝒟</mi><mtext id="S1.p3.3.m3.1.1.3.3" xref="S1.p3.3.m3.1.1.3.3a.cmml">test</mtext></msup><mo id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml">=</mo><mrow id="S1.p3.3.m3.1.1.1.1" xref="S1.p3.3.m3.1.1.1.2.cmml"><mo id="S1.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S1.p3.3.m3.1.1.1.2.cmml">{</mo><msub id="S1.p3.3.m3.1.1.1.1.1" xref="S1.p3.3.m3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.1.1.1.1.1.2" xref="S1.p3.3.m3.1.1.1.1.1.2.cmml">𝒮</mi><mn id="S1.p3.3.m3.1.1.1.1.1.3" xref="S1.p3.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S1.p3.3.m3.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><apply id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1"><eq id="S1.p3.3.m3.1.1.2.cmml" xref="S1.p3.3.m3.1.1.2"></eq><apply id="S1.p3.3.m3.1.1.3.cmml" xref="S1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.3.1.cmml" xref="S1.p3.3.m3.1.1.3">superscript</csymbol><ci id="S1.p3.3.m3.1.1.3.2.cmml" xref="S1.p3.3.m3.1.1.3.2">𝒟</ci><ci id="S1.p3.3.m3.1.1.3.3a.cmml" xref="S1.p3.3.m3.1.1.3.3"><mtext id="S1.p3.3.m3.1.1.3.3.cmml" mathsize="70%" xref="S1.p3.3.m3.1.1.3.3">test</mtext></ci></apply><set id="S1.p3.3.m3.1.1.1.2.cmml" xref="S1.p3.3.m3.1.1.1.1"><apply id="S1.p3.3.m3.1.1.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.1.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.p3.3.m3.1.1.1.1.1.2.cmml" xref="S1.p3.3.m3.1.1.1.1.1.2">𝒮</ci><cn id="S1.p3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S1.p3.3.m3.1.1.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">\mathcal{D}^{\text{test}}=\{\mathcal{S}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT = { caligraphic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. In this work, we use <em class="ltx_emph ltx_font_italic" id="S1.p3.5.3">nominal</em> as a synonym for <em class="ltx_emph ltx_font_italic" id="S1.p3.5.4">normal</em> or <em class="ltx_emph ltx_font_italic" id="S1.p3.5.5">anomaly-free</em> to avoid confusion with Gaussian distributions. Discrete-sequence anomaly detection, in contrast, is defined as detecting anomalies in <math alttext="N" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">N</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_N</annotation></semantics></math> chunks of processes that happen independently of each other, such as automotive test benches, where several tests may occur sequentially but are not temporally contiguous and hence provide a time series for each test, i.e. <math alttext="\mathcal{D}^{\text{test}}=\{\mathcal{S}_{1},...,\mathcal{S}_{n},...,\mathcal{S% }_{N}\}" class="ltx_Math" display="inline" id="S1.p3.5.m5.5"><semantics id="S1.p3.5.m5.5a"><mrow id="S1.p3.5.m5.5.5" 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Here, the testees, i.e. the test subjects, are not monitored over a continuous period of time, but are instead monitored solely during each process chunk. Automotive testing is not the only use for discrete-sequence anomaly detection, however. Another discrete-sequence problem could also include the analysis of the flight behaviour of an aeroplane, where the time while it is docked is irrelevant and may not be recorded. Therefore, datasets for discrete-sequence anomaly detection consist of several nominal and anomalous time series, where a given anomalous time series may be entirely anomalous or only partly.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Depending on the system, the time series data may also feature <em class="ltx_emph ltx_font_italic" id="S1.p4.1.1">variable states</em>, meaning the recorded signals appear slightly different if certain external conditions change but are still considered nominal. One example of a variable-state system is a battery, where the voltage response to current changes depending on states like the battery temperature and the battery state of charge (SoC). A problem involving such a system requires the distinction between behaviour changes due to different states and behaviour changes due to an anomaly, further complicating detection.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">In addition to that, detecting anomalous behaviour in a timely manner is also advantageous because the source of anomalous behaviour may bring about damage to said system. Such problems where the detection delay plays a role are referred to as <em class="ltx_emph ltx_font_italic" id="S1.p5.1.1">online</em> time series anomaly detection.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Analogous to types of learning, there is supervised, semi-supervised, and unsupervised anomaly detection. <em class="ltx_emph ltx_font_italic" id="S1.p6.1.1">Supervised</em> anomaly detection is essentially imbalanced binary time series classification and is only rarely found in the literature. This is most likely because, in real-world use cases, possible anomaly types are rarely known a priori. In addition to that, labelling data is expensive, which is why unsupervised and semi-supervised anomaly detection are more relevant in both literature and the real world. <em class="ltx_emph ltx_font_italic" id="S1.p6.1.2">Unsupervised</em> anomaly detection is independent of any labels, i.e. any available data for model training contains both anomalous and nominal time series, and it is not known which is which <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">chandola_anomaly_2012</span>]</cite>. In contrast, <em class="ltx_emph ltx_font_italic" id="S1.p6.1.3">semi-supervised</em> anomaly detection can be considered a more relaxed setting, where anomalous time series are absent from the training subset <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">chandola_anomaly_2012</span>]</cite>. In the real world, semi-supervised problems still require some labelling to ensure an entirely nominal training subset, which is not always given.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">Recently, time series anomaly detection has mostly been attempted using deep learning, with simpler statistical methods remaining largely unexplored. Several researchers <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">audibert_deep_2022</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">rewicki_is_2023</span>]</cite> have hence raised concerns on whether deep learning-based methods are warranted for the complexity of publicly available datasets and even challenge any anomaly detection performance claims made because such datasets are deemed as flawed <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite>. The lack of a standardised procedure to benchmark approaches on a high-quality dataset is considered a fundamental issue, which prevents any tangible and objective advances in the field.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">Our contribution is a novel, non-trivial, and high-quality dataset consisting of multivariate time series for online anomaly detection, named the Powertrain Anomaly Time series bencHmark (PATH) dataset. While primarily aimed at unsupervised anomaly detection, we provide versions for semi-supervised anomaly detection and time series generation and forecasting as well. Despite the data being generated using simulation, the electric vehicle simulation model is strongly motivated by the real world and is therefore complex and variable-state.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">This paper is structured as follows. First, we introduce the related work in the area of benchmarking time series anomaly detection approaches. It includes discussion on the datasets used to evaluate time series anomaly detection approaches in the past, why they are no longer suitable, and a summary of the work dedicated to outlining the status quo in benchmarking time series anomaly detection approaches. Then, we introduce the PATH dataset in detail, outlining the generation process and a few benchmarking considerations. Following that, we provide some baseline results for a selection of deep learning-based approaches, as well as a non-parametric method. Finally, we conclude our work and outline an outlook on future work. The source code corresponding to this paper and the simulation model can be found under <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/lcs-crr/PATH" title="">https://github.com/lcs-crr/PATH</a>, and the dataset can be downloaded from <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://zenodo.org/records/13255121" title="">https://zenodo.org/records/13255121</a> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">data_set</span>]</cite>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related Work</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Publicly Available Datasets</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">Over the last few years, five benchmark datasets have emerged as by far the most popular, with at least one of them being cited in the vast majority of publications on multivariate time series anomaly detection. 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Note that the SMD dataset consists of 28 time series in the training subset and 28 time series in the test subset, hence one time series for each of the 28 different machines. 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id="S2.T1.12.4.4.m1.1a"><mrow id="S2.T1.12.4.4.m1.1b"><mo id="S2.T1.12.4.4.m1.1.1">%</mo><mi id="S2.T1.12.4.4.m1.1.2">A</mi></mrow><annotation encoding="application/x-tex" id="S2.T1.12.4.4.m1.1c">\%A</annotation><annotation encoding="application/x-llamapun" id="S2.T1.12.4.4.m1.1d">% italic_A</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.12.5.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S2.T1.12.5.1.1"><a class="ltx_ref ltx_href" href="https://itrust.sutd.edu.sg/itrust-labs_data%20sets/data%20set_info/" title="">SWaT</a></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T1.12.5.1.2">51</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T1.12.5.1.3">1</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T1.12.5.1.4">1</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T1.12.5.1.5"> <span class="ltx_ERROR undefined" id="S2.T1.12.5.1.5.1">\qty</span>12</td> </tr> <tr class="ltx_tr" id="S2.T1.12.6.2"> <td class="ltx_td ltx_align_left" id="S2.T1.12.6.2.1"><a class="ltx_ref ltx_href" href="https://itrust.sutd.edu.sg/itrust-labs_data%20sets/data%20set_info/" title="">WADI</a></td> <td class="ltx_td ltx_align_center" id="S2.T1.12.6.2.2">127</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.6.2.3">1</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.6.2.4">1</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.6.2.5"> <span class="ltx_ERROR undefined" id="S2.T1.12.6.2.5.1">\qty</span>6</td> </tr> <tr class="ltx_tr" id="S2.T1.12.7.3"> <td class="ltx_td ltx_align_left" id="S2.T1.12.7.3.1"><a class="ltx_ref ltx_href" href="https://github.com/khundman/telemanom" title="">SMAP</a></td> <td class="ltx_td ltx_align_center" id="S2.T1.12.7.3.2">25</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.7.3.3">54</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.7.3.4">54</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.7.3.5"> <span class="ltx_ERROR undefined" id="S2.T1.12.7.3.5.1">\qty</span>13</td> </tr> <tr class="ltx_tr" id="S2.T1.12.8.4"> <td class="ltx_td ltx_align_left" id="S2.T1.12.8.4.1"><a class="ltx_ref ltx_href" href="https://github.com/khundman/telemanom" title="">MSL</a></td> <td class="ltx_td ltx_align_center" id="S2.T1.12.8.4.2">55</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.8.4.3">27</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.8.4.4">27</td> <td class="ltx_td ltx_align_center" id="S2.T1.12.8.4.5"> <span class="ltx_ERROR undefined" id="S2.T1.12.8.4.5.1">\qty</span>11</td> </tr> <tr class="ltx_tr" id="S2.T1.12.9.5"> <td class="ltx_td ltx_align_left ltx_border_b" id="S2.T1.12.9.5.1"><a class="ltx_ref ltx_href" href="https://github.com/NetManAIOps/OmniAnomaly" title="">SMD</a></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S2.T1.12.9.5.2">38</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S2.T1.12.9.5.3">1</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S2.T1.12.9.5.4">1</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S2.T1.12.9.5.5"> <span class="ltx_ERROR undefined" id="S2.T1.12.9.5.5.1">\qty</span>4</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.1">The MSL <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">hundman_detecting_2018</span>]</cite>, SMAP <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">hundman_detecting_2018</span>]</cite>, and SMD <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">su_robust_2019</span>]</cite> have already been thoroughly analysed by Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite>, who point out several <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.1.1">issues with the datasets</em>. The first issue observed in the datasets is triviality, defined by being solvable using so-called <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.1.2">one-line code</em>, such as the moving standard deviation over a subset of the dataset features. Moreover, all of them suffer from what Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite> have called unrealistic anomaly density, meaning that they have sub-sequences with a very high anomaly share and hence do not match the assumption that anomalies are rare events. In addition to that, Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite> suspect possible mislabelling present in the MSL dataset. They base their suspicion on the fact that the dataset contains sub-sequences with static behaviour in an evidently dynamic channel, which is labelled as nominal. Many of the issues pointed out by Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite> can also be extended to the SWaT and WADI datasets, as discussed by Correia et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_online_2024</span>]</cite>.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">Certain real-world applications like automotive testing present <em class="ltx_emph ltx_font_italic" id="S2.SS1.p3.1.1">complexities</em> previously unseen in public datasets. As outlined by Correia et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite>, such applications comprise much more diverse discrete-sequence datasets, owed to the presence of both highly dynamic and mostly static features, as well as variable states. The presence of variable states leads to features exhibiting a different pattern depending on the time it is observed. In the context of automotive testing, an example of such a feature would be the state of charge of a battery, which discharges with time and hence shows different behaviour for the same test done twice in a row.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.1">Hence, we construct a new high-quality dataset that is non-trivial and reflects real-world complexity.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Doubts Regarding Applicability of Deep Learning</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">Recently, there has been growing doubt on whether deep learning (DL) algorithms are definitively the better choice for time series anomaly detection. For the purpose of this publication, classical methods refer to all approaches not based on DL, including non-parametric and statistical approaches, as well as simpler machine learning methods like clustering.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite> claim that the superiority of DL in anomaly detection is assumed to be a given, despite a lack of clear evidence for the need for DL. They stress that existing classical methods should be considered, given their generally simpler and faster nature.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">To investigate the comparative performance of classical methods and DL-based methods, Audibert et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">audibert_deep_2022</span>]</cite> analyse a variety of different models on five of the most popular benchmark datasets, shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S2.T1" title="Table 1 ‣ 2.1 Publicly Available Datasets ‣ 2 Related Work ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">1</span></a>. They conclude that, across the datasets considered, there is no algorithm that dominates all the other ones, arguing that there is no reason to omit classical methods from benchmarking.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.1">Rewicki et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">rewicki_is_2023</span>]</cite> also conduct a comparative study of classical and DL-based methods, though on the UCR Anomaly Archive benchmark proposed by Wu and Keogh <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">wu_current_2021</span>]</cite>, which exclusively contains univariate time series and therefore lacks correlations between channels present in multivariate time series. They conclude that classical methods perform better than their DL counterparts, although this is to be expected given the simpler, univariate nature of the dataset.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.1">While the findings and doubts of the above-mentioned are valid, they are limited by the lack of large, high-quality multivariate datasets. In this paper, we purposefully include results from a state-of-the-art classical method to find out whether doubts on DL are still justified for extensive and complex real-world datasets. See Section <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4" title="4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">4</span></a> for results and discussion.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Proposed Dataset</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Simulation Model</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">To create an extensive and diverse dataset, we propose to use a physically-inspired model from which we can generate data using simulation. MathWorks offers reference models for a variety of dynamic systems, one of which is the full electric vehicle (FEV) model <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mathworks_build_2024</span>]</cite> from the powertrain block set in Simulink. This choice is based on our familiarity with the domain, as generating data blindly without any background may lead to systematic errors. The FEV model offered by MathWorks consists of six main subsystems: the drive cycle block, the driver block, the environment block, the controllers block and the vehicle block. The topology of the FEV model is illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F1" title="Figure 1 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="229" id="S3.F1.g1" src="x1.png" width="581"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Simplified schematic of the FEV model used for the generation of the PATH dataset. Numbers represent the indices of signal flow, reference is shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.1">To represent system behaviour, <math alttext="d_{\mathcal{D}}=16" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mrow id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml"><msub id="S3.SS1.p2.1.m1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.2.cmml"><mi id="S3.SS1.p2.1.m1.1.1.2.2" xref="S3.SS1.p2.1.m1.1.1.2.2.cmml">d</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.1.m1.1.1.2.3" xref="S3.SS1.p2.1.m1.1.1.2.3.cmml">𝒟</mi></msub><mo id="S3.SS1.p2.1.m1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.p2.1.m1.1.1.3" xref="S3.SS1.p2.1.m1.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><apply id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1"><eq id="S3.SS1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1"></eq><apply id="S3.SS1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.1.1.2.1.cmml" xref="S3.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.1.m1.1.1.2.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2.2">𝑑</ci><ci id="S3.SS1.p2.1.m1.1.1.2.3.cmml" xref="S3.SS1.p2.1.m1.1.1.2.3">𝒟</ci></apply><cn id="S3.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p2.1.m1.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">d_{\mathcal{D}}=16</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_d start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT = 16</annotation></semantics></math> signals are chosen to be logged during simulation and are summarised in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>. We chose these signals based on domain knowledge, with the goal of picking the features that are most representative of powertrain behaviour.</p> </div> <figure class="ltx_table" id="S3.T2"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 2: </span>Signals included in the PATH dataset, along with their physical units and persistent indices.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T2.14"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T2.14.15.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T2.14.15.1.1">Index</th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column" id="S3.T2.14.15.1.2">Signal Name</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T2.14.15.1.3">Unit</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T2.1.1"> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T2.1.1.2">1</td> <td class="ltx_td 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class="ltx_td ltx_align_center" id="S3.T2.12.12.1"><math alttext="\mathrm{\SIUnitSymbolCelsius}" class="ltx_Math" display="inline" id="S3.T2.12.12.1.m1.1"><semantics id="S3.T2.12.12.1.m1.1a"><mi class="ltx_unit" id="S3.T2.12.12.1.m1.1.1" xref="S3.T2.12.12.1.m1.1.1.cmml">°C</mi><annotation-xml encoding="MathML-Content" id="S3.T2.12.12.1.m1.1b"><csymbol cd="latexml" id="S3.T2.12.12.1.m1.1.1.cmml" xref="S3.T2.12.12.1.m1.1.1">degreeCelsius</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.12.12.1.m1.1c">\mathrm{\SIUnitSymbolCelsius}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.12.12.1.m1.1d">°C</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T2.13.13"> <td class="ltx_td ltx_align_center" id="S3.T2.13.13.2">15</td> <td class="ltx_td ltx_align_left" id="S3.T2.13.13.3">Cooling Pump Power</td> <td class="ltx_td ltx_align_center" id="S3.T2.13.13.1"><math alttext="\mathrm{W}" class="ltx_Math" display="inline" id="S3.T2.13.13.1.m1.1"><semantics id="S3.T2.13.13.1.m1.1a"><mi class="ltx_unit" id="S3.T2.13.13.1.m1.1.1" mathvariant="normal" xref="S3.T2.13.13.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.T2.13.13.1.m1.1b"><csymbol cd="latexml" id="S3.T2.13.13.1.m1.1.1.cmml" xref="S3.T2.13.13.1.m1.1.1">watt</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.13.13.1.m1.1c">\mathrm{W}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.13.13.1.m1.1d">roman_W</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T2.14.14"> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.14.14.2">16</td> <td class="ltx_td ltx_align_left ltx_border_b" id="S3.T2.14.14.3">Refrigerator Power</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.14.14.1"><math alttext="\mathrm{W}" class="ltx_Math" display="inline" id="S3.T2.14.14.1.m1.1"><semantics id="S3.T2.14.14.1.m1.1a"><mi class="ltx_unit" id="S3.T2.14.14.1.m1.1.1" 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From the list of speed profiles available in the original FEV model <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mathworks_build_2024</span>]</cite>, a subset of them is eliminated due to their unrealistic nature, e.g. high linearity or duplicity, as many cycles are present as sub-sequences in others. Our analysis shows that, for example, the presence of the FTP72 drive cycle within FTP75 or the presence of the LA92Short drive cycle within LA92. In addition to that, drive cycles aimed at heavy vehicles, like trucks or buses, are also eliminated. The resulting subset of drive cycles chosen for simulation contains <math alttext="33" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mn id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">33</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><cn id="S3.SS1.p3.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1">33</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">33</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">33</annotation></semantics></math> different speed profiles of varying length, shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T3" title="Table 3 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">3</span></a>, along with their lengths in seconds. Some drive cycles may be designed for specific types of powertrains such as diesel ones, but given that they only represent vehicle speed profiles, there is no reason why they cannot be driven by a vehicle with an electric powertrain, like the one modelled in this case.</p> </div> <figure class="ltx_table" id="S3.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 3: </span>Drive cycles used for the PATH dataset generation, along with their respective lengths in seconds.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T3.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T3.1.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column" id="S3.T3.1.1.1.1">Drive Cycle</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T3.1.1.1.2">Length [s]</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T3.1.2.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S3.T3.1.2.1.1">FTP75</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T3.1.2.1.2">2474</td> </tr> <tr class="ltx_tr" id="S3.T3.1.3.2"> <td class="ltx_td ltx_align_left" id="S3.T3.1.3.2.1">US06</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.3.2.2">600</td> </tr> <tr class="ltx_tr" id="S3.T3.1.4.3"> <td class="ltx_td ltx_align_left" id="S3.T3.1.4.3.1">SC03</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.4.3.2">600</td> </tr> <tr class="ltx_tr" id="S3.T3.1.5.4"> <td class="ltx_td ltx_align_left" id="S3.T3.1.5.4.1">HWFET</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.5.4.2">765</td> </tr> <tr class="ltx_tr" id="S3.T3.1.6.5"> <td class="ltx_td ltx_align_left" id="S3.T3.1.6.5.1">NYCC</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.6.5.2">598</td> </tr> <tr class="ltx_tr" id="S3.T3.1.7.6"> <td class="ltx_td ltx_align_left" id="S3.T3.1.7.6.1">HUDDS</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.7.6.2">1060</td> </tr> <tr class="ltx_tr" id="S3.T3.1.8.7"> <td class="ltx_td ltx_align_left" id="S3.T3.1.8.7.1">LA92</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.8.7.2">1435</td> </tr> <tr class="ltx_tr" id="S3.T3.1.9.8"> <td class="ltx_td ltx_align_left" id="S3.T3.1.9.8.1">IM240</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.9.8.2">240</td> </tr> <tr class="ltx_tr" id="S3.T3.1.10.9"> <td class="ltx_td ltx_align_left" id="S3.T3.1.10.9.1">UDDS</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.10.9.2">1369</td> </tr> <tr class="ltx_tr" id="S3.T3.1.11.10"> <td class="ltx_td ltx_align_left" id="S3.T3.1.11.10.1">WLTP Class 1</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.11.10.2">1022</td> </tr> <tr class="ltx_tr" id="S3.T3.1.12.11"> <td class="ltx_td ltx_align_left" id="S3.T3.1.12.11.1">WLTP Class 2</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.12.11.2">1477</td> </tr> <tr class="ltx_tr" id="S3.T3.1.13.12"> <td class="ltx_td ltx_align_left" id="S3.T3.1.13.12.1">WLTP Class 3</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.13.12.2">1800</td> </tr> <tr class="ltx_tr" id="S3.T3.1.14.13"> <td class="ltx_td ltx_align_left" id="S3.T3.1.14.13.1">Artemis Urban</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.14.13.2">993</td> </tr> <tr class="ltx_tr" id="S3.T3.1.15.14"> <td class="ltx_td ltx_align_left" id="S3.T3.1.15.14.1">Artemis Rural Road</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.15.14.2">1082</td> </tr> <tr class="ltx_tr" id="S3.T3.1.16.15"> <td class="ltx_td ltx_align_left" id="S3.T3.1.16.15.1">Artemis Motorway 130 kmph</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.16.15.2">1068</td> </tr> <tr class="ltx_tr" id="S3.T3.1.17.16"> <td class="ltx_td ltx_align_left" id="S3.T3.1.17.16.1">Artemis Motorway 150 kmph</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.17.16.2">1068</td> </tr> <tr class="ltx_tr" id="S3.T3.1.18.17"> <td class="ltx_td ltx_align_left" id="S3.T3.1.18.17.1">JC08</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.18.17.2">1204</td> </tr> <tr class="ltx_tr" id="S3.T3.1.19.18"> <td class="ltx_td ltx_align_left" id="S3.T3.1.19.18.1">JC08 Hot</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.19.18.2">1376</td> </tr> <tr class="ltx_tr" id="S3.T3.1.20.19"> <td class="ltx_td ltx_align_left" id="S3.T3.1.20.19.1">World Harmonized Vehicle Cycle (WHVC)</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.20.19.2">900</td> </tr> <tr class="ltx_tr" id="S3.T3.1.21.20"> <td class="ltx_td ltx_align_left" id="S3.T3.1.21.20.1">Braunschweig City Driving Cycle</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.21.20.2">1740</td> </tr> <tr class="ltx_tr" id="S3.T3.1.22.21"> <td class="ltx_td ltx_align_left" id="S3.T3.1.22.21.1">RTS 95</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.22.21.2">886</td> </tr> <tr class="ltx_tr" id="S3.T3.1.23.22"> <td class="ltx_td ltx_align_left" id="S3.T3.1.23.22.1">ETC FIGE Version 4</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.23.22.2">1800</td> </tr> <tr class="ltx_tr" id="S3.T3.1.24.23"> <td class="ltx_td ltx_align_left" id="S3.T3.1.24.23.1">CUEDC Petrol cycle</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.24.23.2">499</td> </tr> <tr class="ltx_tr" id="S3.T3.1.25.24"> <td class="ltx_td ltx_align_left" id="S3.T3.1.25.24.1">CUEDC SPC240 cycle</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.25.24.2">240</td> </tr> <tr class="ltx_tr" id="S3.T3.1.26.25"> <td class="ltx_td ltx_align_left" id="S3.T3.1.26.25.1">CUEDC diesel cycle - MC</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.26.25.2">1723</td> </tr> <tr class="ltx_tr" id="S3.T3.1.27.26"> <td class="ltx_td ltx_align_left" id="S3.T3.1.27.26.1">CUEDC diesel cycle - NA</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.27.26.2">1795</td> </tr> <tr class="ltx_tr" id="S3.T3.1.28.27"> <td class="ltx_td ltx_align_left" id="S3.T3.1.28.27.1">CUEDC diesel cycle - NB</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.28.27.2">1706</td> </tr> <tr class="ltx_tr" id="S3.T3.1.29.28"> <td class="ltx_td ltx_align_left" id="S3.T3.1.29.28.1">CUEDC diesel cycle - ME</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.29.28.2">1678</td> </tr> <tr class="ltx_tr" id="S3.T3.1.30.29"> <td class="ltx_td ltx_align_left" id="S3.T3.1.30.29.1">CUEDC diesel cycle - NC</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.30.29.2">1797</td> </tr> <tr class="ltx_tr" id="S3.T3.1.31.30"> <td class="ltx_td ltx_align_left" id="S3.T3.1.31.30.1">CUEDC diesel cycle - NCH</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.31.30.2">1676</td> </tr> <tr class="ltx_tr" id="S3.T3.1.32.31"> <td class="ltx_td ltx_align_left" id="S3.T3.1.32.31.1">China Light-Duty Vehicle Test Cycle for Passenger Cars</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.32.31.2">1800</td> </tr> <tr class="ltx_tr" id="S3.T3.1.33.32"> <td class="ltx_td ltx_align_left" id="S3.T3.1.33.32.1">China Light-Duty Vehicle Test Cycle for Commercial Vehicles</td> <td class="ltx_td ltx_align_center" id="S3.T3.1.33.32.2">1800</td> </tr> <tr class="ltx_tr" id="S3.T3.1.34.33"> <td class="ltx_td ltx_align_left ltx_border_b" id="S3.T3.1.34.33.1">China Worldwide Transient Vehicle Cycle</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T3.1.34.33.2">1799</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.3">The driver block of the FEV model regulates the dynamic system to maintain the target speed. Its inputs are the target vehicle speed and the actual vehicle speed, and its outputs are the acceleration and deceleration control commands, index <math alttext="12" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mn id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml">12</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><cn id="S3.SS1.p4.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1">12</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">12</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">12</annotation></semantics></math> and <math alttext="13" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mn id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml">13</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><cn id="S3.SS1.p4.2.m2.1.1.cmml" type="integer" xref="S3.SS1.p4.2.m2.1.1">13</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">13</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">13</annotation></semantics></math> in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively, which are fed into the controllers block of the FEV model. This block takes said accelerator and brake pedal commands stemming as well as vehicle states like actual vehicle speed, electric motor speed and battery signals to calculate request signals for the powertrain, like the required electric motor torque and the brake signal, as well as battery management system signals like the battery SoC, index <math alttext="5" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mn id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><cn id="S3.SS1.p4.3.m3.1.1.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">5</annotation></semantics></math> in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>. Electric vehicles are capable of regenerative braking, meaning, the electric motor is used to decelerate the vehicle by acting as a generator, thereby charging the battery if it is not already fully charged.</p> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.13">Following is the vehicle block of the FEV model, which outputs how the vehicle reacts to any inputs and contains the electric plant subsystem and the drivetrain subsystem. Both take inputs from the controllers block, including the battery SoC, and the environment block, as well as from each other, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F2" title="Figure 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>. The electric plant subsystem outputs the electric motor torque, the battery current and power, the battery temperature and the cooling pump and refrigerator powers, which correspond to indices <math alttext="2" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><mn id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><cn id="S3.SS1.p5.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p5.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">2</annotation></semantics></math>, <math alttext="6" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><mn id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><cn id="S3.SS1.p5.2.m2.1.1.cmml" type="integer" xref="S3.SS1.p5.2.m2.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">6</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">6</annotation></semantics></math>, <math alttext="7" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><mn id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml">7</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><cn id="S3.SS1.p5.3.m3.1.1.cmml" type="integer" xref="S3.SS1.p5.3.m3.1.1">7</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">7</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">7</annotation></semantics></math>, <math alttext="14" class="ltx_Math" display="inline" id="S3.SS1.p5.4.m4.1"><semantics id="S3.SS1.p5.4.m4.1a"><mn id="S3.SS1.p5.4.m4.1.1" xref="S3.SS1.p5.4.m4.1.1.cmml">14</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.4.m4.1b"><cn id="S3.SS1.p5.4.m4.1.1.cmml" type="integer" xref="S3.SS1.p5.4.m4.1.1">14</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.4.m4.1c">14</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.4.m4.1d">14</annotation></semantics></math>, <math alttext="15" class="ltx_Math" display="inline" id="S3.SS1.p5.5.m5.1"><semantics id="S3.SS1.p5.5.m5.1a"><mn id="S3.SS1.p5.5.m5.1.1" xref="S3.SS1.p5.5.m5.1.1.cmml">15</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.5.m5.1b"><cn id="S3.SS1.p5.5.m5.1.1.cmml" type="integer" xref="S3.SS1.p5.5.m5.1.1">15</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.5.m5.1c">15</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.5.m5.1d">15</annotation></semantics></math> and <math alttext="16" class="ltx_Math" display="inline" id="S3.SS1.p5.6.m6.1"><semantics id="S3.SS1.p5.6.m6.1a"><mn id="S3.SS1.p5.6.m6.1.1" xref="S3.SS1.p5.6.m6.1.1.cmml">16</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.6.m6.1b"><cn id="S3.SS1.p5.6.m6.1.1.cmml" type="integer" xref="S3.SS1.p5.6.m6.1.1">16</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.6.m6.1c">16</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.6.m6.1d">16</annotation></semantics></math> in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively. The electric motor torque is input into the drivetrain subsystem, which in turn outputs the electric motor speed, and front and rear axle torques, forces and speeds, corresponding to indices <math alttext="1" class="ltx_Math" display="inline" id="S3.SS1.p5.7.m7.1"><semantics id="S3.SS1.p5.7.m7.1a"><mn id="S3.SS1.p5.7.m7.1.1" xref="S3.SS1.p5.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.7.m7.1b"><cn id="S3.SS1.p5.7.m7.1.1.cmml" type="integer" xref="S3.SS1.p5.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.7.m7.1d">1</annotation></semantics></math>, <math alttext="3" class="ltx_Math" display="inline" id="S3.SS1.p5.8.m8.1"><semantics id="S3.SS1.p5.8.m8.1a"><mn id="S3.SS1.p5.8.m8.1.1" xref="S3.SS1.p5.8.m8.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.8.m8.1b"><cn id="S3.SS1.p5.8.m8.1.1.cmml" type="integer" xref="S3.SS1.p5.8.m8.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.8.m8.1c">3</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.8.m8.1d">3</annotation></semantics></math>, <math alttext="4" class="ltx_Math" display="inline" id="S3.SS1.p5.9.m9.1"><semantics id="S3.SS1.p5.9.m9.1a"><mn id="S3.SS1.p5.9.m9.1.1" xref="S3.SS1.p5.9.m9.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.9.m9.1b"><cn id="S3.SS1.p5.9.m9.1.1.cmml" type="integer" xref="S3.SS1.p5.9.m9.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.9.m9.1c">4</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.9.m9.1d">4</annotation></semantics></math>, <math alttext="8" class="ltx_Math" display="inline" id="S3.SS1.p5.10.m10.1"><semantics id="S3.SS1.p5.10.m10.1a"><mn id="S3.SS1.p5.10.m10.1.1" xref="S3.SS1.p5.10.m10.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.10.m10.1b"><cn id="S3.SS1.p5.10.m10.1.1.cmml" type="integer" xref="S3.SS1.p5.10.m10.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.10.m10.1c">8</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.10.m10.1d">8</annotation></semantics></math>, <math alttext="9" class="ltx_Math" display="inline" id="S3.SS1.p5.11.m11.1"><semantics id="S3.SS1.p5.11.m11.1a"><mn id="S3.SS1.p5.11.m11.1.1" xref="S3.SS1.p5.11.m11.1.1.cmml">9</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.11.m11.1b"><cn id="S3.SS1.p5.11.m11.1.1.cmml" type="integer" xref="S3.SS1.p5.11.m11.1.1">9</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.11.m11.1c">9</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.11.m11.1d">9</annotation></semantics></math>, <math alttext="10" class="ltx_Math" display="inline" id="S3.SS1.p5.12.m12.1"><semantics id="S3.SS1.p5.12.m12.1a"><mn id="S3.SS1.p5.12.m12.1.1" xref="S3.SS1.p5.12.m12.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.12.m12.1b"><cn id="S3.SS1.p5.12.m12.1.1.cmml" type="integer" xref="S3.SS1.p5.12.m12.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.12.m12.1c">10</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.12.m12.1d">10</annotation></semantics></math>, <math alttext="11" class="ltx_Math" display="inline" id="S3.SS1.p5.13.m13.1"><semantics id="S3.SS1.p5.13.m13.1a"><mn id="S3.SS1.p5.13.m13.1.1" xref="S3.SS1.p5.13.m13.1.1.cmml">11</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.13.m13.1b"><cn id="S3.SS1.p5.13.m13.1.1.cmml" type="integer" xref="S3.SS1.p5.13.m13.1.1">11</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.13.m13.1c">11</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.13.m13.1d">11</annotation></semantics></math> in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively. The motor speed is also fed back into the electric plant model, completing the control loop. Both subsystems also contain further subsystems within them which uncover the causal relationships between their respective signals, and diving as deep as the lowest abstraction level of the model is outside the scope of this paper. Readers interested in more detail can refer to the Simulink model available in the provided repository. By default, the battery model features a static temperature model, however, to increase system complexity, a dynamic temperature model is added to the FEV model. The model used is the EV Battery Cooling System <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mathworks_ev_2024</span>]</cite>, also from MathWorks.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="314" id="S3.F2.g1" src="x2.png" width="540"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>A more detailed schematic of the vehicle model depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F1" title="Figure 1 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">1</span></a>. Numbers represent the indices of signal flow, reference is shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>. Output signals of the vehicle model, which are not fed back into other subsystems, are not shown, for simplicity.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.1">The environment block of the FEV model dictates environmental conditions that affect the longitudinal dynamics of the FEV model. Parameters like atmospheric pressure, wind speed, road grade and coefficient of friction can be set within this subsystem.</p> </div> <div class="ltx_para" id="S3.SS1.p7"> <p class="ltx_p" id="S3.SS1.p7.1">By default, the signals are logged at a sampling frequency of <span class="ltx_ERROR undefined" id="S3.SS1.p7.1.1">\qty</span>10 and the solver used is the differential algebraic equations’ solver for Simscape (daessc). Physical simulations may encounter numerical under- and overflow, which slow down simulations drastically. To avoid this, a timeout counter of one hour is set in place to skip the current simulation if triggered. Simulation time depends on the length of the drive cycle, however, for the computer hardware used simulations never take longer than 20 minutes, and hence one hour is considered sufficient for problem-free simulations.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Dataset Generation</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.5">To generate a dataset that is not only extensive but also diverse, <math alttext="100" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mn id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><cn id="S3.SS2.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">100</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">100</annotation></semantics></math> simulations have been undertaken for each of the <math alttext="33" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mn id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">33</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><cn id="S3.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1">33</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">33</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">33</annotation></semantics></math> drive cycles, each with random initial battery temperatures and battery SoCs. At this stage, all model properties have been left as default, and hence all simulation results have been considered <em class="ltx_emph ltx_font_italic" id="S3.SS2.p1.5.1">nominal</em>. 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)</annotation></semantics></math>, respectively, to ensure no bias is introduced. Sampling from uniform distributions also eliminates and reduces the effectiveness of simple threshold and control chart methods because the battery temperature and state of charge, but also, by extension, other channels, exhibit nominal but high deviation from the average behaviour. As a precautionary measure, drive cycles with a minimum SoC value lower than <span class="ltx_ERROR undefined" id="S3.SS2.p1.5.2">\qty</span>5 have been removed, as very low values have been observed to result in abnormal behaviour. After simulation, <math alttext="N_{\text{n}}=3273" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.1"><semantics id="S3.SS2.p1.5.m5.1a"><mrow id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml"><msub id="S3.SS2.p1.5.m5.1.1.2" xref="S3.SS2.p1.5.m5.1.1.2.cmml"><mi id="S3.SS2.p1.5.m5.1.1.2.2" xref="S3.SS2.p1.5.m5.1.1.2.2.cmml">N</mi><mtext id="S3.SS2.p1.5.m5.1.1.2.3" xref="S3.SS2.p1.5.m5.1.1.2.3a.cmml">n</mtext></msub><mo id="S3.SS2.p1.5.m5.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS2.p1.5.m5.1.1.3" xref="S3.SS2.p1.5.m5.1.1.3.cmml">3273</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><apply id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1"><eq id="S3.SS2.p1.5.m5.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1"></eq><apply id="S3.SS2.p1.5.m5.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.1.1.2.1.cmml" xref="S3.SS2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS2.p1.5.m5.1.1.2.2.cmml" xref="S3.SS2.p1.5.m5.1.1.2.2">𝑁</ci><ci id="S3.SS2.p1.5.m5.1.1.2.3a.cmml" xref="S3.SS2.p1.5.m5.1.1.2.3"><mtext id="S3.SS2.p1.5.m5.1.1.2.3.cmml" mathsize="70%" xref="S3.SS2.p1.5.m5.1.1.2.3">n</mtext></ci></apply><cn id="S3.SS2.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.SS2.p1.5.m5.1.1.3">3273</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">N_{\text{n}}=3273</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.1d">italic_N start_POSTSUBSCRIPT n end_POSTSUBSCRIPT = 3273</annotation></semantics></math> highly diverse and unique nominal time series have been collected. For illustration purposes, a nominal time series is plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F3" title="Figure 3 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="883" id="S3.F3.g1" src="x3.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Sample plot of a <em class="ltx_emph ltx_font_italic" id="S3.F3.2.1">nominal</em> sequence with added noise and undergone trimming. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">For the generation of <em class="ltx_emph ltx_font_italic" id="S3.SS2.p2.1.1">anomalous</em> time series, six types of anomalies have been considered. Some types can occur as both sub-sequence anomalies and sequence anomalies <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_online_2024</span>]</cite>, while others only in sequence anomaly form, due to simulation model limitations. To better distinguish the two anomaly forms, we refer to sequence anomalies as whole-sequence anomalies henceforth. The distribution of sub-sequence and whole-sequence anomalies across the different anomaly types is shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T4" title="Table 4 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">4</span></a>. Anomalies are caused by changing certain model properties <em class="ltx_emph ltx_font_italic" id="S3.SS2.p2.1.2">prior</em> to simulation, ensuring that any observed anomalous behaviour results from simulation rather than manual tampering of the data, like in the UCR dataset <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">dau_ucr_2019</span>]</cite>, which eliminates any bias. We ran all simulations with a fixed seed of <math alttext="1" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mn id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><cn id="S3.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p2.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">1</annotation></semantics></math>.</p> </div> <figure class="ltx_table" id="S3.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 4: </span>Number of sub-sequence anomalies <math alttext="N_{\text{ss}}" class="ltx_Math" display="inline" id="S3.T4.3.m1.1"><semantics id="S3.T4.3.m1.1b"><msub id="S3.T4.3.m1.1.1" xref="S3.T4.3.m1.1.1.cmml"><mi id="S3.T4.3.m1.1.1.2" xref="S3.T4.3.m1.1.1.2.cmml">N</mi><mtext id="S3.T4.3.m1.1.1.3" xref="S3.T4.3.m1.1.1.3a.cmml">ss</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T4.3.m1.1c"><apply id="S3.T4.3.m1.1.1.cmml" xref="S3.T4.3.m1.1.1"><csymbol cd="ambiguous" id="S3.T4.3.m1.1.1.1.cmml" xref="S3.T4.3.m1.1.1">subscript</csymbol><ci id="S3.T4.3.m1.1.1.2.cmml" xref="S3.T4.3.m1.1.1.2">𝑁</ci><ci id="S3.T4.3.m1.1.1.3a.cmml" xref="S3.T4.3.m1.1.1.3"><mtext id="S3.T4.3.m1.1.1.3.cmml" mathsize="70%" xref="S3.T4.3.m1.1.1.3">ss</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.3.m1.1d">N_{\text{ss}}</annotation><annotation encoding="application/x-llamapun" id="S3.T4.3.m1.1e">italic_N start_POSTSUBSCRIPT ss end_POSTSUBSCRIPT</annotation></semantics></math> and number of whole-sequence anomalies <math alttext="N_{\text{ws}}" class="ltx_Math" display="inline" id="S3.T4.4.m2.1"><semantics id="S3.T4.4.m2.1b"><msub id="S3.T4.4.m2.1.1" xref="S3.T4.4.m2.1.1.cmml"><mi id="S3.T4.4.m2.1.1.2" xref="S3.T4.4.m2.1.1.2.cmml">N</mi><mtext id="S3.T4.4.m2.1.1.3" xref="S3.T4.4.m2.1.1.3a.cmml">ws</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T4.4.m2.1c"><apply id="S3.T4.4.m2.1.1.cmml" xref="S3.T4.4.m2.1.1"><csymbol cd="ambiguous" id="S3.T4.4.m2.1.1.1.cmml" xref="S3.T4.4.m2.1.1">subscript</csymbol><ci id="S3.T4.4.m2.1.1.2.cmml" xref="S3.T4.4.m2.1.1.2">𝑁</ci><ci id="S3.T4.4.m2.1.1.3a.cmml" xref="S3.T4.4.m2.1.1.3"><mtext id="S3.T4.4.m2.1.1.3.cmml" mathsize="70%" xref="S3.T4.4.m2.1.1.3">ws</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.4.m2.1d">N_{\text{ws}}</annotation><annotation encoding="application/x-llamapun" id="S3.T4.4.m2.1e">italic_N start_POSTSUBSCRIPT ws end_POSTSUBSCRIPT</annotation></semantics></math> by anomaly type.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T4.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T4.6.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column" id="S3.T4.6.2.3">Anomaly Types</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T4.5.1.1"><math alttext="N_{\text{ss}}" class="ltx_Math" display="inline" id="S3.T4.5.1.1.m1.1"><semantics id="S3.T4.5.1.1.m1.1a"><msub id="S3.T4.5.1.1.m1.1.1" xref="S3.T4.5.1.1.m1.1.1.cmml"><mi id="S3.T4.5.1.1.m1.1.1.2" xref="S3.T4.5.1.1.m1.1.1.2.cmml">N</mi><mtext id="S3.T4.5.1.1.m1.1.1.3" xref="S3.T4.5.1.1.m1.1.1.3a.cmml">ss</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T4.5.1.1.m1.1b"><apply id="S3.T4.5.1.1.m1.1.1.cmml" xref="S3.T4.5.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T4.5.1.1.m1.1.1.1.cmml" xref="S3.T4.5.1.1.m1.1.1">subscript</csymbol><ci id="S3.T4.5.1.1.m1.1.1.2.cmml" xref="S3.T4.5.1.1.m1.1.1.2">𝑁</ci><ci id="S3.T4.5.1.1.m1.1.1.3a.cmml" xref="S3.T4.5.1.1.m1.1.1.3"><mtext id="S3.T4.5.1.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T4.5.1.1.m1.1.1.3">ss</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.5.1.1.m1.1c">N_{\text{ss}}</annotation><annotation encoding="application/x-llamapun" id="S3.T4.5.1.1.m1.1d">italic_N start_POSTSUBSCRIPT ss end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T4.6.2.2"><math alttext="N_{\text{ws}}" class="ltx_Math" display="inline" id="S3.T4.6.2.2.m1.1"><semantics id="S3.T4.6.2.2.m1.1a"><msub id="S3.T4.6.2.2.m1.1.1" xref="S3.T4.6.2.2.m1.1.1.cmml"><mi id="S3.T4.6.2.2.m1.1.1.2" xref="S3.T4.6.2.2.m1.1.1.2.cmml">N</mi><mtext id="S3.T4.6.2.2.m1.1.1.3" xref="S3.T4.6.2.2.m1.1.1.3a.cmml">ws</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T4.6.2.2.m1.1b"><apply id="S3.T4.6.2.2.m1.1.1.cmml" xref="S3.T4.6.2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.T4.6.2.2.m1.1.1.1.cmml" xref="S3.T4.6.2.2.m1.1.1">subscript</csymbol><ci id="S3.T4.6.2.2.m1.1.1.2.cmml" xref="S3.T4.6.2.2.m1.1.1.2">𝑁</ci><ci id="S3.T4.6.2.2.m1.1.1.3a.cmml" xref="S3.T4.6.2.2.m1.1.1.3"><mtext id="S3.T4.6.2.2.m1.1.1.3.cmml" mathsize="70%" xref="S3.T4.6.2.2.m1.1.1.3">ws</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T4.6.2.2.m1.1c">N_{\text{ws}}</annotation><annotation encoding="application/x-llamapun" id="S3.T4.6.2.2.m1.1d">italic_N start_POSTSUBSCRIPT ws end_POSTSUBSCRIPT</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T4.6.3.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S3.T4.6.3.1.1">Regenerative Braking Off</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T4.6.3.1.2">33</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T4.6.3.1.3">32</td> </tr> <tr class="ltx_tr" id="S3.T4.6.4.2"> <td class="ltx_td ltx_align_left" id="S3.T4.6.4.2.1">Increased Headwind</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.4.2.2">33</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.4.2.3">31</td> </tr> <tr class="ltx_tr" id="S3.T4.6.5.3"> <td class="ltx_td ltx_align_left" id="S3.T4.6.5.3.1">Reduced Pump Displacement</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.5.3.2">1</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.5.3.3">23</td> </tr> <tr class="ltx_tr" id="S3.T4.6.6.4"> <td class="ltx_td ltx_align_left" id="S3.T4.6.6.4.1">Reduced Motor Torque Request</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.6.4.2">32</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.6.4.3">33</td> </tr> <tr class="ltx_tr" id="S3.T4.6.7.5"> <td class="ltx_td ltx_align_left" id="S3.T4.6.7.5.1">Increased Wheel Diameter</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.7.5.2">0</td> <td class="ltx_td ltx_align_center" id="S3.T4.6.7.5.3">33</td> </tr> <tr class="ltx_tr" id="S3.T4.6.8.6"> <td class="ltx_td ltx_align_left ltx_border_b" id="S3.T4.6.8.6.1">Increased Driver Reaction Time</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T4.6.8.6.2">0</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T4.6.8.6.3">33</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.2">For the first kind of anomaly, we turn the regenerative braking off, which leads to visibly different motor and axle torques, as well as battery current and power, as these can no longer assume negative values. When regenerative braking is off, the battery SoC now has an exclusively negative gradient as it is no longer recharged via regenerative braking, and hence it decreases at a faster rate. The brake pedal is also used more to compensate for the missing braking motor torque. For each of the cycles, this anomaly type is simulated in two different ways: without regenerative braking from the beginning and from a random point in time within the drive cycle. This random point in time is sampled from a uniform distribution <math alttext="\mathcal{U}(0.2T,0.8T)" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.2"><semantics id="S3.SS2.p3.1.m1.2a"><mrow id="S3.SS2.p3.1.m1.2.2" xref="S3.SS2.p3.1.m1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.1.m1.2.2.4" xref="S3.SS2.p3.1.m1.2.2.4.cmml">𝒰</mi><mo id="S3.SS2.p3.1.m1.2.2.3" xref="S3.SS2.p3.1.m1.2.2.3.cmml">⁢</mo><mrow id="S3.SS2.p3.1.m1.2.2.2.2" xref="S3.SS2.p3.1.m1.2.2.2.3.cmml"><mo id="S3.SS2.p3.1.m1.2.2.2.2.3" stretchy="false" xref="S3.SS2.p3.1.m1.2.2.2.3.cmml">(</mo><mrow id="S3.SS2.p3.1.m1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.cmml"><mn id="S3.SS2.p3.1.m1.1.1.1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.1.1.1.2.cmml">0.2</mn><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.1.1.1.3.cmml">T</mi></mrow><mo id="S3.SS2.p3.1.m1.2.2.2.2.4" xref="S3.SS2.p3.1.m1.2.2.2.3.cmml">,</mo><mrow id="S3.SS2.p3.1.m1.2.2.2.2.2" xref="S3.SS2.p3.1.m1.2.2.2.2.2.cmml"><mn id="S3.SS2.p3.1.m1.2.2.2.2.2.2" xref="S3.SS2.p3.1.m1.2.2.2.2.2.2.cmml">0.8</mn><mo id="S3.SS2.p3.1.m1.2.2.2.2.2.1" xref="S3.SS2.p3.1.m1.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.SS2.p3.1.m1.2.2.2.2.2.3" xref="S3.SS2.p3.1.m1.2.2.2.2.2.3.cmml">T</mi></mrow><mo id="S3.SS2.p3.1.m1.2.2.2.2.5" stretchy="false" xref="S3.SS2.p3.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.2b"><apply id="S3.SS2.p3.1.m1.2.2.cmml" xref="S3.SS2.p3.1.m1.2.2"><times id="S3.SS2.p3.1.m1.2.2.3.cmml" xref="S3.SS2.p3.1.m1.2.2.3"></times><ci id="S3.SS2.p3.1.m1.2.2.4.cmml" xref="S3.SS2.p3.1.m1.2.2.4">𝒰</ci><interval closure="open" id="S3.SS2.p3.1.m1.2.2.2.3.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2"><apply id="S3.SS2.p3.1.m1.1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1"><times id="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1"></times><cn id="S3.SS2.p3.1.m1.1.1.1.1.1.2.cmml" type="float" xref="S3.SS2.p3.1.m1.1.1.1.1.1.2">0.2</cn><ci id="S3.SS2.p3.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1.3">𝑇</ci></apply><apply id="S3.SS2.p3.1.m1.2.2.2.2.2.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2.2"><times id="S3.SS2.p3.1.m1.2.2.2.2.2.1.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2.2.1"></times><cn id="S3.SS2.p3.1.m1.2.2.2.2.2.2.cmml" type="float" xref="S3.SS2.p3.1.m1.2.2.2.2.2.2">0.8</cn><ci id="S3.SS2.p3.1.m1.2.2.2.2.2.3.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2.2.3">𝑇</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.2c">\mathcal{U}(0.2T,0.8T)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.2d">caligraphic_U ( 0.2 italic_T , 0.8 italic_T )</annotation></semantics></math>, where <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mi id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><ci id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_T</annotation></semantics></math> denotes the temporal length of the drive cycle in question, see Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T3" title="Table 3 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">3</span></a>. This statistical distribution is used for all sub-sequence anomaly types. One of the anomalous time series for the CADC130 drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F4" title="Figure 4 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="881" id="S3.F4.g1" src="x4.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F4.3.1">anomalous</em> sequence without regenerative braking (in red) and its control counterpart (in black), both with added noise and undergone trimming. The anomalous sub-sequence starts after <span class="ltx_ERROR undefined" id="S3.F4.4.2">\qty</span>384.6. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.1">In the case of the next anomaly type, we introduce a headwind of <span class="ltx_ERROR undefined" id="S3.SS2.p4.1.1">\qty</span>5<span class="ltx_ERROR undefined" id="S3.SS2.p4.1.2">\per</span> to the model. This headwind acts as a force on the frontal area of the vehicle and needs to be overcome to maintain the target vehicle speed by using the accelerator pedal more than the norm, which leads to higher motor and axle torques and therefore axle forces. The higher motor torque requires a higher battery current and power, which also causes accelerated discharging. Like previously, this anomaly type is simulated for each drive cycle, both from the beginning and from a random point in time within the cycle. One of the anomalous time series for the CLTCPassenger drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F5" title="Figure 5 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="883" id="S3.F5.g1" src="x5.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F5.3.1">anomalous</em> sequence with an added headwind (in red) and its control counterpart (in black), both with added noise and undergone trimming. The anomalous sub-sequence starts after <span class="ltx_ERROR undefined" id="S3.F5.4.2">\qty</span>738.0. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.1">Following that, we reduce the displacement of the cooling pump by <span class="ltx_ERROR undefined" id="S3.SS2.p5.1.1">\qty</span>10 to simulate another anomaly type. Evidently, this change leads to a higher battery temperature as the cooling capacity is reduced. This reduction is also visible in the pump power. Like with the previous two anomaly types, this anomaly type can start from the beginning and from a random point in time within the cycle. One of the anomalous time series for the CUEDCDieselME drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F6" title="Figure 6 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">6</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="874" id="S3.F6.g1" src="x6.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F6.2.1">anomalous</em> sequence with a reduced cooling pump displacement (in red) and its control counterpart (in black), both with added noise and undergone trimming. It is a whole-sequence anomaly, and hence the anomalous behaviour starts from the first time step. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.1">For the next anomaly type, we reduce the requested motor torque value output by the powertrain control module by <span class="ltx_ERROR undefined" id="S3.SS2.p6.1.1">\qty</span>10. As a response to the change, the driver model requests a higher acceleration pedal value and consequently a different brake pedal values as well. This anomaly type can also start from the beginning and from a random point in time within the cycle. One of the anomalous time series for the FTP75 drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F7" title="Figure 7 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">7</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="883" id="S3.F7.g1" src="x7.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F7.3.1">anomalous</em> sequence with a reduced requested motor torque (in red) and its control counterpart (in black), both with added noise and undergone trimming. The anomalous sub-sequence starts after <span class="ltx_ERROR undefined" id="S3.F7.4.2">\qty</span>940.2. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p7"> <p class="ltx_p" id="S3.SS2.p7.1">In the next case, we increase the loaded wheel diameter by <span class="ltx_ERROR undefined" id="S3.SS2.p7.1.1">\qty</span>10 which, for the same target vehicle speed, leads to a lower motor and axle angular velocity. Furthermore, a larger wheel diameter leads to higher motor and axle torques, which are achieved using higher accelerator and brake pedal values. Here, the wheel diameter also has an effect on the battery temperature, which, depending on its absolute magnitude, may also affect the cooling system. Due to model limitations, this anomaly can only be simulated for whole-sequence anomalies. One of the anomalous time series for the HUDDS drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F8" title="Figure 8 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">8</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="879" id="S3.F8.g1" src="x8.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F8.2.1">anomalous</em> sequence with an increased loaded wheel diameter (in red) and its control counterpart (in black), both with added noise and undergone trimming. It is a whole-sequence anomaly, and hence the anomalous behaviour starts from the first time step. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p8"> <p class="ltx_p" id="S3.SS2.p8.1">The last anomaly is recorded after increasing the driver response time by a factor of <math alttext="4" class="ltx_Math" display="inline" id="S3.SS2.p8.1.m1.1"><semantics id="S3.SS2.p8.1.m1.1a"><mn id="S3.SS2.p8.1.m1.1.1" xref="S3.SS2.p8.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.1.m1.1b"><cn id="S3.SS2.p8.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p8.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.1.m1.1c">4</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.1.m1.1d">4</annotation></semantics></math>. This is one of the more subtle anomalies types, but manifests itself in all channels, except for the battery temperature and cooling. Like for the wheel diameter anomaly, this anomaly can only be simulated for whole-sequence anomalies. One of the anomalous time series for the LA92 drive cycle and its control counterpart are plotted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.F9" title="Figure 9 ‣ 3.2 Dataset Generation ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">9</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="883" id="S3.F9.g1" src="x9.png" width="822"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Plot of an <em class="ltx_emph ltx_font_italic" id="S3.F9.2.1">anomalous</em> sequence with an increased driver response time (in red) and its control counterpart (in black), both with added noise and undergone trimming. It is a whole-sequence anomaly, and hence the anomalous behaviour starts from the first time step. The channel legend can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3.T2" title="Table 2 ‣ 3.1 Simulation Model ‣ 3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">2</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p9"> <p class="ltx_p" id="S3.SS2.p9.1">To ensure that the different anomaly types actually lead to anomalous behaviour, we run control simulations with the same initial battery states but with otherwise nominal model properties. Given the uniform distribution from which the battery temperature is sampled from, half of the simulated anomaly types start with a battery temperature below <span class="ltx_ERROR undefined" id="S3.SS2.p9.1.1">\qty</span>20. In these cases, the battery will naturally heat up as it is being used and hence the cooling system does not play a role. Therefore, in the case of the reduced cooling pump displacement, often no anomalous behaviour can be observed because the simulated anomaly is identical with the corresponding control simulation. For these cases, the simulated anomaly is discarded.</p> </div> <div class="ltx_para" id="S3.SS2.p10"> <p class="ltx_p" id="S3.SS2.p10.18">Finally, this results in <math alttext="N_{\text{a}}=284" class="ltx_Math" display="inline" id="S3.SS2.p10.1.m1.1"><semantics id="S3.SS2.p10.1.m1.1a"><mrow id="S3.SS2.p10.1.m1.1.1" xref="S3.SS2.p10.1.m1.1.1.cmml"><msub id="S3.SS2.p10.1.m1.1.1.2" xref="S3.SS2.p10.1.m1.1.1.2.cmml"><mi id="S3.SS2.p10.1.m1.1.1.2.2" xref="S3.SS2.p10.1.m1.1.1.2.2.cmml">N</mi><mtext id="S3.SS2.p10.1.m1.1.1.2.3" xref="S3.SS2.p10.1.m1.1.1.2.3a.cmml">a</mtext></msub><mo id="S3.SS2.p10.1.m1.1.1.1" xref="S3.SS2.p10.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS2.p10.1.m1.1.1.3" xref="S3.SS2.p10.1.m1.1.1.3.cmml">284</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.1.m1.1b"><apply id="S3.SS2.p10.1.m1.1.1.cmml" xref="S3.SS2.p10.1.m1.1.1"><eq id="S3.SS2.p10.1.m1.1.1.1.cmml" xref="S3.SS2.p10.1.m1.1.1.1"></eq><apply id="S3.SS2.p10.1.m1.1.1.2.cmml" xref="S3.SS2.p10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p10.1.m1.1.1.2.1.cmml" xref="S3.SS2.p10.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS2.p10.1.m1.1.1.2.2.cmml" xref="S3.SS2.p10.1.m1.1.1.2.2">𝑁</ci><ci id="S3.SS2.p10.1.m1.1.1.2.3a.cmml" xref="S3.SS2.p10.1.m1.1.1.2.3"><mtext id="S3.SS2.p10.1.m1.1.1.2.3.cmml" mathsize="70%" xref="S3.SS2.p10.1.m1.1.1.2.3">a</mtext></ci></apply><cn id="S3.SS2.p10.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p10.1.m1.1.1.3">284</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.1.m1.1c">N_{\text{a}}=284</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.1.m1.1d">italic_N start_POSTSUBSCRIPT a end_POSTSUBSCRIPT = 284</annotation></semantics></math> successful anomalous simulations, where <math alttext="N_{\text{a}}=N_{\text{ss}}+N_{\text{ws}}" class="ltx_Math" display="inline" id="S3.SS2.p10.2.m2.1"><semantics id="S3.SS2.p10.2.m2.1a"><mrow id="S3.SS2.p10.2.m2.1.1" xref="S3.SS2.p10.2.m2.1.1.cmml"><msub id="S3.SS2.p10.2.m2.1.1.2" xref="S3.SS2.p10.2.m2.1.1.2.cmml"><mi id="S3.SS2.p10.2.m2.1.1.2.2" xref="S3.SS2.p10.2.m2.1.1.2.2.cmml">N</mi><mtext id="S3.SS2.p10.2.m2.1.1.2.3" xref="S3.SS2.p10.2.m2.1.1.2.3a.cmml">a</mtext></msub><mo id="S3.SS2.p10.2.m2.1.1.1" xref="S3.SS2.p10.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS2.p10.2.m2.1.1.3" xref="S3.SS2.p10.2.m2.1.1.3.cmml"><msub id="S3.SS2.p10.2.m2.1.1.3.2" xref="S3.SS2.p10.2.m2.1.1.3.2.cmml"><mi id="S3.SS2.p10.2.m2.1.1.3.2.2" xref="S3.SS2.p10.2.m2.1.1.3.2.2.cmml">N</mi><mtext id="S3.SS2.p10.2.m2.1.1.3.2.3" xref="S3.SS2.p10.2.m2.1.1.3.2.3a.cmml">ss</mtext></msub><mo id="S3.SS2.p10.2.m2.1.1.3.1" xref="S3.SS2.p10.2.m2.1.1.3.1.cmml">+</mo><msub id="S3.SS2.p10.2.m2.1.1.3.3" xref="S3.SS2.p10.2.m2.1.1.3.3.cmml"><mi id="S3.SS2.p10.2.m2.1.1.3.3.2" xref="S3.SS2.p10.2.m2.1.1.3.3.2.cmml">N</mi><mtext id="S3.SS2.p10.2.m2.1.1.3.3.3" xref="S3.SS2.p10.2.m2.1.1.3.3.3a.cmml">ws</mtext></msub></mrow></mrow><annotation-xml 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italic_N start_POSTSUBSCRIPT ws end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, the entire dataset <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S3.SS2.p10.3.m3.1"><semantics id="S3.SS2.p10.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.3.m3.1.1" xref="S3.SS2.p10.3.m3.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.3.m3.1b"><ci id="S3.SS2.p10.3.m3.1.1.cmml" xref="S3.SS2.p10.3.m3.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.3.m3.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.3.m3.1d">caligraphic_D</annotation></semantics></math> consists of <math alttext="N_{\text{n}}+N_{\text{ss}}+N_{\text{ws}}=3557" class="ltx_Math" display="inline" id="S3.SS2.p10.4.m4.1"><semantics id="S3.SS2.p10.4.m4.1a"><mrow id="S3.SS2.p10.4.m4.1.1" xref="S3.SS2.p10.4.m4.1.1.cmml"><mrow id="S3.SS2.p10.4.m4.1.1.2" xref="S3.SS2.p10.4.m4.1.1.2.cmml"><msub id="S3.SS2.p10.4.m4.1.1.2.2" xref="S3.SS2.p10.4.m4.1.1.2.2.cmml"><mi id="S3.SS2.p10.4.m4.1.1.2.2.2" xref="S3.SS2.p10.4.m4.1.1.2.2.2.cmml">N</mi><mtext id="S3.SS2.p10.4.m4.1.1.2.2.3" xref="S3.SS2.p10.4.m4.1.1.2.2.3a.cmml">n</mtext></msub><mo id="S3.SS2.p10.4.m4.1.1.2.1" xref="S3.SS2.p10.4.m4.1.1.2.1.cmml">+</mo><msub id="S3.SS2.p10.4.m4.1.1.2.3" xref="S3.SS2.p10.4.m4.1.1.2.3.cmml"><mi id="S3.SS2.p10.4.m4.1.1.2.3.2" xref="S3.SS2.p10.4.m4.1.1.2.3.2.cmml">N</mi><mtext id="S3.SS2.p10.4.m4.1.1.2.3.3" xref="S3.SS2.p10.4.m4.1.1.2.3.3a.cmml">ss</mtext></msub><mo id="S3.SS2.p10.4.m4.1.1.2.1a" xref="S3.SS2.p10.4.m4.1.1.2.1.cmml">+</mo><msub id="S3.SS2.p10.4.m4.1.1.2.4" xref="S3.SS2.p10.4.m4.1.1.2.4.cmml"><mi id="S3.SS2.p10.4.m4.1.1.2.4.2" xref="S3.SS2.p10.4.m4.1.1.2.4.2.cmml">N</mi><mtext id="S3.SS2.p10.4.m4.1.1.2.4.3" xref="S3.SS2.p10.4.m4.1.1.2.4.3a.cmml">ws</mtext></msub></mrow><mo id="S3.SS2.p10.4.m4.1.1.1" xref="S3.SS2.p10.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS2.p10.4.m4.1.1.3" xref="S3.SS2.p10.4.m4.1.1.3.cmml">3557</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.4.m4.1b"><apply id="S3.SS2.p10.4.m4.1.1.cmml" xref="S3.SS2.p10.4.m4.1.1"><eq id="S3.SS2.p10.4.m4.1.1.1.cmml" xref="S3.SS2.p10.4.m4.1.1.1"></eq><apply id="S3.SS2.p10.4.m4.1.1.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2"><plus id="S3.SS2.p10.4.m4.1.1.2.1.cmml" xref="S3.SS2.p10.4.m4.1.1.2.1"></plus><apply id="S3.SS2.p10.4.m4.1.1.2.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p10.4.m4.1.1.2.2.1.cmml" xref="S3.SS2.p10.4.m4.1.1.2.2">subscript</csymbol><ci id="S3.SS2.p10.4.m4.1.1.2.2.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2.2.2">𝑁</ci><ci id="S3.SS2.p10.4.m4.1.1.2.2.3a.cmml" xref="S3.SS2.p10.4.m4.1.1.2.2.3"><mtext id="S3.SS2.p10.4.m4.1.1.2.2.3.cmml" mathsize="70%" xref="S3.SS2.p10.4.m4.1.1.2.2.3">n</mtext></ci></apply><apply id="S3.SS2.p10.4.m4.1.1.2.3.cmml" xref="S3.SS2.p10.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS2.p10.4.m4.1.1.2.3.1.cmml" xref="S3.SS2.p10.4.m4.1.1.2.3">subscript</csymbol><ci id="S3.SS2.p10.4.m4.1.1.2.3.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2.3.2">𝑁</ci><ci id="S3.SS2.p10.4.m4.1.1.2.3.3a.cmml" xref="S3.SS2.p10.4.m4.1.1.2.3.3"><mtext id="S3.SS2.p10.4.m4.1.1.2.3.3.cmml" mathsize="70%" xref="S3.SS2.p10.4.m4.1.1.2.3.3">ss</mtext></ci></apply><apply id="S3.SS2.p10.4.m4.1.1.2.4.cmml" xref="S3.SS2.p10.4.m4.1.1.2.4"><csymbol cd="ambiguous" id="S3.SS2.p10.4.m4.1.1.2.4.1.cmml" xref="S3.SS2.p10.4.m4.1.1.2.4">subscript</csymbol><ci id="S3.SS2.p10.4.m4.1.1.2.4.2.cmml" xref="S3.SS2.p10.4.m4.1.1.2.4.2">𝑁</ci><ci id="S3.SS2.p10.4.m4.1.1.2.4.3a.cmml" xref="S3.SS2.p10.4.m4.1.1.2.4.3"><mtext id="S3.SS2.p10.4.m4.1.1.2.4.3.cmml" mathsize="70%" xref="S3.SS2.p10.4.m4.1.1.2.4.3">ws</mtext></ci></apply></apply><cn id="S3.SS2.p10.4.m4.1.1.3.cmml" type="integer" xref="S3.SS2.p10.4.m4.1.1.3">3557</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.4.m4.1c">N_{\text{n}}+N_{\text{ss}}+N_{\text{ws}}=3557</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.4.m4.1d">italic_N start_POSTSUBSCRIPT n end_POSTSUBSCRIPT + italic_N start_POSTSUBSCRIPT ss end_POSTSUBSCRIPT + italic_N start_POSTSUBSCRIPT ws end_POSTSUBSCRIPT = 3557</annotation></semantics></math> unique (nominal and anomalous) multivariate time series, with an anomalous sequence ratio of <math alttext="284/3557\approx 8\%" class="ltx_Math" display="inline" id="S3.SS2.p10.5.m5.1"><semantics id="S3.SS2.p10.5.m5.1a"><mrow id="S3.SS2.p10.5.m5.1.1" xref="S3.SS2.p10.5.m5.1.1.cmml"><mrow id="S3.SS2.p10.5.m5.1.1.2" xref="S3.SS2.p10.5.m5.1.1.2.cmml"><mn id="S3.SS2.p10.5.m5.1.1.2.2" xref="S3.SS2.p10.5.m5.1.1.2.2.cmml">284</mn><mo id="S3.SS2.p10.5.m5.1.1.2.1" xref="S3.SS2.p10.5.m5.1.1.2.1.cmml">/</mo><mn id="S3.SS2.p10.5.m5.1.1.2.3" xref="S3.SS2.p10.5.m5.1.1.2.3.cmml">3557</mn></mrow><mo id="S3.SS2.p10.5.m5.1.1.1" xref="S3.SS2.p10.5.m5.1.1.1.cmml">≈</mo><mrow id="S3.SS2.p10.5.m5.1.1.3" xref="S3.SS2.p10.5.m5.1.1.3.cmml"><mn id="S3.SS2.p10.5.m5.1.1.3.2" xref="S3.SS2.p10.5.m5.1.1.3.2.cmml">8</mn><mo id="S3.SS2.p10.5.m5.1.1.3.1" xref="S3.SS2.p10.5.m5.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.5.m5.1b"><apply id="S3.SS2.p10.5.m5.1.1.cmml" xref="S3.SS2.p10.5.m5.1.1"><approx id="S3.SS2.p10.5.m5.1.1.1.cmml" xref="S3.SS2.p10.5.m5.1.1.1"></approx><apply id="S3.SS2.p10.5.m5.1.1.2.cmml" xref="S3.SS2.p10.5.m5.1.1.2"><divide id="S3.SS2.p10.5.m5.1.1.2.1.cmml" xref="S3.SS2.p10.5.m5.1.1.2.1"></divide><cn id="S3.SS2.p10.5.m5.1.1.2.2.cmml" type="integer" xref="S3.SS2.p10.5.m5.1.1.2.2">284</cn><cn id="S3.SS2.p10.5.m5.1.1.2.3.cmml" type="integer" xref="S3.SS2.p10.5.m5.1.1.2.3">3557</cn></apply><apply id="S3.SS2.p10.5.m5.1.1.3.cmml" xref="S3.SS2.p10.5.m5.1.1.3"><csymbol cd="latexml" id="S3.SS2.p10.5.m5.1.1.3.1.cmml" xref="S3.SS2.p10.5.m5.1.1.3.1">percent</csymbol><cn id="S3.SS2.p10.5.m5.1.1.3.2.cmml" type="integer" xref="S3.SS2.p10.5.m5.1.1.3.2">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.5.m5.1c">284/3557\approx 8\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.5.m5.1d">284 / 3557 ≈ 8 %</annotation></semantics></math>. <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S3.SS2.p10.6.m6.1"><semantics id="S3.SS2.p10.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.6.m6.1.1" xref="S3.SS2.p10.6.m6.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.6.m6.1b"><ci id="S3.SS2.p10.6.m6.1.1.cmml" xref="S3.SS2.p10.6.m6.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.6.m6.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.6.m6.1d">caligraphic_D</annotation></semantics></math> is then shuffled and divided into three separate folds for cross-validation, which corresponds to <math alttext="2/3" class="ltx_Math" display="inline" id="S3.SS2.p10.7.m7.1"><semantics id="S3.SS2.p10.7.m7.1a"><mrow id="S3.SS2.p10.7.m7.1.1" xref="S3.SS2.p10.7.m7.1.1.cmml"><mn id="S3.SS2.p10.7.m7.1.1.2" xref="S3.SS2.p10.7.m7.1.1.2.cmml">2</mn><mo id="S3.SS2.p10.7.m7.1.1.1" xref="S3.SS2.p10.7.m7.1.1.1.cmml">/</mo><mn id="S3.SS2.p10.7.m7.1.1.3" xref="S3.SS2.p10.7.m7.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.7.m7.1b"><apply id="S3.SS2.p10.7.m7.1.1.cmml" xref="S3.SS2.p10.7.m7.1.1"><divide id="S3.SS2.p10.7.m7.1.1.1.cmml" xref="S3.SS2.p10.7.m7.1.1.1"></divide><cn id="S3.SS2.p10.7.m7.1.1.2.cmml" type="integer" xref="S3.SS2.p10.7.m7.1.1.2">2</cn><cn id="S3.SS2.p10.7.m7.1.1.3.cmml" type="integer" xref="S3.SS2.p10.7.m7.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.7.m7.1c">2/3</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.7.m7.1d">2 / 3</annotation></semantics></math> and <math alttext="1/3" class="ltx_Math" display="inline" id="S3.SS2.p10.8.m8.1"><semantics id="S3.SS2.p10.8.m8.1a"><mrow id="S3.SS2.p10.8.m8.1.1" xref="S3.SS2.p10.8.m8.1.1.cmml"><mn id="S3.SS2.p10.8.m8.1.1.2" xref="S3.SS2.p10.8.m8.1.1.2.cmml">1</mn><mo id="S3.SS2.p10.8.m8.1.1.1" xref="S3.SS2.p10.8.m8.1.1.1.cmml">/</mo><mn id="S3.SS2.p10.8.m8.1.1.3" xref="S3.SS2.p10.8.m8.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.8.m8.1b"><apply id="S3.SS2.p10.8.m8.1.1.cmml" xref="S3.SS2.p10.8.m8.1.1"><divide id="S3.SS2.p10.8.m8.1.1.1.cmml" xref="S3.SS2.p10.8.m8.1.1.1"></divide><cn id="S3.SS2.p10.8.m8.1.1.2.cmml" type="integer" xref="S3.SS2.p10.8.m8.1.1.2">1</cn><cn id="S3.SS2.p10.8.m8.1.1.3.cmml" type="integer" xref="S3.SS2.p10.8.m8.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.8.m8.1c">1/3</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.8.m8.1d">1 / 3</annotation></semantics></math> split training and test subsets, respectively. 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xref="S3.SS2.p10.9.m9.5.5.3.3.3.2.cmml">𝒮</mi><mi id="S3.SS2.p10.9.m9.5.5.3.3.3.3" xref="S3.SS2.p10.9.m9.5.5.3.3.3.3.cmml">M</mi></msub><mo id="S3.SS2.p10.9.m9.5.5.3.3.9" stretchy="false" xref="S3.SS2.p10.9.m9.5.5.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.9.m9.5b"><apply id="S3.SS2.p10.9.m9.5.5.cmml" xref="S3.SS2.p10.9.m9.5.5"><eq id="S3.SS2.p10.9.m9.5.5.4.cmml" xref="S3.SS2.p10.9.m9.5.5.4"></eq><apply id="S3.SS2.p10.9.m9.5.5.5.cmml" xref="S3.SS2.p10.9.m9.5.5.5"><csymbol cd="ambiguous" id="S3.SS2.p10.9.m9.5.5.5.1.cmml" xref="S3.SS2.p10.9.m9.5.5.5">superscript</csymbol><ci id="S3.SS2.p10.9.m9.5.5.5.2.cmml" xref="S3.SS2.p10.9.m9.5.5.5.2">𝒟</ci><ci id="S3.SS2.p10.9.m9.5.5.5.3a.cmml" xref="S3.SS2.p10.9.m9.5.5.5.3"><mtext id="S3.SS2.p10.9.m9.5.5.5.3.cmml" mathsize="70%" xref="S3.SS2.p10.9.m9.5.5.5.3">train</mtext></ci></apply><set id="S3.SS2.p10.9.m9.5.5.3.4.cmml" xref="S3.SS2.p10.9.m9.5.5.3.3"><apply id="S3.SS2.p10.9.m9.3.3.1.1.1.cmml" 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xref="S3.SS2.p10.9.m9.5.5.3.3.3">subscript</csymbol><ci id="S3.SS2.p10.9.m9.5.5.3.3.3.2.cmml" xref="S3.SS2.p10.9.m9.5.5.3.3.3.2">𝒮</ci><ci id="S3.SS2.p10.9.m9.5.5.3.3.3.3.cmml" xref="S3.SS2.p10.9.m9.5.5.3.3.3.3">𝑀</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.9.m9.5c">\mathcal{D}^{\text{train}}=\{\mathcal{S}_{1},...,\mathcal{S}_{m},...,\mathcal{% S}_{M}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.9.m9.5d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT = { caligraphic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , caligraphic_S start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , … , caligraphic_S start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT }</annotation></semantics></math> then consists of <math alttext="M=2371" class="ltx_Math" display="inline" id="S3.SS2.p10.10.m10.1"><semantics id="S3.SS2.p10.10.m10.1a"><mrow id="S3.SS2.p10.10.m10.1.1" xref="S3.SS2.p10.10.m10.1.1.cmml"><mi id="S3.SS2.p10.10.m10.1.1.2" xref="S3.SS2.p10.10.m10.1.1.2.cmml">M</mi><mo id="S3.SS2.p10.10.m10.1.1.1" xref="S3.SS2.p10.10.m10.1.1.1.cmml">=</mo><mn id="S3.SS2.p10.10.m10.1.1.3" xref="S3.SS2.p10.10.m10.1.1.3.cmml">2371</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.10.m10.1b"><apply id="S3.SS2.p10.10.m10.1.1.cmml" xref="S3.SS2.p10.10.m10.1.1"><eq id="S3.SS2.p10.10.m10.1.1.1.cmml" xref="S3.SS2.p10.10.m10.1.1.1"></eq><ci id="S3.SS2.p10.10.m10.1.1.2.cmml" xref="S3.SS2.p10.10.m10.1.1.2">𝑀</ci><cn id="S3.SS2.p10.10.m10.1.1.3.cmml" type="integer" xref="S3.SS2.p10.10.m10.1.1.3">2371</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.10.m10.1c">M=2371</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.10.m10.1d">italic_M = 2371</annotation></semantics></math> multivariate time series on average, where <math alttext="\mathcal{S}_{m}\in\mathbb{R}^{T_{m}\times d_{\mathcal{D}}}" class="ltx_Math" display="inline" id="S3.SS2.p10.11.m11.1"><semantics id="S3.SS2.p10.11.m11.1a"><mrow id="S3.SS2.p10.11.m11.1.1" xref="S3.SS2.p10.11.m11.1.1.cmml"><msub id="S3.SS2.p10.11.m11.1.1.2" xref="S3.SS2.p10.11.m11.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.11.m11.1.1.2.2" xref="S3.SS2.p10.11.m11.1.1.2.2.cmml">𝒮</mi><mi id="S3.SS2.p10.11.m11.1.1.2.3" xref="S3.SS2.p10.11.m11.1.1.2.3.cmml">m</mi></msub><mo id="S3.SS2.p10.11.m11.1.1.1" xref="S3.SS2.p10.11.m11.1.1.1.cmml">∈</mo><msup id="S3.SS2.p10.11.m11.1.1.3" xref="S3.SS2.p10.11.m11.1.1.3.cmml"><mi id="S3.SS2.p10.11.m11.1.1.3.2" xref="S3.SS2.p10.11.m11.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p10.11.m11.1.1.3.3" xref="S3.SS2.p10.11.m11.1.1.3.3.cmml"><msub id="S3.SS2.p10.11.m11.1.1.3.3.2" xref="S3.SS2.p10.11.m11.1.1.3.3.2.cmml"><mi id="S3.SS2.p10.11.m11.1.1.3.3.2.2" xref="S3.SS2.p10.11.m11.1.1.3.3.2.2.cmml">T</mi><mi id="S3.SS2.p10.11.m11.1.1.3.3.2.3" xref="S3.SS2.p10.11.m11.1.1.3.3.2.3.cmml">m</mi></msub><mo id="S3.SS2.p10.11.m11.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p10.11.m11.1.1.3.3.1.cmml">×</mo><msub id="S3.SS2.p10.11.m11.1.1.3.3.3" xref="S3.SS2.p10.11.m11.1.1.3.3.3.cmml"><mi id="S3.SS2.p10.11.m11.1.1.3.3.3.2" xref="S3.SS2.p10.11.m11.1.1.3.3.3.2.cmml">d</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.11.m11.1.1.3.3.3.3" xref="S3.SS2.p10.11.m11.1.1.3.3.3.3.cmml">𝒟</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.11.m11.1b"><apply id="S3.SS2.p10.11.m11.1.1.cmml" xref="S3.SS2.p10.11.m11.1.1"><in id="S3.SS2.p10.11.m11.1.1.1.cmml" xref="S3.SS2.p10.11.m11.1.1.1"></in><apply id="S3.SS2.p10.11.m11.1.1.2.cmml" xref="S3.SS2.p10.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p10.11.m11.1.1.2.1.cmml" xref="S3.SS2.p10.11.m11.1.1.2">subscript</csymbol><ci id="S3.SS2.p10.11.m11.1.1.2.2.cmml" xref="S3.SS2.p10.11.m11.1.1.2.2">𝒮</ci><ci id="S3.SS2.p10.11.m11.1.1.2.3.cmml" xref="S3.SS2.p10.11.m11.1.1.2.3">𝑚</ci></apply><apply id="S3.SS2.p10.11.m11.1.1.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p10.11.m11.1.1.3.1.cmml" xref="S3.SS2.p10.11.m11.1.1.3">superscript</csymbol><ci id="S3.SS2.p10.11.m11.1.1.3.2.cmml" xref="S3.SS2.p10.11.m11.1.1.3.2">ℝ</ci><apply id="S3.SS2.p10.11.m11.1.1.3.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3"><times id="S3.SS2.p10.11.m11.1.1.3.3.1.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.1"></times><apply id="S3.SS2.p10.11.m11.1.1.3.3.2.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p10.11.m11.1.1.3.3.2.1.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.2">subscript</csymbol><ci id="S3.SS2.p10.11.m11.1.1.3.3.2.2.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.2.2">𝑇</ci><ci id="S3.SS2.p10.11.m11.1.1.3.3.2.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.2.3">𝑚</ci></apply><apply id="S3.SS2.p10.11.m11.1.1.3.3.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p10.11.m11.1.1.3.3.3.1.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.3">subscript</csymbol><ci id="S3.SS2.p10.11.m11.1.1.3.3.3.2.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.3.2">𝑑</ci><ci id="S3.SS2.p10.11.m11.1.1.3.3.3.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3.3.3.3">𝒟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.11.m11.1c">\mathcal{S}_{m}\in\mathbb{R}^{T_{m}\times d_{\mathcal{D}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.11.m11.1d">caligraphic_S start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="T_{m}" class="ltx_Math" display="inline" id="S3.SS2.p10.12.m12.1"><semantics id="S3.SS2.p10.12.m12.1a"><msub id="S3.SS2.p10.12.m12.1.1" xref="S3.SS2.p10.12.m12.1.1.cmml"><mi id="S3.SS2.p10.12.m12.1.1.2" xref="S3.SS2.p10.12.m12.1.1.2.cmml">T</mi><mi id="S3.SS2.p10.12.m12.1.1.3" xref="S3.SS2.p10.12.m12.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.12.m12.1b"><apply id="S3.SS2.p10.12.m12.1.1.cmml" xref="S3.SS2.p10.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.12.m12.1.1.1.cmml" xref="S3.SS2.p10.12.m12.1.1">subscript</csymbol><ci id="S3.SS2.p10.12.m12.1.1.2.cmml" xref="S3.SS2.p10.12.m12.1.1.2">𝑇</ci><ci id="S3.SS2.p10.12.m12.1.1.3.cmml" xref="S3.SS2.p10.12.m12.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.12.m12.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.12.m12.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> is the number of time steps in sequence <math alttext="\mathcal{S}_{m}" class="ltx_Math" display="inline" id="S3.SS2.p10.13.m13.1"><semantics id="S3.SS2.p10.13.m13.1a"><msub id="S3.SS2.p10.13.m13.1.1" xref="S3.SS2.p10.13.m13.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.13.m13.1.1.2" xref="S3.SS2.p10.13.m13.1.1.2.cmml">𝒮</mi><mi id="S3.SS2.p10.13.m13.1.1.3" xref="S3.SS2.p10.13.m13.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.13.m13.1b"><apply id="S3.SS2.p10.13.m13.1.1.cmml" xref="S3.SS2.p10.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.13.m13.1.1.1.cmml" xref="S3.SS2.p10.13.m13.1.1">subscript</csymbol><ci id="S3.SS2.p10.13.m13.1.1.2.cmml" xref="S3.SS2.p10.13.m13.1.1.2">𝒮</ci><ci id="S3.SS2.p10.13.m13.1.1.3.cmml" xref="S3.SS2.p10.13.m13.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.13.m13.1c">\mathcal{S}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.13.m13.1d">caligraphic_S start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. Likewise, the test subset <math alttext="\mathcal{D}^{\text{test}}=\{\mathcal{S}_{1},...,\mathcal{S}_{n},...,\mathcal{S% }_{N}\}" class="ltx_Math" display="inline" id="S3.SS2.p10.14.m14.5"><semantics id="S3.SS2.p10.14.m14.5a"><mrow id="S3.SS2.p10.14.m14.5.5" xref="S3.SS2.p10.14.m14.5.5.cmml"><msup id="S3.SS2.p10.14.m14.5.5.5" xref="S3.SS2.p10.14.m14.5.5.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.14.m14.5.5.5.2" xref="S3.SS2.p10.14.m14.5.5.5.2.cmml">𝒟</mi><mtext id="S3.SS2.p10.14.m14.5.5.5.3" xref="S3.SS2.p10.14.m14.5.5.5.3a.cmml">test</mtext></msup><mo id="S3.SS2.p10.14.m14.5.5.4" xref="S3.SS2.p10.14.m14.5.5.4.cmml">=</mo><mrow id="S3.SS2.p10.14.m14.5.5.3.3" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml"><mo id="S3.SS2.p10.14.m14.5.5.3.3.4" stretchy="false" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">{</mo><msub id="S3.SS2.p10.14.m14.3.3.1.1.1" xref="S3.SS2.p10.14.m14.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.14.m14.3.3.1.1.1.2" xref="S3.SS2.p10.14.m14.3.3.1.1.1.2.cmml">𝒮</mi><mn id="S3.SS2.p10.14.m14.3.3.1.1.1.3" xref="S3.SS2.p10.14.m14.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p10.14.m14.5.5.3.3.5" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">,</mo><mi id="S3.SS2.p10.14.m14.1.1" mathvariant="normal" xref="S3.SS2.p10.14.m14.1.1.cmml">…</mi><mo id="S3.SS2.p10.14.m14.5.5.3.3.6" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">,</mo><msub id="S3.SS2.p10.14.m14.4.4.2.2.2" xref="S3.SS2.p10.14.m14.4.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.14.m14.4.4.2.2.2.2" xref="S3.SS2.p10.14.m14.4.4.2.2.2.2.cmml">𝒮</mi><mi id="S3.SS2.p10.14.m14.4.4.2.2.2.3" xref="S3.SS2.p10.14.m14.4.4.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.p10.14.m14.5.5.3.3.7" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">,</mo><mi id="S3.SS2.p10.14.m14.2.2" mathvariant="normal" xref="S3.SS2.p10.14.m14.2.2.cmml">…</mi><mo id="S3.SS2.p10.14.m14.5.5.3.3.8" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">,</mo><msub id="S3.SS2.p10.14.m14.5.5.3.3.3" xref="S3.SS2.p10.14.m14.5.5.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.14.m14.5.5.3.3.3.2" xref="S3.SS2.p10.14.m14.5.5.3.3.3.2.cmml">𝒮</mi><mi id="S3.SS2.p10.14.m14.5.5.3.3.3.3" xref="S3.SS2.p10.14.m14.5.5.3.3.3.3.cmml">N</mi></msub><mo id="S3.SS2.p10.14.m14.5.5.3.3.9" stretchy="false" xref="S3.SS2.p10.14.m14.5.5.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.14.m14.5b"><apply id="S3.SS2.p10.14.m14.5.5.cmml" xref="S3.SS2.p10.14.m14.5.5"><eq id="S3.SS2.p10.14.m14.5.5.4.cmml" xref="S3.SS2.p10.14.m14.5.5.4"></eq><apply id="S3.SS2.p10.14.m14.5.5.5.cmml" xref="S3.SS2.p10.14.m14.5.5.5"><csymbol cd="ambiguous" id="S3.SS2.p10.14.m14.5.5.5.1.cmml" xref="S3.SS2.p10.14.m14.5.5.5">superscript</csymbol><ci id="S3.SS2.p10.14.m14.5.5.5.2.cmml" xref="S3.SS2.p10.14.m14.5.5.5.2">𝒟</ci><ci id="S3.SS2.p10.14.m14.5.5.5.3a.cmml" xref="S3.SS2.p10.14.m14.5.5.5.3"><mtext id="S3.SS2.p10.14.m14.5.5.5.3.cmml" mathsize="70%" xref="S3.SS2.p10.14.m14.5.5.5.3">test</mtext></ci></apply><set id="S3.SS2.p10.14.m14.5.5.3.4.cmml" xref="S3.SS2.p10.14.m14.5.5.3.3"><apply id="S3.SS2.p10.14.m14.3.3.1.1.1.cmml" xref="S3.SS2.p10.14.m14.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.14.m14.3.3.1.1.1.1.cmml" xref="S3.SS2.p10.14.m14.3.3.1.1.1">subscript</csymbol><ci id="S3.SS2.p10.14.m14.3.3.1.1.1.2.cmml" xref="S3.SS2.p10.14.m14.3.3.1.1.1.2">𝒮</ci><cn id="S3.SS2.p10.14.m14.3.3.1.1.1.3.cmml" type="integer" xref="S3.SS2.p10.14.m14.3.3.1.1.1.3">1</cn></apply><ci id="S3.SS2.p10.14.m14.1.1.cmml" xref="S3.SS2.p10.14.m14.1.1">…</ci><apply id="S3.SS2.p10.14.m14.4.4.2.2.2.cmml" xref="S3.SS2.p10.14.m14.4.4.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p10.14.m14.4.4.2.2.2.1.cmml" xref="S3.SS2.p10.14.m14.4.4.2.2.2">subscript</csymbol><ci id="S3.SS2.p10.14.m14.4.4.2.2.2.2.cmml" xref="S3.SS2.p10.14.m14.4.4.2.2.2.2">𝒮</ci><ci id="S3.SS2.p10.14.m14.4.4.2.2.2.3.cmml" xref="S3.SS2.p10.14.m14.4.4.2.2.2.3">𝑛</ci></apply><ci id="S3.SS2.p10.14.m14.2.2.cmml" xref="S3.SS2.p10.14.m14.2.2">…</ci><apply id="S3.SS2.p10.14.m14.5.5.3.3.3.cmml" xref="S3.SS2.p10.14.m14.5.5.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p10.14.m14.5.5.3.3.3.1.cmml" xref="S3.SS2.p10.14.m14.5.5.3.3.3">subscript</csymbol><ci id="S3.SS2.p10.14.m14.5.5.3.3.3.2.cmml" xref="S3.SS2.p10.14.m14.5.5.3.3.3.2">𝒮</ci><ci id="S3.SS2.p10.14.m14.5.5.3.3.3.3.cmml" xref="S3.SS2.p10.14.m14.5.5.3.3.3.3">𝑁</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.14.m14.5c">\mathcal{D}^{\text{test}}=\{\mathcal{S}_{1},...,\mathcal{S}_{n},...,\mathcal{S% }_{N}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.14.m14.5d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT = { caligraphic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , caligraphic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , … , caligraphic_S start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT }</annotation></semantics></math> consists of <math alttext="N=1186" class="ltx_Math" display="inline" id="S3.SS2.p10.15.m15.1"><semantics id="S3.SS2.p10.15.m15.1a"><mrow id="S3.SS2.p10.15.m15.1.1" xref="S3.SS2.p10.15.m15.1.1.cmml"><mi id="S3.SS2.p10.15.m15.1.1.2" xref="S3.SS2.p10.15.m15.1.1.2.cmml">N</mi><mo id="S3.SS2.p10.15.m15.1.1.1" xref="S3.SS2.p10.15.m15.1.1.1.cmml">=</mo><mn id="S3.SS2.p10.15.m15.1.1.3" xref="S3.SS2.p10.15.m15.1.1.3.cmml">1186</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.15.m15.1b"><apply id="S3.SS2.p10.15.m15.1.1.cmml" xref="S3.SS2.p10.15.m15.1.1"><eq id="S3.SS2.p10.15.m15.1.1.1.cmml" xref="S3.SS2.p10.15.m15.1.1.1"></eq><ci id="S3.SS2.p10.15.m15.1.1.2.cmml" xref="S3.SS2.p10.15.m15.1.1.2">𝑁</ci><cn id="S3.SS2.p10.15.m15.1.1.3.cmml" type="integer" xref="S3.SS2.p10.15.m15.1.1.3">1186</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.15.m15.1c">N=1186</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.15.m15.1d">italic_N = 1186</annotation></semantics></math> multivariate time series on average, where <math alttext="\mathcal{S}_{n}\in\mathbb{R}^{T_{n}\times d_{\mathcal{D}}}" class="ltx_Math" display="inline" id="S3.SS2.p10.16.m16.1"><semantics id="S3.SS2.p10.16.m16.1a"><mrow id="S3.SS2.p10.16.m16.1.1" xref="S3.SS2.p10.16.m16.1.1.cmml"><msub id="S3.SS2.p10.16.m16.1.1.2" xref="S3.SS2.p10.16.m16.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.16.m16.1.1.2.2" xref="S3.SS2.p10.16.m16.1.1.2.2.cmml">𝒮</mi><mi id="S3.SS2.p10.16.m16.1.1.2.3" xref="S3.SS2.p10.16.m16.1.1.2.3.cmml">n</mi></msub><mo id="S3.SS2.p10.16.m16.1.1.1" xref="S3.SS2.p10.16.m16.1.1.1.cmml">∈</mo><msup id="S3.SS2.p10.16.m16.1.1.3" xref="S3.SS2.p10.16.m16.1.1.3.cmml"><mi id="S3.SS2.p10.16.m16.1.1.3.2" xref="S3.SS2.p10.16.m16.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS2.p10.16.m16.1.1.3.3" xref="S3.SS2.p10.16.m16.1.1.3.3.cmml"><msub id="S3.SS2.p10.16.m16.1.1.3.3.2" xref="S3.SS2.p10.16.m16.1.1.3.3.2.cmml"><mi id="S3.SS2.p10.16.m16.1.1.3.3.2.2" xref="S3.SS2.p10.16.m16.1.1.3.3.2.2.cmml">T</mi><mi id="S3.SS2.p10.16.m16.1.1.3.3.2.3" xref="S3.SS2.p10.16.m16.1.1.3.3.2.3.cmml">n</mi></msub><mo id="S3.SS2.p10.16.m16.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p10.16.m16.1.1.3.3.1.cmml">×</mo><msub id="S3.SS2.p10.16.m16.1.1.3.3.3" xref="S3.SS2.p10.16.m16.1.1.3.3.3.cmml"><mi id="S3.SS2.p10.16.m16.1.1.3.3.3.2" xref="S3.SS2.p10.16.m16.1.1.3.3.3.2.cmml">d</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.16.m16.1.1.3.3.3.3" xref="S3.SS2.p10.16.m16.1.1.3.3.3.3.cmml">𝒟</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.16.m16.1b"><apply id="S3.SS2.p10.16.m16.1.1.cmml" xref="S3.SS2.p10.16.m16.1.1"><in id="S3.SS2.p10.16.m16.1.1.1.cmml" xref="S3.SS2.p10.16.m16.1.1.1"></in><apply id="S3.SS2.p10.16.m16.1.1.2.cmml" xref="S3.SS2.p10.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p10.16.m16.1.1.2.1.cmml" xref="S3.SS2.p10.16.m16.1.1.2">subscript</csymbol><ci id="S3.SS2.p10.16.m16.1.1.2.2.cmml" xref="S3.SS2.p10.16.m16.1.1.2.2">𝒮</ci><ci id="S3.SS2.p10.16.m16.1.1.2.3.cmml" xref="S3.SS2.p10.16.m16.1.1.2.3">𝑛</ci></apply><apply id="S3.SS2.p10.16.m16.1.1.3.cmml" xref="S3.SS2.p10.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p10.16.m16.1.1.3.1.cmml" xref="S3.SS2.p10.16.m16.1.1.3">superscript</csymbol><ci id="S3.SS2.p10.16.m16.1.1.3.2.cmml" xref="S3.SS2.p10.16.m16.1.1.3.2">ℝ</ci><apply id="S3.SS2.p10.16.m16.1.1.3.3.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3"><times id="S3.SS2.p10.16.m16.1.1.3.3.1.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.1"></times><apply id="S3.SS2.p10.16.m16.1.1.3.3.2.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p10.16.m16.1.1.3.3.2.1.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.2">subscript</csymbol><ci id="S3.SS2.p10.16.m16.1.1.3.3.2.2.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.2.2">𝑇</ci><ci id="S3.SS2.p10.16.m16.1.1.3.3.2.3.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.2.3">𝑛</ci></apply><apply id="S3.SS2.p10.16.m16.1.1.3.3.3.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p10.16.m16.1.1.3.3.3.1.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.3">subscript</csymbol><ci id="S3.SS2.p10.16.m16.1.1.3.3.3.2.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.3.2">𝑑</ci><ci id="S3.SS2.p10.16.m16.1.1.3.3.3.3.cmml" xref="S3.SS2.p10.16.m16.1.1.3.3.3.3">𝒟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.16.m16.1c">\mathcal{S}_{n}\in\mathbb{R}^{T_{n}\times d_{\mathcal{D}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.16.m16.1d">caligraphic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="T_{n}" class="ltx_Math" display="inline" id="S3.SS2.p10.17.m17.1"><semantics id="S3.SS2.p10.17.m17.1a"><msub id="S3.SS2.p10.17.m17.1.1" xref="S3.SS2.p10.17.m17.1.1.cmml"><mi id="S3.SS2.p10.17.m17.1.1.2" xref="S3.SS2.p10.17.m17.1.1.2.cmml">T</mi><mi id="S3.SS2.p10.17.m17.1.1.3" xref="S3.SS2.p10.17.m17.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.17.m17.1b"><apply id="S3.SS2.p10.17.m17.1.1.cmml" xref="S3.SS2.p10.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.17.m17.1.1.1.cmml" xref="S3.SS2.p10.17.m17.1.1">subscript</csymbol><ci id="S3.SS2.p10.17.m17.1.1.2.cmml" xref="S3.SS2.p10.17.m17.1.1.2">𝑇</ci><ci id="S3.SS2.p10.17.m17.1.1.3.cmml" xref="S3.SS2.p10.17.m17.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.17.m17.1c">T_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.17.m17.1d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is the number of time steps in sequence <math alttext="\mathcal{S}_{n}" class="ltx_Math" display="inline" id="S3.SS2.p10.18.m18.1"><semantics id="S3.SS2.p10.18.m18.1a"><msub id="S3.SS2.p10.18.m18.1.1" xref="S3.SS2.p10.18.m18.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.18.m18.1.1.2" xref="S3.SS2.p10.18.m18.1.1.2.cmml">𝒮</mi><mi id="S3.SS2.p10.18.m18.1.1.3" xref="S3.SS2.p10.18.m18.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.18.m18.1b"><apply id="S3.SS2.p10.18.m18.1.1.cmml" xref="S3.SS2.p10.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.18.m18.1.1.1.cmml" xref="S3.SS2.p10.18.m18.1.1">subscript</csymbol><ci id="S3.SS2.p10.18.m18.1.1.2.cmml" xref="S3.SS2.p10.18.m18.1.1.2">𝒮</ci><ci id="S3.SS2.p10.18.m18.1.1.3.cmml" xref="S3.SS2.p10.18.m18.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.18.m18.1c">\mathcal{S}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.18.m18.1d">caligraphic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. For benchmarking purposes, we suggest the users use the prescribed training and test split to ensure comparable results.</p> </div> <div class="ltx_para" id="S3.SS2.p11"> <p class="ltx_p" id="S3.SS2.p11.4">To add further complexity and to reflect real-world properties, we undertake some <em class="ltx_emph ltx_font_italic" id="S3.SS2.p11.4.1">post-processing</em>. First, we trim the beginning of each time series in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S3.SS2.p11.1.m1.1"><semantics id="S3.SS2.p11.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p11.1.m1.1.1" xref="S3.SS2.p11.1.m1.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p11.1.m1.1b"><ci id="S3.SS2.p11.1.m1.1.1.cmml" xref="S3.SS2.p11.1.m1.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p11.1.m1.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p11.1.m1.1d">caligraphic_D</annotation></semantics></math> by random amounts so that time series representing the same drive cycle are rarely in sync. The amount by which a given time series is trimmed is sampled from uniform distribution <math alttext="\mathcal{U}(0,0.1T)" class="ltx_Math" display="inline" id="S3.SS2.p11.2.m2.2"><semantics id="S3.SS2.p11.2.m2.2a"><mrow id="S3.SS2.p11.2.m2.2.2" xref="S3.SS2.p11.2.m2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p11.2.m2.2.2.3" xref="S3.SS2.p11.2.m2.2.2.3.cmml">𝒰</mi><mo id="S3.SS2.p11.2.m2.2.2.2" xref="S3.SS2.p11.2.m2.2.2.2.cmml">⁢</mo><mrow id="S3.SS2.p11.2.m2.2.2.1.1" xref="S3.SS2.p11.2.m2.2.2.1.2.cmml"><mo id="S3.SS2.p11.2.m2.2.2.1.1.2" stretchy="false" xref="S3.SS2.p11.2.m2.2.2.1.2.cmml">(</mo><mn id="S3.SS2.p11.2.m2.1.1" xref="S3.SS2.p11.2.m2.1.1.cmml">0</mn><mo id="S3.SS2.p11.2.m2.2.2.1.1.3" xref="S3.SS2.p11.2.m2.2.2.1.2.cmml">,</mo><mrow id="S3.SS2.p11.2.m2.2.2.1.1.1" xref="S3.SS2.p11.2.m2.2.2.1.1.1.cmml"><mn id="S3.SS2.p11.2.m2.2.2.1.1.1.2" xref="S3.SS2.p11.2.m2.2.2.1.1.1.2.cmml">0.1</mn><mo id="S3.SS2.p11.2.m2.2.2.1.1.1.1" xref="S3.SS2.p11.2.m2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p11.2.m2.2.2.1.1.1.3" xref="S3.SS2.p11.2.m2.2.2.1.1.1.3.cmml">T</mi></mrow><mo id="S3.SS2.p11.2.m2.2.2.1.1.4" stretchy="false" xref="S3.SS2.p11.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p11.2.m2.2b"><apply id="S3.SS2.p11.2.m2.2.2.cmml" xref="S3.SS2.p11.2.m2.2.2"><times id="S3.SS2.p11.2.m2.2.2.2.cmml" xref="S3.SS2.p11.2.m2.2.2.2"></times><ci id="S3.SS2.p11.2.m2.2.2.3.cmml" xref="S3.SS2.p11.2.m2.2.2.3">𝒰</ci><interval closure="open" id="S3.SS2.p11.2.m2.2.2.1.2.cmml" xref="S3.SS2.p11.2.m2.2.2.1.1"><cn id="S3.SS2.p11.2.m2.1.1.cmml" type="integer" xref="S3.SS2.p11.2.m2.1.1">0</cn><apply id="S3.SS2.p11.2.m2.2.2.1.1.1.cmml" xref="S3.SS2.p11.2.m2.2.2.1.1.1"><times id="S3.SS2.p11.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS2.p11.2.m2.2.2.1.1.1.1"></times><cn id="S3.SS2.p11.2.m2.2.2.1.1.1.2.cmml" type="float" xref="S3.SS2.p11.2.m2.2.2.1.1.1.2">0.1</cn><ci id="S3.SS2.p11.2.m2.2.2.1.1.1.3.cmml" xref="S3.SS2.p11.2.m2.2.2.1.1.1.3">𝑇</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p11.2.m2.2c">\mathcal{U}(0,0.1T)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p11.2.m2.2d">caligraphic_U ( 0 , 0.1 italic_T )</annotation></semantics></math>. This artefact can happen in the real world and means that, for the same drive cycle, any given time step is not comparable across different sequences, eliminating the viability of simple statistical methods such as control charts. 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end_POSTSUBSCRIPT</annotation></semantics></math> is the feature-wise standard deviation of the dataset.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Usability of the Dataset</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.3">Clearly, both <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><msup id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1">superscript</csymbol><ci 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This is because the dataset is aimed at <em class="ltx_emph ltx_font_italic" id="S3.SS3.p1.3.1">unsupervised</em> time series anomaly detection, which requires approaches especially robust to contaminated training data.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.9">We believe the underlying properties of the dataset can be useful in other research areas too. The same <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><msup id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.1.m1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">superscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2">𝒟</ci><ci id="S3.SS3.p2.1.m1.1.1.3a.cmml" xref="S3.SS3.p2.1.m1.1.1.3"><mtext id="S3.SS3.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.1.m1.1.1.3">train</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\mathcal{D}^{\text{train}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msup id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.2.m2.1.1.3" xref="S3.SS3.p2.2.m2.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2">𝒟</ci><ci id="S3.SS3.p2.2.m2.1.1.3a.cmml" xref="S3.SS3.p2.2.m2.1.1.3"><mtext id="S3.SS3.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.2.m2.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math> subsets can also be used for <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.9.1">imbalanced time series classification</em> if the labels are considered. Additionally, we provide a number of different subset variations for other tasks. For <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.9.2">semi-supervised</em> anomaly detection, we provide a clean <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.1"><semantics id="S3.SS3.p2.3.m3.1a"><msup id="S3.SS3.p2.3.m3.1.1" xref="S3.SS3.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.3.m3.1.1.2" xref="S3.SS3.p2.3.m3.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.3.m3.1.1.3" xref="S3.SS3.p2.3.m3.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.1b"><apply id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.3.m3.1.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1">superscript</csymbol><ci id="S3.SS3.p2.3.m3.1.1.2.cmml" xref="S3.SS3.p2.3.m3.1.1.2">𝒟</ci><ci id="S3.SS3.p2.3.m3.1.1.3a.cmml" xref="S3.SS3.p2.3.m3.1.1.3"><mtext id="S3.SS3.p2.3.m3.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.3.m3.1.1.3">train</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.1c">\mathcal{D}^{\text{train}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.1d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT</annotation></semantics></math> with on average <math alttext="M=2182" class="ltx_Math" display="inline" id="S3.SS3.p2.4.m4.1"><semantics id="S3.SS3.p2.4.m4.1a"><mrow id="S3.SS3.p2.4.m4.1.1" xref="S3.SS3.p2.4.m4.1.1.cmml"><mi id="S3.SS3.p2.4.m4.1.1.2" xref="S3.SS3.p2.4.m4.1.1.2.cmml">M</mi><mo id="S3.SS3.p2.4.m4.1.1.1" xref="S3.SS3.p2.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS3.p2.4.m4.1.1.3" xref="S3.SS3.p2.4.m4.1.1.3.cmml">2182</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.4.m4.1b"><apply id="S3.SS3.p2.4.m4.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1"><eq id="S3.SS3.p2.4.m4.1.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1.1"></eq><ci id="S3.SS3.p2.4.m4.1.1.2.cmml" xref="S3.SS3.p2.4.m4.1.1.2">𝑀</ci><cn id="S3.SS3.p2.4.m4.1.1.3.cmml" type="integer" xref="S3.SS3.p2.4.m4.1.1.3">2182</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.m4.1c">M=2182</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.m4.1d">italic_M = 2182</annotation></semantics></math> nominal time series and the same labelled <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.1"><semantics id="S3.SS3.p2.5.m5.1a"><msup id="S3.SS3.p2.5.m5.1.1" xref="S3.SS3.p2.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.5.m5.1.1.2" xref="S3.SS3.p2.5.m5.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.5.m5.1.1.3" xref="S3.SS3.p2.5.m5.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m5.1b"><apply id="S3.SS3.p2.5.m5.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1">superscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2">𝒟</ci><ci id="S3.SS3.p2.5.m5.1.1.3a.cmml" xref="S3.SS3.p2.5.m5.1.1.3"><mtext id="S3.SS3.p2.5.m5.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.5.m5.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math> in the dataset, where clean refers to the absence of anomalous sequences in the subset. Furthermore, for time series <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.9.3">forecasting</em> or <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.9.4">generation</em>, we supply clean versions of both <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m6.1"><semantics id="S3.SS3.p2.6.m6.1a"><msup id="S3.SS3.p2.6.m6.1.1" xref="S3.SS3.p2.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.6.m6.1.1.2" xref="S3.SS3.p2.6.m6.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.6.m6.1.1.3" xref="S3.SS3.p2.6.m6.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m6.1b"><apply id="S3.SS3.p2.6.m6.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.6.m6.1.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1">superscript</csymbol><ci id="S3.SS3.p2.6.m6.1.1.2.cmml" xref="S3.SS3.p2.6.m6.1.1.2">𝒟</ci><ci id="S3.SS3.p2.6.m6.1.1.3a.cmml" xref="S3.SS3.p2.6.m6.1.1.3"><mtext id="S3.SS3.p2.6.m6.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.6.m6.1.1.3">train</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m6.1c">\mathcal{D}^{\text{train}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m6.1d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S3.SS3.p2.7.m7.1"><semantics id="S3.SS3.p2.7.m7.1a"><msup id="S3.SS3.p2.7.m7.1.1" xref="S3.SS3.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.7.m7.1.1.2" xref="S3.SS3.p2.7.m7.1.1.2.cmml">𝒟</mi><mtext id="S3.SS3.p2.7.m7.1.1.3" xref="S3.SS3.p2.7.m7.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.7.m7.1b"><apply id="S3.SS3.p2.7.m7.1.1.cmml" xref="S3.SS3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.7.m7.1.1.1.cmml" xref="S3.SS3.p2.7.m7.1.1">superscript</csymbol><ci id="S3.SS3.p2.7.m7.1.1.2.cmml" xref="S3.SS3.p2.7.m7.1.1.2">𝒟</ci><ci id="S3.SS3.p2.7.m7.1.1.3a.cmml" xref="S3.SS3.p2.7.m7.1.1.3"><mtext id="S3.SS3.p2.7.m7.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p2.7.m7.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.7.m7.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.7.m7.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="M=2182" class="ltx_Math" display="inline" id="S3.SS3.p2.8.m8.1"><semantics id="S3.SS3.p2.8.m8.1a"><mrow id="S3.SS3.p2.8.m8.1.1" xref="S3.SS3.p2.8.m8.1.1.cmml"><mi id="S3.SS3.p2.8.m8.1.1.2" xref="S3.SS3.p2.8.m8.1.1.2.cmml">M</mi><mo id="S3.SS3.p2.8.m8.1.1.1" xref="S3.SS3.p2.8.m8.1.1.1.cmml">=</mo><mn id="S3.SS3.p2.8.m8.1.1.3" xref="S3.SS3.p2.8.m8.1.1.3.cmml">2182</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.8.m8.1b"><apply id="S3.SS3.p2.8.m8.1.1.cmml" xref="S3.SS3.p2.8.m8.1.1"><eq id="S3.SS3.p2.8.m8.1.1.1.cmml" xref="S3.SS3.p2.8.m8.1.1.1"></eq><ci id="S3.SS3.p2.8.m8.1.1.2.cmml" xref="S3.SS3.p2.8.m8.1.1.2">𝑀</ci><cn id="S3.SS3.p2.8.m8.1.1.3.cmml" type="integer" xref="S3.SS3.p2.8.m8.1.1.3">2182</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.8.m8.1c">M=2182</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.8.m8.1d">italic_M = 2182</annotation></semantics></math> and <math alttext="N=1091" class="ltx_Math" display="inline" id="S3.SS3.p2.9.m9.1"><semantics id="S3.SS3.p2.9.m9.1a"><mrow id="S3.SS3.p2.9.m9.1.1" xref="S3.SS3.p2.9.m9.1.1.cmml"><mi id="S3.SS3.p2.9.m9.1.1.2" xref="S3.SS3.p2.9.m9.1.1.2.cmml">N</mi><mo id="S3.SS3.p2.9.m9.1.1.1" xref="S3.SS3.p2.9.m9.1.1.1.cmml">=</mo><mn id="S3.SS3.p2.9.m9.1.1.3" xref="S3.SS3.p2.9.m9.1.1.3.cmml">1091</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.9.m9.1b"><apply id="S3.SS3.p2.9.m9.1.1.cmml" xref="S3.SS3.p2.9.m9.1.1"><eq id="S3.SS3.p2.9.m9.1.1.1.cmml" xref="S3.SS3.p2.9.m9.1.1.1"></eq><ci id="S3.SS3.p2.9.m9.1.1.2.cmml" xref="S3.SS3.p2.9.m9.1.1.2">𝑁</ci><cn id="S3.SS3.p2.9.m9.1.1.3.cmml" type="integer" xref="S3.SS3.p2.9.m9.1.1.3">1091</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.9.m9.1c">N=1091</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.9.m9.1d">italic_N = 1091</annotation></semantics></math> on average, respectively. Despite being targeted at <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.9.5">online</em> time series anomaly detection, the PATH dataset can just as well be used in offline time series anomaly detection.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Baseline Results on the Dataset</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Methodology</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">The evaluation metrics used to quantify anomaly detection performance by Correia et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> are adopted, as they provide a parameter-free way to quantify <em class="ltx_emph ltx_font_italic" id="S4.SS1.p1.1.1">online</em> anomaly detection performance in an interpretable way.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">In this work, we consider OmniAnomaly <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">su_robust_2019</span>]</cite>, TCN-AE <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">thill_temporal_2021</span>]</cite>, SISVAE <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">li_anomaly_2021</span>]</cite>, LW-VAE <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">fahrmann_lightweight_2022</span>]</cite>, TSADIS <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">tafazoli_matrix_2023</span>]</cite>, and TeVAE <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> when conducting experiments. The hyperparameters for each approach are set as specified in the respective publication, though early stopping is applied to all that require a training procedure. Early stopping is parameterised such that the respective reconstruction error is monitored and training is stopped once validation loss has stopped decreasing for 250 epochs.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.3">The validation subset <math alttext="\mathcal{D}^{\text{val}}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><msup id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.1.1.2" xref="S4.SS1.p3.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S4.SS1.p3.1.m1.1.1.3" xref="S4.SS1.p3.1.m1.1.1.3a.cmml">val</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2">𝒟</ci><ci id="S4.SS1.p3.1.m1.1.1.3a.cmml" xref="S4.SS1.p3.1.m1.1.1.3"><mtext id="S4.SS1.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS1.p3.1.m1.1.1.3">val</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\mathcal{D}^{\text{val}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT val end_POSTSUPERSCRIPT</annotation></semantics></math> is obtained by further splitting <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><msup id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml">𝒟</mi><mtext id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2">𝒟</ci><ci id="S4.SS1.p3.2.m2.1.1.3a.cmml" xref="S4.SS1.p3.2.m2.1.1.3"><mtext id="S4.SS1.p3.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS1.p3.2.m2.1.1.3">train</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\mathcal{D}^{\text{train}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT</annotation></semantics></math> and hence is also unlabelled. As future work may require a validation subset, it is left to the individual to extract it from the training subset if needed. The test subset <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><msup id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.3.m3.1.1.2" xref="S4.SS1.p3.3.m3.1.1.2.cmml">𝒟</mi><mtext id="S4.SS1.p3.3.m3.1.1.3" xref="S4.SS1.p3.3.m3.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><apply id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.1.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">superscript</csymbol><ci id="S4.SS1.p3.3.m3.1.1.2.cmml" xref="S4.SS1.p3.3.m3.1.1.2">𝒟</ci><ci id="S4.SS1.p3.3.m3.1.1.3a.cmml" xref="S4.SS1.p3.3.m3.1.1.3"><mtext id="S4.SS1.p3.3.m3.1.1.3.cmml" mathsize="70%" xref="S4.SS1.p3.3.m3.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math> should be the same as the one provided to ensure comparable results.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1">As mentioned in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S3" title="3 Proposed Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">3</span></a> the simulation signals are sampled at <span class="ltx_ERROR undefined" id="S4.SS1.p4.1.1">\qty</span>10 by default, however, to reduce the computational load in our experiments, we downsample the data to <span class="ltx_ERROR undefined" id="S4.SS1.p4.1.2">\qty</span>2 with a low-pass filter with a cut-off frequency of <span class="ltx_ERROR undefined" id="S4.SS1.p4.1.3">\qty</span>1, as it is consistent with the Whittaker–Nyquist–Shannon theorem <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">shannon_communication_1949</span>]</cite>. This downsampling procedure is considered as part of the approaches tested and is optional for any future work, which may alternatively use the raw time series data or perhaps even correlation matrices <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">tayeh_attention-based_2022</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">zhang_deep_2019</span>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.1">To bring all channels to a common magnitude, the dataset features are z-score normalised.</p> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.2">Finally, we segment the time series data into fixed-length sub-sequences, also referred to as <em class="ltx_emph ltx_font_italic" id="S4.SS1.p6.2.1">windows</em>. The rationale for using windows instead of full-length sequences is that the dynamics present in the time series data tend to occur quickly and only influence the data for a brief duration. Modelling entire variable-length sequences is possible, but it would lead to inefficient use of the model’s learning capacity, as it would have to maintain information over unnecessarily long periods. By focusing on windows that are just long enough to capture the existing dynamics in the data, model training should be more effective. To determine the optimal window length at <span class="ltx_ERROR undefined" id="S4.SS1.p6.2.2">\qty</span>2, especially to capture the slowest dynamics present in a signal, we perform an autocorrelation analysis <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> for each drive cycle and for every feature within those cycles, yielding a window size of <math alttext="256" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.1"><semantics id="S4.SS1.p6.1.m1.1a"><mn id="S4.SS1.p6.1.m1.1.1" xref="S4.SS1.p6.1.m1.1.1.cmml">256</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.m1.1b"><cn id="S4.SS1.p6.1.m1.1.1.cmml" type="integer" xref="S4.SS1.p6.1.m1.1.1">256</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.1c">256</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.1d">256</annotation></semantics></math> time steps. In the literature, the window size is often treated as a hyperparameter <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">fahrmann_lightweight_2022</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">tuli_tranad_2022</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">tafazoli_matrix_2023</span>]</cite> or provided without reasoning <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">pereira_unsupervised_2018</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">chen_unsupervised_2020</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">li_anomaly_2021</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">thill_temporal_2021</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">doshi_reward_2022</span>]</cite>. However, it is not possible to tune hyperparameters outside a supervised setting, and therefore such methods might not be applicable in real-world settings. In contrast, finding a suitable window length using autocorrelation is completely unsupervised. TSADIS takes window size as a hyperparameter before calling, hence a window size of <math alttext="256" class="ltx_Math" display="inline" id="S4.SS1.p6.2.m2.1"><semantics id="S4.SS1.p6.2.m2.1a"><mn id="S4.SS1.p6.2.m2.1.1" xref="S4.SS1.p6.2.m2.1.1.cmml">256</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.2.m2.1b"><cn id="S4.SS1.p6.2.m2.1.1.cmml" type="integer" xref="S4.SS1.p6.2.m2.1.1">256</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.m2.1c">256</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.m2.1d">256</annotation></semantics></math> time steps is also used. To map the individual windows back to continuous sequences, mean-type reverse-windowing <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> is used where applicable.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Reproducibility and Benchmarking Considerations</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">While perhaps sounding similar, repeatability, reproducibility, and replicability are defined differently according to the Association for Computing Machinery <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">association_for_computing_machinery_artifact_nodate</span>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">Several position papers <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">drummond_replicability_2009</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">peng_reproducible_2011</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">munafo_manifesto_2017</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">gundersen_state_2018</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">bartz-beielstein_benchmarking_2020</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">kapoor_leakage_2023</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">semmelrock_reproducibility_2023</span>]</cite> call for greater attention to be paid to reproducibility and replicability in computer science. Additionally, some conferences focus on reproduction, like the Machine Learning Reproducibility Challenge <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">machine_learning_reproducibility_challenge</span>]</cite>, or make specific calls for reproducibility and replicability papers, like the European Conference on Information Retrieval <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">european_conference_on_information_retrieval</span>]</cite>. To enable future work to reproduce the results in this paper, we aim to be as transparent as possible by providing publicly available, clean and thoroughly commented source code for all experiments and the Simulink model under <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/lcs-crr/PATH" title="">https://github.com/lcs-crr/PATH</a>, as is suggested in literature <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">peng_reproducible_2011</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">munafo_manifesto_2017</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">semmelrock_reproducibility_2023</span>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.4">The seed for random operations has an impact on model training, given that processes like sampling and weight initialisation rely on it. To increase robustness of the results and to eliminate the possibility of the results owing to a specific fold and seed combination rather from the characteristics of the model <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">semmelrock_reproducibility_2023</span>]</cite>, all three folds are trained on seeds <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mn id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><cn id="S4.SS2.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS2.p3.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">1</annotation></semantics></math> through <math alttext="5" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mn id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><cn id="S4.SS2.p3.2.m2.1.1.cmml" type="integer" xref="S4.SS2.p3.2.m2.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">5</annotation></semantics></math>, yielding <math alttext="15" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><mn id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml">15</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><cn id="S4.SS2.p3.3.m3.1.1.cmml" type="integer" xref="S4.SS2.p3.3.m3.1.1">15</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">15</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">15</annotation></semantics></math> different combinations. The final result is then given as the average of the <math alttext="15" class="ltx_Math" display="inline" id="S4.SS2.p3.4.m4.1"><semantics id="S4.SS2.p3.4.m4.1a"><mn id="S4.SS2.p3.4.m4.1.1" xref="S4.SS2.p3.4.m4.1.1.cmml">15</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m4.1b"><cn id="S4.SS2.p3.4.m4.1.1.cmml" type="integer" xref="S4.SS2.p3.4.m4.1.1">15</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m4.1c">15</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.m4.1d">15</annotation></semantics></math> different combinations. As mentioned, TSADIS does not require training, and hence its results are simply the average over all three folds.</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">In case future work aims to replicate the results of this paper, we encourage deviating from the experimental setup outlined in this paper <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">goodman_what_2016</span>]</cite>, though, as Bartz-Beielstein et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">bartz-beielstein_benchmarking_2020</span>]</cite> point out, there is no definition for how different an experimental setup needs to be for results to be considered replicable. Using a different set of seeds, splitting the dataset into different folds, using different implementations of the approaches or even by using different software and hardware are some of the variables that could be changed in the setup, for example. In the case of replicability, the above-mentioned changes should not change the outcome <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">gundersen_state_2018</span>]</cite>. Moreover, it is just as important that future work provides the same level of transparency regarding the experimental setup and documentation.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.2">It should be noted that the test subset <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><msup id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.1.m1.1.1.2" xref="S4.SS2.p5.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S4.SS2.p5.1.m1.1.1.3" xref="S4.SS2.p5.1.m1.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><apply id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.p5.1.m1.1.1.2.cmml" xref="S4.SS2.p5.1.m1.1.1.2">𝒟</ci><ci id="S4.SS2.p5.1.m1.1.1.3a.cmml" xref="S4.SS2.p5.1.m1.1.1.3"><mtext id="S4.SS2.p5.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS2.p5.1.m1.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math> is often not available in the real world, so we strongly discourage approaches performing supervised threshold search using the labelled test data in <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.1"><semantics id="S4.SS2.p5.2.m2.1a"><msup id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.2.m2.1.1.2" xref="S4.SS2.p5.2.m2.1.1.2.cmml">𝒟</mi><mtext id="S4.SS2.p5.2.m2.1.1.3" xref="S4.SS2.p5.2.m2.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><apply id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.2.m2.1.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.p5.2.m2.1.1.2.cmml" xref="S4.SS2.p5.2.m2.1.1.2">𝒟</ci><ci id="S4.SS2.p5.2.m2.1.1.3a.cmml" xref="S4.SS2.p5.2.m2.1.1.3"><mtext id="S4.SS2.p5.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS2.p5.2.m2.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.1">Furthermore, there is no way to stop future research from performing hyperparameter tuning using <math alttext="\mathcal{D}^{\text{test}}" class="ltx_Math" display="inline" id="S4.SS2.p6.1.m1.1"><semantics id="S4.SS2.p6.1.m1.1a"><msup id="S4.SS2.p6.1.m1.1.1" xref="S4.SS2.p6.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p6.1.m1.1.1.2" xref="S4.SS2.p6.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S4.SS2.p6.1.m1.1.1.3" xref="S4.SS2.p6.1.m1.1.1.3a.cmml">test</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.1.m1.1b"><apply id="S4.SS2.p6.1.m1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p6.1.m1.1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.p6.1.m1.1.1.2.cmml" xref="S4.SS2.p6.1.m1.1.1.2">𝒟</ci><ci id="S4.SS2.p6.1.m1.1.1.3a.cmml" xref="S4.SS2.p6.1.m1.1.1.3"><mtext id="S4.SS2.p6.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS2.p6.1.m1.1.1.3">test</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.1.m1.1c">\mathcal{D}^{\text{test}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT test end_POSTSUPERSCRIPT</annotation></semantics></math>, hence any results obtained for this dataset should be considered as the theoretical maximum anomaly detection performance achievable by the model, not as a realistic anomaly detection performance observable in the real world.</p> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.1">We run all simulations that generate the PATH dataset on a workstation equipped with an Intel Xeon Gold 6234 CPU running Windows 10 Enterprise LTSC version 21H2 with MATLAB version 23.2. The framework used for model training is TensorFlow 2.15.1 and TensorFlow Probability 0.23 on Python 3.10 on a workstation running Ubuntu 22.04.5 LTS, equipped with two Nvidia RTX A6000 GPUs. All work involving TSADIS is done in a separate environment with the latest compatible Python version of 3.9. Further information on library versions used can be found in the <em class="ltx_emph ltx_font_italic" id="S4.SS2.p7.1.1">requirements.txt</em> file in the repository.</p> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Results and Discussion</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">To provide baseline results for the version of the PATH dataset for <em class="ltx_emph ltx_font_italic" id="S4.SS3.p1.1.1">unsupervised</em> anomaly detection, we test several approaches. The corresponding results are shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.T5" title="Table 5 ‣ 4.3 Results and Discussion ‣ 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <figure class="ltx_table" id="S4.T5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 5: </span><math alttext="F_{1}" class="ltx_Math" display="inline" id="S4.T5.5.m1.1"><semantics id="S4.T5.5.m1.1b"><msub id="S4.T5.5.m1.1.1" xref="S4.T5.5.m1.1.1.cmml"><mi id="S4.T5.5.m1.1.1.2" xref="S4.T5.5.m1.1.1.2.cmml">F</mi><mn id="S4.T5.5.m1.1.1.3" xref="S4.T5.5.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T5.5.m1.1c"><apply id="S4.T5.5.m1.1.1.cmml" xref="S4.T5.5.m1.1.1"><csymbol cd="ambiguous" id="S4.T5.5.m1.1.1.1.cmml" xref="S4.T5.5.m1.1.1">subscript</csymbol><ci id="S4.T5.5.m1.1.1.2.cmml" xref="S4.T5.5.m1.1.1.2">𝐹</ci><cn id="S4.T5.5.m1.1.1.3.cmml" type="integer" xref="S4.T5.5.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.5.m1.1d">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.5.m1.1e">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> score, precision <math alttext="P" class="ltx_Math" display="inline" id="S4.T5.6.m2.1"><semantics id="S4.T5.6.m2.1b"><mi id="S4.T5.6.m2.1.1" xref="S4.T5.6.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.T5.6.m2.1c"><ci id="S4.T5.6.m2.1.1.cmml" xref="S4.T5.6.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.6.m2.1d">P</annotation><annotation encoding="application/x-llamapun" id="S4.T5.6.m2.1e">italic_P</annotation></semantics></math>, recall <math alttext="R" class="ltx_Math" display="inline" id="S4.T5.7.m3.1"><semantics id="S4.T5.7.m3.1b"><mi id="S4.T5.7.m3.1.1" xref="S4.T5.7.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.T5.7.m3.1c"><ci id="S4.T5.7.m3.1.1.cmml" xref="S4.T5.7.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.7.m3.1d">R</annotation><annotation encoding="application/x-llamapun" id="S4.T5.7.m3.1e">italic_R</annotation></semantics></math>, and average detection delay <math alttext="\bar{\delta}" class="ltx_Math" display="inline" id="S4.T5.8.m4.1"><semantics id="S4.T5.8.m4.1b"><mover accent="true" id="S4.T5.8.m4.1.1" xref="S4.T5.8.m4.1.1.cmml"><mi id="S4.T5.8.m4.1.1.2" xref="S4.T5.8.m4.1.1.2.cmml">δ</mi><mo id="S4.T5.8.m4.1.1.1" xref="S4.T5.8.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.T5.8.m4.1c"><apply id="S4.T5.8.m4.1.1.cmml" xref="S4.T5.8.m4.1.1"><ci id="S4.T5.8.m4.1.1.1.cmml" xref="S4.T5.8.m4.1.1.1">¯</ci><ci id="S4.T5.8.m4.1.1.2.cmml" xref="S4.T5.8.m4.1.1.2">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.8.m4.1d">\bar{\delta}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.8.m4.1e">over¯ start_ARG italic_δ end_ARG</annotation></semantics></math> using the <em class="ltx_emph ltx_font_italic" id="S4.T5.64.1">unsupervised</em> threshold (top half) and <em class="ltx_emph ltx_font_italic" id="S4.T5.65.2">theoretical best</em> threshold (bottom half) for a range of models applied to the unsupervised anomaly detection version of the PATH dataset. The best values for each metric are given in <span class="ltx_text ltx_font_bold" id="S4.T5.66.3">bold</span>.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T5.60"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T5.12.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S4.T5.12.4.5" style="padding-left:3.0pt;padding-right:3.0pt;">Model</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T5.9.1.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="F_{1}" class="ltx_Math" display="inline" id="S4.T5.9.1.1.m1.1"><semantics id="S4.T5.9.1.1.m1.1a"><msub id="S4.T5.9.1.1.m1.1.1" xref="S4.T5.9.1.1.m1.1.1.cmml"><mi id="S4.T5.9.1.1.m1.1.1.2" xref="S4.T5.9.1.1.m1.1.1.2.cmml">F</mi><mn id="S4.T5.9.1.1.m1.1.1.3" xref="S4.T5.9.1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T5.9.1.1.m1.1b"><apply id="S4.T5.9.1.1.m1.1.1.cmml" xref="S4.T5.9.1.1.m1.1.1"><csymbol 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xref="S4.T5.12.4.4.m1.1.2.1"></times><apply id="S4.T5.12.4.4.m1.1.2.2.cmml" xref="S4.T5.12.4.4.m1.1.2.2"><ci id="S4.T5.12.4.4.m1.1.2.2.1.cmml" xref="S4.T5.12.4.4.m1.1.2.2.1">¯</ci><ci id="S4.T5.12.4.4.m1.1.2.2.2.cmml" xref="S4.T5.12.4.4.m1.1.2.2.2">𝛿</ci></apply><apply id="S4.T5.12.4.4.m1.1.2.3.1.cmml" xref="S4.T5.12.4.4.m1.1.2.3.2"><csymbol cd="latexml" id="S4.T5.12.4.4.m1.1.2.3.1.1.cmml" xref="S4.T5.12.4.4.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.T5.12.4.4.m1.1.1.cmml" xref="S4.T5.12.4.4.m1.1.1">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.12.4.4.m1.1c">\bar{\delta}\ [s]</annotation><annotation encoding="application/x-llamapun" id="S4.T5.12.4.4.m1.1d">over¯ start_ARG italic_δ end_ARG [ italic_s ]</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T5.16.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S4.T5.16.8.5" style="padding-left:3.0pt;padding-right:3.0pt;">OmniA</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T5.13.5.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.02\pm 0.02" class="ltx_Math" display="inline" id="S4.T5.13.5.1.m1.1"><semantics id="S4.T5.13.5.1.m1.1a"><mrow id="S4.T5.13.5.1.m1.1.1" xref="S4.T5.13.5.1.m1.1.1.cmml"><mn id="S4.T5.13.5.1.m1.1.1.2" xref="S4.T5.13.5.1.m1.1.1.2.cmml">0.02</mn><mo id="S4.T5.13.5.1.m1.1.1.1" xref="S4.T5.13.5.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.13.5.1.m1.1.1.3" xref="S4.T5.13.5.1.m1.1.1.3.cmml">0.02</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.13.5.1.m1.1b"><apply id="S4.T5.13.5.1.m1.1.1.cmml" xref="S4.T5.13.5.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.13.5.1.m1.1.1.1.cmml" xref="S4.T5.13.5.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.13.5.1.m1.1.1.2.cmml" type="float" xref="S4.T5.13.5.1.m1.1.1.2">0.02</cn><cn id="S4.T5.13.5.1.m1.1.1.3.cmml" type="float" xref="S4.T5.13.5.1.m1.1.1.3">0.02</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.13.5.1.m1.1c">0.02\pm 0.02</annotation><annotation encoding="application/x-llamapun" id="S4.T5.13.5.1.m1.1d">0.02 ± 0.02</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T5.14.6.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.21\pm 0.29" class="ltx_Math" display="inline" id="S4.T5.14.6.2.m1.1"><semantics id="S4.T5.14.6.2.m1.1a"><mrow id="S4.T5.14.6.2.m1.1.1" xref="S4.T5.14.6.2.m1.1.1.cmml"><mn id="S4.T5.14.6.2.m1.1.1.2" xref="S4.T5.14.6.2.m1.1.1.2.cmml">0.21</mn><mo id="S4.T5.14.6.2.m1.1.1.1" xref="S4.T5.14.6.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.14.6.2.m1.1.1.3" xref="S4.T5.14.6.2.m1.1.1.3.cmml">0.29</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.14.6.2.m1.1b"><apply id="S4.T5.14.6.2.m1.1.1.cmml" xref="S4.T5.14.6.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.14.6.2.m1.1.1.1.cmml" xref="S4.T5.14.6.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.14.6.2.m1.1.1.2.cmml" type="float" xref="S4.T5.14.6.2.m1.1.1.2">0.21</cn><cn id="S4.T5.14.6.2.m1.1.1.3.cmml" type="float" xref="S4.T5.14.6.2.m1.1.1.3">0.29</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.14.6.2.m1.1c">0.21\pm 0.29</annotation><annotation encoding="application/x-llamapun" id="S4.T5.14.6.2.m1.1d">0.21 ± 0.29</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T5.15.7.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.12" class="ltx_Math" display="inline" id="S4.T5.15.7.3.m1.1"><semantics id="S4.T5.15.7.3.m1.1a"><mrow id="S4.T5.15.7.3.m1.1.1" xref="S4.T5.15.7.3.m1.1.1.cmml"><mn id="S4.T5.15.7.3.m1.1.1.2" xref="S4.T5.15.7.3.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.15.7.3.m1.1.1.1" xref="S4.T5.15.7.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.15.7.3.m1.1.1.3" xref="S4.T5.15.7.3.m1.1.1.3.cmml">0.12</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.15.7.3.m1.1b"><apply id="S4.T5.15.7.3.m1.1.1.cmml" xref="S4.T5.15.7.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.15.7.3.m1.1.1.1.cmml" xref="S4.T5.15.7.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.15.7.3.m1.1.1.2.cmml" type="float" xref="S4.T5.15.7.3.m1.1.1.2">0.01</cn><cn id="S4.T5.15.7.3.m1.1.1.3.cmml" type="float" xref="S4.T5.15.7.3.m1.1.1.3">0.12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.15.7.3.m1.1c">0.01\pm 0.12</annotation><annotation encoding="application/x-llamapun" id="S4.T5.15.7.3.m1.1d">0.01 ± 0.12</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T5.16.8.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}992.7\pm 69.9\phantom{x}" class="ltx_Math" display="inline" id="S4.T5.16.8.4.m1.1"><semantics id="S4.T5.16.8.4.m1.1a"><mrow id="S4.T5.16.8.4.m1.1.1" xref="S4.T5.16.8.4.m1.1.1.cmml"><mn id="S4.T5.16.8.4.m1.1.1.2" xref="S4.T5.16.8.4.m1.1.1.2.cmml">992.7</mn><mo id="S4.T5.16.8.4.m1.1.1.1" xref="S4.T5.16.8.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.16.8.4.m1.1.1.3" xref="S4.T5.16.8.4.m1.1.1.3.cmml">69.9</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.16.8.4.m1.1b"><apply id="S4.T5.16.8.4.m1.1.1.cmml" xref="S4.T5.16.8.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.16.8.4.m1.1.1.1.cmml" xref="S4.T5.16.8.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.16.8.4.m1.1.1.2.cmml" type="float" xref="S4.T5.16.8.4.m1.1.1.2">992.7</cn><cn id="S4.T5.16.8.4.m1.1.1.3.cmml" type="float" xref="S4.T5.16.8.4.m1.1.1.3">69.9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.16.8.4.m1.1c">\phantom{0}992.7\pm 69.9\phantom{x}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.16.8.4.m1.1d">992.7 ± 69.9</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.20.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.20.12.5" style="padding-left:3.0pt;padding-right:3.0pt;">TCN-AE</th> <td class="ltx_td ltx_align_center" id="S4.T5.17.9.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.02\pm 0.02" class="ltx_Math" display="inline" id="S4.T5.17.9.1.m1.1"><semantics id="S4.T5.17.9.1.m1.1a"><mrow id="S4.T5.17.9.1.m1.1.1" xref="S4.T5.17.9.1.m1.1.1.cmml"><mn id="S4.T5.17.9.1.m1.1.1.2" xref="S4.T5.17.9.1.m1.1.1.2.cmml">0.02</mn><mo id="S4.T5.17.9.1.m1.1.1.1" xref="S4.T5.17.9.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.17.9.1.m1.1.1.3" xref="S4.T5.17.9.1.m1.1.1.3.cmml">0.02</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.17.9.1.m1.1b"><apply id="S4.T5.17.9.1.m1.1.1.cmml" xref="S4.T5.17.9.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.17.9.1.m1.1.1.1.cmml" xref="S4.T5.17.9.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.17.9.1.m1.1.1.2.cmml" type="float" xref="S4.T5.17.9.1.m1.1.1.2">0.02</cn><cn id="S4.T5.17.9.1.m1.1.1.3.cmml" type="float" xref="S4.T5.17.9.1.m1.1.1.3">0.02</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.17.9.1.m1.1c">0.02\pm 0.02</annotation><annotation encoding="application/x-llamapun" id="S4.T5.17.9.1.m1.1d">0.02 ± 0.02</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.18.10.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.40\pm 0.41" class="ltx_Math" display="inline" id="S4.T5.18.10.2.m1.1"><semantics id="S4.T5.18.10.2.m1.1a"><mrow id="S4.T5.18.10.2.m1.1.1" xref="S4.T5.18.10.2.m1.1.1.cmml"><mn id="S4.T5.18.10.2.m1.1.1.2" xref="S4.T5.18.10.2.m1.1.1.2.cmml">0.40</mn><mo id="S4.T5.18.10.2.m1.1.1.1" xref="S4.T5.18.10.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.18.10.2.m1.1.1.3" xref="S4.T5.18.10.2.m1.1.1.3.cmml">0.41</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.18.10.2.m1.1b"><apply id="S4.T5.18.10.2.m1.1.1.cmml" xref="S4.T5.18.10.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.18.10.2.m1.1.1.1.cmml" xref="S4.T5.18.10.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.18.10.2.m1.1.1.2.cmml" type="float" xref="S4.T5.18.10.2.m1.1.1.2">0.40</cn><cn id="S4.T5.18.10.2.m1.1.1.3.cmml" type="float" xref="S4.T5.18.10.2.m1.1.1.3">0.41</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.18.10.2.m1.1c">0.40\pm 0.41</annotation><annotation encoding="application/x-llamapun" id="S4.T5.18.10.2.m1.1d">0.40 ± 0.41</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.19.11.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.19.11.3.m1.1"><semantics id="S4.T5.19.11.3.m1.1a"><mrow id="S4.T5.19.11.3.m1.1.1" xref="S4.T5.19.11.3.m1.1.1.cmml"><mn id="S4.T5.19.11.3.m1.1.1.2" xref="S4.T5.19.11.3.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.19.11.3.m1.1.1.1" xref="S4.T5.19.11.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.19.11.3.m1.1.1.3" xref="S4.T5.19.11.3.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.19.11.3.m1.1b"><apply id="S4.T5.19.11.3.m1.1.1.cmml" xref="S4.T5.19.11.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.19.11.3.m1.1.1.1.cmml" xref="S4.T5.19.11.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.19.11.3.m1.1.1.2.cmml" type="float" xref="S4.T5.19.11.3.m1.1.1.2">0.01</cn><cn id="S4.T5.19.11.3.m1.1.1.3.cmml" type="float" xref="S4.T5.19.11.3.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.19.11.3.m1.1c">0.01\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.19.11.3.m1.1d">0.01 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.20.12.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}990.5\pm 68.7\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.20.12.4.m1.1"><semantics id="S4.T5.20.12.4.m1.1a"><mrow id="S4.T5.20.12.4.m1.1.1" xref="S4.T5.20.12.4.m1.1.1.cmml"><mn id="S4.T5.20.12.4.m1.1.1.2" xref="S4.T5.20.12.4.m1.1.1.2.cmml">990.5</mn><mo id="S4.T5.20.12.4.m1.1.1.1" xref="S4.T5.20.12.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.20.12.4.m1.1.1.3" xref="S4.T5.20.12.4.m1.1.1.3.cmml">68.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.20.12.4.m1.1b"><apply id="S4.T5.20.12.4.m1.1.1.cmml" xref="S4.T5.20.12.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.20.12.4.m1.1.1.1.cmml" xref="S4.T5.20.12.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.20.12.4.m1.1.1.2.cmml" type="float" xref="S4.T5.20.12.4.m1.1.1.2">990.5</cn><cn id="S4.T5.20.12.4.m1.1.1.3.cmml" type="float" xref="S4.T5.20.12.4.m1.1.1.3">68.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.20.12.4.m1.1c">\phantom{0}990.5\pm 68.7\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.20.12.4.m1.1d">990.5 ± 68.7</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.24.16"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.24.16.5" style="padding-left:3.0pt;padding-right:3.0pt;">SISVAE</th> <td class="ltx_td ltx_align_center" id="S4.T5.21.13.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.21.13.1.m1.1"><semantics id="S4.T5.21.13.1.m1.1a"><mrow id="S4.T5.21.13.1.m1.1.1" xref="S4.T5.21.13.1.m1.1.1.cmml"><mn id="S4.T5.21.13.1.m1.1.1.2" xref="S4.T5.21.13.1.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.21.13.1.m1.1.1.1" xref="S4.T5.21.13.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.21.13.1.m1.1.1.3" xref="S4.T5.21.13.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.21.13.1.m1.1b"><apply id="S4.T5.21.13.1.m1.1.1.cmml" xref="S4.T5.21.13.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.21.13.1.m1.1.1.1.cmml" xref="S4.T5.21.13.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.21.13.1.m1.1.1.2.cmml" type="float" xref="S4.T5.21.13.1.m1.1.1.2">0.01</cn><cn id="S4.T5.21.13.1.m1.1.1.3.cmml" type="float" xref="S4.T5.21.13.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.21.13.1.m1.1c">0.01\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.21.13.1.m1.1d">0.01 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.22.14.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.19\pm 0.19" class="ltx_Math" display="inline" id="S4.T5.22.14.2.m1.1"><semantics id="S4.T5.22.14.2.m1.1a"><mrow id="S4.T5.22.14.2.m1.1.1" xref="S4.T5.22.14.2.m1.1.1.cmml"><mn id="S4.T5.22.14.2.m1.1.1.2" xref="S4.T5.22.14.2.m1.1.1.2.cmml">0.19</mn><mo id="S4.T5.22.14.2.m1.1.1.1" xref="S4.T5.22.14.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.22.14.2.m1.1.1.3" xref="S4.T5.22.14.2.m1.1.1.3.cmml">0.19</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.22.14.2.m1.1b"><apply id="S4.T5.22.14.2.m1.1.1.cmml" xref="S4.T5.22.14.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.22.14.2.m1.1.1.1.cmml" xref="S4.T5.22.14.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.22.14.2.m1.1.1.2.cmml" type="float" xref="S4.T5.22.14.2.m1.1.1.2">0.19</cn><cn id="S4.T5.22.14.2.m1.1.1.3.cmml" type="float" xref="S4.T5.22.14.2.m1.1.1.3">0.19</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.22.14.2.m1.1c">0.19\pm 0.19</annotation><annotation encoding="application/x-llamapun" id="S4.T5.22.14.2.m1.1d">0.19 ± 0.19</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.23.15.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.23.15.3.m1.1"><semantics id="S4.T5.23.15.3.m1.1a"><mrow id="S4.T5.23.15.3.m1.1.1" xref="S4.T5.23.15.3.m1.1.1.cmml"><mn id="S4.T5.23.15.3.m1.1.1.2" xref="S4.T5.23.15.3.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.23.15.3.m1.1.1.1" xref="S4.T5.23.15.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.23.15.3.m1.1.1.3" xref="S4.T5.23.15.3.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.23.15.3.m1.1b"><apply id="S4.T5.23.15.3.m1.1.1.cmml" xref="S4.T5.23.15.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.23.15.3.m1.1.1.1.cmml" xref="S4.T5.23.15.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.23.15.3.m1.1.1.2.cmml" type="float" xref="S4.T5.23.15.3.m1.1.1.2">0.01</cn><cn id="S4.T5.23.15.3.m1.1.1.3.cmml" type="float" xref="S4.T5.23.15.3.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.23.15.3.m1.1c">0.01\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.23.15.3.m1.1d">0.01 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.24.16.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{\phantom{0}990.4}\pm 67.7\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.24.16.4.m1.1"><semantics id="S4.T5.24.16.4.m1.1a"><mrow id="S4.T5.24.16.4.m1.1.1" xref="S4.T5.24.16.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.24.16.4.m1.1.1.2" mathvariant="bold" xref="S4.T5.24.16.4.m1.1.1.2.cmml">990.4</mn><mo id="S4.T5.24.16.4.m1.1.1.1" xref="S4.T5.24.16.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.24.16.4.m1.1.1.3" xref="S4.T5.24.16.4.m1.1.1.3.cmml">67.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.24.16.4.m1.1b"><apply id="S4.T5.24.16.4.m1.1.1.cmml" xref="S4.T5.24.16.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.24.16.4.m1.1.1.1.cmml" xref="S4.T5.24.16.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.24.16.4.m1.1.1.2.cmml" type="float" xref="S4.T5.24.16.4.m1.1.1.2">990.4</cn><cn id="S4.T5.24.16.4.m1.1.1.3.cmml" type="float" xref="S4.T5.24.16.4.m1.1.1.3">67.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.24.16.4.m1.1c">\mathbf{\phantom{0}990.4}\pm 67.7\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.24.16.4.m1.1d">bold_990.4 ± 67.7</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.28.20"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.28.20.5" style="padding-left:3.0pt;padding-right:3.0pt;">LW-VAE</th> <td class="ltx_td ltx_align_center" id="S4.T5.25.17.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.02" class="ltx_Math" display="inline" id="S4.T5.25.17.1.m1.1"><semantics id="S4.T5.25.17.1.m1.1a"><mrow id="S4.T5.25.17.1.m1.1.1" xref="S4.T5.25.17.1.m1.1.1.cmml"><mn id="S4.T5.25.17.1.m1.1.1.2" xref="S4.T5.25.17.1.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.25.17.1.m1.1.1.1" xref="S4.T5.25.17.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.25.17.1.m1.1.1.3" xref="S4.T5.25.17.1.m1.1.1.3.cmml">0.02</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.25.17.1.m1.1b"><apply id="S4.T5.25.17.1.m1.1.1.cmml" xref="S4.T5.25.17.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.25.17.1.m1.1.1.1.cmml" xref="S4.T5.25.17.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.25.17.1.m1.1.1.2.cmml" type="float" xref="S4.T5.25.17.1.m1.1.1.2">0.01</cn><cn id="S4.T5.25.17.1.m1.1.1.3.cmml" type="float" xref="S4.T5.25.17.1.m1.1.1.3">0.02</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.25.17.1.m1.1c">0.01\pm 0.02</annotation><annotation encoding="application/x-llamapun" id="S4.T5.25.17.1.m1.1d">0.01 ± 0.02</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.26.18.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.27\pm 0.41" class="ltx_Math" display="inline" id="S4.T5.26.18.2.m1.1"><semantics id="S4.T5.26.18.2.m1.1a"><mrow id="S4.T5.26.18.2.m1.1.1" xref="S4.T5.26.18.2.m1.1.1.cmml"><mn id="S4.T5.26.18.2.m1.1.1.2" xref="S4.T5.26.18.2.m1.1.1.2.cmml">0.27</mn><mo id="S4.T5.26.18.2.m1.1.1.1" xref="S4.T5.26.18.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.26.18.2.m1.1.1.3" xref="S4.T5.26.18.2.m1.1.1.3.cmml">0.41</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.26.18.2.m1.1b"><apply id="S4.T5.26.18.2.m1.1.1.cmml" xref="S4.T5.26.18.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.26.18.2.m1.1.1.1.cmml" xref="S4.T5.26.18.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.26.18.2.m1.1.1.2.cmml" type="float" xref="S4.T5.26.18.2.m1.1.1.2">0.27</cn><cn id="S4.T5.26.18.2.m1.1.1.3.cmml" type="float" xref="S4.T5.26.18.2.m1.1.1.3">0.41</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.26.18.2.m1.1c">0.27\pm 0.41</annotation><annotation encoding="application/x-llamapun" id="S4.T5.26.18.2.m1.1d">0.27 ± 0.41</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.27.19.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.01\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.27.19.3.m1.1"><semantics id="S4.T5.27.19.3.m1.1a"><mrow id="S4.T5.27.19.3.m1.1.1" xref="S4.T5.27.19.3.m1.1.1.cmml"><mn id="S4.T5.27.19.3.m1.1.1.2" xref="S4.T5.27.19.3.m1.1.1.2.cmml">0.01</mn><mo id="S4.T5.27.19.3.m1.1.1.1" xref="S4.T5.27.19.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.27.19.3.m1.1.1.3" xref="S4.T5.27.19.3.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.27.19.3.m1.1b"><apply id="S4.T5.27.19.3.m1.1.1.cmml" xref="S4.T5.27.19.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.27.19.3.m1.1.1.1.cmml" xref="S4.T5.27.19.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.27.19.3.m1.1.1.2.cmml" type="float" xref="S4.T5.27.19.3.m1.1.1.2">0.01</cn><cn id="S4.T5.27.19.3.m1.1.1.3.cmml" type="float" xref="S4.T5.27.19.3.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.27.19.3.m1.1c">0.01\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.27.19.3.m1.1d">0.01 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.28.20.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}992.4\pm 70.6\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.28.20.4.m1.1"><semantics id="S4.T5.28.20.4.m1.1a"><mrow id="S4.T5.28.20.4.m1.1.1" xref="S4.T5.28.20.4.m1.1.1.cmml"><mn id="S4.T5.28.20.4.m1.1.1.2" xref="S4.T5.28.20.4.m1.1.1.2.cmml">992.4</mn><mo id="S4.T5.28.20.4.m1.1.1.1" xref="S4.T5.28.20.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.28.20.4.m1.1.1.3" xref="S4.T5.28.20.4.m1.1.1.3.cmml">70.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.28.20.4.m1.1b"><apply id="S4.T5.28.20.4.m1.1.1.cmml" xref="S4.T5.28.20.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.28.20.4.m1.1.1.1.cmml" xref="S4.T5.28.20.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.28.20.4.m1.1.1.2.cmml" type="float" xref="S4.T5.28.20.4.m1.1.1.2">992.4</cn><cn id="S4.T5.28.20.4.m1.1.1.3.cmml" type="float" xref="S4.T5.28.20.4.m1.1.1.3">70.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.28.20.4.m1.1c">\phantom{0}992.4\pm 70.6\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.28.20.4.m1.1d">992.4 ± 70.6</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.32.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.32.24.5" style="padding-left:3.0pt;padding-right:3.0pt;">TSADIS</th> <td class="ltx_td ltx_align_center" id="S4.T5.29.21.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.00\pm 0.00" class="ltx_Math" display="inline" id="S4.T5.29.21.1.m1.1"><semantics id="S4.T5.29.21.1.m1.1a"><mrow id="S4.T5.29.21.1.m1.1.1" xref="S4.T5.29.21.1.m1.1.1.cmml"><mn id="S4.T5.29.21.1.m1.1.1.2" xref="S4.T5.29.21.1.m1.1.1.2.cmml">0.00</mn><mo id="S4.T5.29.21.1.m1.1.1.1" xref="S4.T5.29.21.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.29.21.1.m1.1.1.3" xref="S4.T5.29.21.1.m1.1.1.3.cmml">0.00</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.29.21.1.m1.1b"><apply id="S4.T5.29.21.1.m1.1.1.cmml" xref="S4.T5.29.21.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.29.21.1.m1.1.1.1.cmml" xref="S4.T5.29.21.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.29.21.1.m1.1.1.2.cmml" type="float" xref="S4.T5.29.21.1.m1.1.1.2">0.00</cn><cn id="S4.T5.29.21.1.m1.1.1.3.cmml" type="float" xref="S4.T5.29.21.1.m1.1.1.3">0.00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.29.21.1.m1.1c">0.00\pm 0.00</annotation><annotation encoding="application/x-llamapun" id="S4.T5.29.21.1.m1.1d">0.00 ± 0.00</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.30.22.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.00\pm 0.00" class="ltx_Math" display="inline" id="S4.T5.30.22.2.m1.1"><semantics id="S4.T5.30.22.2.m1.1a"><mrow id="S4.T5.30.22.2.m1.1.1" xref="S4.T5.30.22.2.m1.1.1.cmml"><mn id="S4.T5.30.22.2.m1.1.1.2" xref="S4.T5.30.22.2.m1.1.1.2.cmml">0.00</mn><mo id="S4.T5.30.22.2.m1.1.1.1" xref="S4.T5.30.22.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.30.22.2.m1.1.1.3" xref="S4.T5.30.22.2.m1.1.1.3.cmml">0.00</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.30.22.2.m1.1b"><apply id="S4.T5.30.22.2.m1.1.1.cmml" xref="S4.T5.30.22.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.30.22.2.m1.1.1.1.cmml" xref="S4.T5.30.22.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.30.22.2.m1.1.1.2.cmml" type="float" xref="S4.T5.30.22.2.m1.1.1.2">0.00</cn><cn id="S4.T5.30.22.2.m1.1.1.3.cmml" type="float" xref="S4.T5.30.22.2.m1.1.1.3">0.00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.30.22.2.m1.1c">0.00\pm 0.00</annotation><annotation encoding="application/x-llamapun" id="S4.T5.30.22.2.m1.1d">0.00 ± 0.00</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.31.23.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.00\pm 0.00" class="ltx_Math" display="inline" id="S4.T5.31.23.3.m1.1"><semantics id="S4.T5.31.23.3.m1.1a"><mrow id="S4.T5.31.23.3.m1.1.1" xref="S4.T5.31.23.3.m1.1.1.cmml"><mn id="S4.T5.31.23.3.m1.1.1.2" xref="S4.T5.31.23.3.m1.1.1.2.cmml">0.00</mn><mo id="S4.T5.31.23.3.m1.1.1.1" xref="S4.T5.31.23.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.31.23.3.m1.1.1.3" xref="S4.T5.31.23.3.m1.1.1.3.cmml">0.00</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.31.23.3.m1.1b"><apply id="S4.T5.31.23.3.m1.1.1.cmml" xref="S4.T5.31.23.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.31.23.3.m1.1.1.1.cmml" xref="S4.T5.31.23.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.31.23.3.m1.1.1.2.cmml" type="float" xref="S4.T5.31.23.3.m1.1.1.2">0.00</cn><cn id="S4.T5.31.23.3.m1.1.1.3.cmml" type="float" xref="S4.T5.31.23.3.m1.1.1.3">0.00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.31.23.3.m1.1c">0.00\pm 0.00</annotation><annotation encoding="application/x-llamapun" id="S4.T5.31.23.3.m1.1d">0.00 ± 0.00</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.32.24.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}995.8\pm 69.1\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.32.24.4.m1.1"><semantics id="S4.T5.32.24.4.m1.1a"><mrow id="S4.T5.32.24.4.m1.1.1" xref="S4.T5.32.24.4.m1.1.1.cmml"><mn id="S4.T5.32.24.4.m1.1.1.2" xref="S4.T5.32.24.4.m1.1.1.2.cmml">995.8</mn><mo id="S4.T5.32.24.4.m1.1.1.1" xref="S4.T5.32.24.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.32.24.4.m1.1.1.3" xref="S4.T5.32.24.4.m1.1.1.3.cmml">69.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.32.24.4.m1.1b"><apply id="S4.T5.32.24.4.m1.1.1.cmml" xref="S4.T5.32.24.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.32.24.4.m1.1.1.1.cmml" xref="S4.T5.32.24.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.32.24.4.m1.1.1.2.cmml" type="float" xref="S4.T5.32.24.4.m1.1.1.2">995.8</cn><cn id="S4.T5.32.24.4.m1.1.1.3.cmml" type="float" xref="S4.T5.32.24.4.m1.1.1.3">69.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.32.24.4.m1.1c">\phantom{0}995.8\pm 69.1\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.32.24.4.m1.1d">995.8 ± 69.1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.36.28"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.36.28.5" style="padding-left:3.0pt;padding-right:3.0pt;">TeVAE</th> <td class="ltx_td ltx_align_center" id="S4.T5.33.25.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.03}\pm 0.03" class="ltx_Math" display="inline" id="S4.T5.33.25.1.m1.1"><semantics id="S4.T5.33.25.1.m1.1a"><mrow id="S4.T5.33.25.1.m1.1.1" xref="S4.T5.33.25.1.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.33.25.1.m1.1.1.2" mathvariant="bold" xref="S4.T5.33.25.1.m1.1.1.2.cmml">0.03</mn><mo id="S4.T5.33.25.1.m1.1.1.1" xref="S4.T5.33.25.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.33.25.1.m1.1.1.3" xref="S4.T5.33.25.1.m1.1.1.3.cmml">0.03</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.33.25.1.m1.1b"><apply id="S4.T5.33.25.1.m1.1.1.cmml" xref="S4.T5.33.25.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.33.25.1.m1.1.1.1.cmml" xref="S4.T5.33.25.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.33.25.1.m1.1.1.2.cmml" type="float" xref="S4.T5.33.25.1.m1.1.1.2">0.03</cn><cn id="S4.T5.33.25.1.m1.1.1.3.cmml" type="float" xref="S4.T5.33.25.1.m1.1.1.3">0.03</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.33.25.1.m1.1c">\mathbf{0.03}\pm 0.03</annotation><annotation encoding="application/x-llamapun" id="S4.T5.33.25.1.m1.1d">bold_0.03 ± 0.03</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.34.26.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.48}\pm 0.43" class="ltx_Math" display="inline" id="S4.T5.34.26.2.m1.1"><semantics id="S4.T5.34.26.2.m1.1a"><mrow id="S4.T5.34.26.2.m1.1.1" xref="S4.T5.34.26.2.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.34.26.2.m1.1.1.2" mathvariant="bold" xref="S4.T5.34.26.2.m1.1.1.2.cmml">0.48</mn><mo id="S4.T5.34.26.2.m1.1.1.1" xref="S4.T5.34.26.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.34.26.2.m1.1.1.3" xref="S4.T5.34.26.2.m1.1.1.3.cmml">0.43</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.34.26.2.m1.1b"><apply id="S4.T5.34.26.2.m1.1.1.cmml" xref="S4.T5.34.26.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.34.26.2.m1.1.1.1.cmml" xref="S4.T5.34.26.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.34.26.2.m1.1.1.2.cmml" type="float" xref="S4.T5.34.26.2.m1.1.1.2">0.48</cn><cn id="S4.T5.34.26.2.m1.1.1.3.cmml" type="float" xref="S4.T5.34.26.2.m1.1.1.3">0.43</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.34.26.2.m1.1c">\mathbf{0.48}\pm 0.43</annotation><annotation encoding="application/x-llamapun" id="S4.T5.34.26.2.m1.1d">bold_0.48 ± 0.43</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.35.27.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.02}\pm 0.02" class="ltx_Math" display="inline" id="S4.T5.35.27.3.m1.1"><semantics id="S4.T5.35.27.3.m1.1a"><mrow id="S4.T5.35.27.3.m1.1.1" xref="S4.T5.35.27.3.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.35.27.3.m1.1.1.2" mathvariant="bold" xref="S4.T5.35.27.3.m1.1.1.2.cmml">0.02</mn><mo id="S4.T5.35.27.3.m1.1.1.1" xref="S4.T5.35.27.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.35.27.3.m1.1.1.3" xref="S4.T5.35.27.3.m1.1.1.3.cmml">0.02</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.35.27.3.m1.1b"><apply id="S4.T5.35.27.3.m1.1.1.cmml" xref="S4.T5.35.27.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.35.27.3.m1.1.1.1.cmml" xref="S4.T5.35.27.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.35.27.3.m1.1.1.2.cmml" type="float" xref="S4.T5.35.27.3.m1.1.1.2">0.02</cn><cn id="S4.T5.35.27.3.m1.1.1.3.cmml" type="float" xref="S4.T5.35.27.3.m1.1.1.3">0.02</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.35.27.3.m1.1c">\mathbf{0.02}\pm 0.02</annotation><annotation encoding="application/x-llamapun" id="S4.T5.35.27.3.m1.1d">bold_0.02 ± 0.02</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.36.28.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}990.6\pm 70.5\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.36.28.4.m1.1"><semantics id="S4.T5.36.28.4.m1.1a"><mrow id="S4.T5.36.28.4.m1.1.1" xref="S4.T5.36.28.4.m1.1.1.cmml"><mn id="S4.T5.36.28.4.m1.1.1.2" xref="S4.T5.36.28.4.m1.1.1.2.cmml">990.6</mn><mo id="S4.T5.36.28.4.m1.1.1.1" xref="S4.T5.36.28.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.36.28.4.m1.1.1.3" xref="S4.T5.36.28.4.m1.1.1.3.cmml">70.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.36.28.4.m1.1b"><apply id="S4.T5.36.28.4.m1.1.1.cmml" xref="S4.T5.36.28.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.36.28.4.m1.1.1.1.cmml" xref="S4.T5.36.28.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.36.28.4.m1.1.1.2.cmml" type="float" xref="S4.T5.36.28.4.m1.1.1.2">990.6</cn><cn id="S4.T5.36.28.4.m1.1.1.3.cmml" type="float" xref="S4.T5.36.28.4.m1.1.1.3">70.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.36.28.4.m1.1c">\phantom{0}990.6\pm 70.5\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.36.28.4.m1.1d">990.6 ± 70.5</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.40.32"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S4.T5.40.32.5" style="padding-left:3.0pt;padding-right:3.0pt;">OmniA</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.37.29.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.24\pm 0.08" class="ltx_Math" display="inline" id="S4.T5.37.29.1.m1.1"><semantics id="S4.T5.37.29.1.m1.1a"><mrow id="S4.T5.37.29.1.m1.1.1" xref="S4.T5.37.29.1.m1.1.1.cmml"><mn id="S4.T5.37.29.1.m1.1.1.2" xref="S4.T5.37.29.1.m1.1.1.2.cmml">0.24</mn><mo id="S4.T5.37.29.1.m1.1.1.1" xref="S4.T5.37.29.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.37.29.1.m1.1.1.3" xref="S4.T5.37.29.1.m1.1.1.3.cmml">0.08</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.37.29.1.m1.1b"><apply id="S4.T5.37.29.1.m1.1.1.cmml" xref="S4.T5.37.29.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.37.29.1.m1.1.1.1.cmml" xref="S4.T5.37.29.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.37.29.1.m1.1.1.2.cmml" type="float" xref="S4.T5.37.29.1.m1.1.1.2">0.24</cn><cn id="S4.T5.37.29.1.m1.1.1.3.cmml" type="float" xref="S4.T5.37.29.1.m1.1.1.3">0.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.37.29.1.m1.1c">0.24\pm 0.08</annotation><annotation encoding="application/x-llamapun" id="S4.T5.37.29.1.m1.1d">0.24 ± 0.08</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.38.30.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.21\pm 0.13" class="ltx_Math" display="inline" id="S4.T5.38.30.2.m1.1"><semantics id="S4.T5.38.30.2.m1.1a"><mrow id="S4.T5.38.30.2.m1.1.1" xref="S4.T5.38.30.2.m1.1.1.cmml"><mn id="S4.T5.38.30.2.m1.1.1.2" xref="S4.T5.38.30.2.m1.1.1.2.cmml">0.21</mn><mo id="S4.T5.38.30.2.m1.1.1.1" xref="S4.T5.38.30.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.38.30.2.m1.1.1.3" xref="S4.T5.38.30.2.m1.1.1.3.cmml">0.13</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.38.30.2.m1.1b"><apply id="S4.T5.38.30.2.m1.1.1.cmml" xref="S4.T5.38.30.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.38.30.2.m1.1.1.1.cmml" xref="S4.T5.38.30.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.38.30.2.m1.1.1.2.cmml" type="float" xref="S4.T5.38.30.2.m1.1.1.2">0.21</cn><cn id="S4.T5.38.30.2.m1.1.1.3.cmml" type="float" xref="S4.T5.38.30.2.m1.1.1.3">0.13</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.38.30.2.m1.1c">0.21\pm 0.13</annotation><annotation encoding="application/x-llamapun" id="S4.T5.38.30.2.m1.1d">0.21 ± 0.13</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.39.31.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.34\pm 0.11" class="ltx_Math" display="inline" id="S4.T5.39.31.3.m1.1"><semantics id="S4.T5.39.31.3.m1.1a"><mrow id="S4.T5.39.31.3.m1.1.1" xref="S4.T5.39.31.3.m1.1.1.cmml"><mn id="S4.T5.39.31.3.m1.1.1.2" xref="S4.T5.39.31.3.m1.1.1.2.cmml">0.34</mn><mo id="S4.T5.39.31.3.m1.1.1.1" xref="S4.T5.39.31.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.39.31.3.m1.1.1.3" xref="S4.T5.39.31.3.m1.1.1.3.cmml">0.11</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.39.31.3.m1.1b"><apply id="S4.T5.39.31.3.m1.1.1.cmml" xref="S4.T5.39.31.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.39.31.3.m1.1.1.1.cmml" xref="S4.T5.39.31.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.39.31.3.m1.1.1.2.cmml" type="float" xref="S4.T5.39.31.3.m1.1.1.2">0.34</cn><cn id="S4.T5.39.31.3.m1.1.1.3.cmml" type="float" xref="S4.T5.39.31.3.m1.1.1.3">0.11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.39.31.3.m1.1c">0.34\pm 0.11</annotation><annotation encoding="application/x-llamapun" id="S4.T5.39.31.3.m1.1d">0.34 ± 0.11</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.40.32.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}808.7\pm 102.0" class="ltx_Math" display="inline" id="S4.T5.40.32.4.m1.1"><semantics id="S4.T5.40.32.4.m1.1a"><mrow id="S4.T5.40.32.4.m1.1.1" xref="S4.T5.40.32.4.m1.1.1.cmml"><mn id="S4.T5.40.32.4.m1.1.1.2" xref="S4.T5.40.32.4.m1.1.1.2.cmml">808.7</mn><mo id="S4.T5.40.32.4.m1.1.1.1" xref="S4.T5.40.32.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.40.32.4.m1.1.1.3" xref="S4.T5.40.32.4.m1.1.1.3.cmml">102.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.40.32.4.m1.1b"><apply id="S4.T5.40.32.4.m1.1.1.cmml" xref="S4.T5.40.32.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.40.32.4.m1.1.1.1.cmml" xref="S4.T5.40.32.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.40.32.4.m1.1.1.2.cmml" type="float" xref="S4.T5.40.32.4.m1.1.1.2">808.7</cn><cn id="S4.T5.40.32.4.m1.1.1.3.cmml" type="float" xref="S4.T5.40.32.4.m1.1.1.3">102.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.40.32.4.m1.1c">\phantom{0}808.7\pm 102.0</annotation><annotation encoding="application/x-llamapun" id="S4.T5.40.32.4.m1.1d">808.7 ± 102.0</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.44.36"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.44.36.5" style="padding-left:3.0pt;padding-right:3.0pt;">TCN-AE</th> <td class="ltx_td ltx_align_center" id="S4.T5.41.33.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.14\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.41.33.1.m1.1"><semantics id="S4.T5.41.33.1.m1.1a"><mrow id="S4.T5.41.33.1.m1.1.1" xref="S4.T5.41.33.1.m1.1.1.cmml"><mn id="S4.T5.41.33.1.m1.1.1.2" xref="S4.T5.41.33.1.m1.1.1.2.cmml">0.14</mn><mo id="S4.T5.41.33.1.m1.1.1.1" xref="S4.T5.41.33.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.41.33.1.m1.1.1.3" xref="S4.T5.41.33.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.41.33.1.m1.1b"><apply id="S4.T5.41.33.1.m1.1.1.cmml" xref="S4.T5.41.33.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.41.33.1.m1.1.1.1.cmml" xref="S4.T5.41.33.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.41.33.1.m1.1.1.2.cmml" type="float" xref="S4.T5.41.33.1.m1.1.1.2">0.14</cn><cn id="S4.T5.41.33.1.m1.1.1.3.cmml" type="float" xref="S4.T5.41.33.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.41.33.1.m1.1c">0.14\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.41.33.1.m1.1d">0.14 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.42.34.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.08\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.42.34.2.m1.1"><semantics id="S4.T5.42.34.2.m1.1a"><mrow id="S4.T5.42.34.2.m1.1.1" xref="S4.T5.42.34.2.m1.1.1.cmml"><mn id="S4.T5.42.34.2.m1.1.1.2" xref="S4.T5.42.34.2.m1.1.1.2.cmml">0.08</mn><mo id="S4.T5.42.34.2.m1.1.1.1" xref="S4.T5.42.34.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.42.34.2.m1.1.1.3" xref="S4.T5.42.34.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.42.34.2.m1.1b"><apply id="S4.T5.42.34.2.m1.1.1.cmml" xref="S4.T5.42.34.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.42.34.2.m1.1.1.1.cmml" xref="S4.T5.42.34.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.42.34.2.m1.1.1.2.cmml" type="float" xref="S4.T5.42.34.2.m1.1.1.2">0.08</cn><cn id="S4.T5.42.34.2.m1.1.1.3.cmml" type="float" xref="S4.T5.42.34.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.42.34.2.m1.1c">0.08\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.42.34.2.m1.1d">0.08 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.43.35.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.44\pm 0.14" class="ltx_Math" display="inline" id="S4.T5.43.35.3.m1.1"><semantics id="S4.T5.43.35.3.m1.1a"><mrow id="S4.T5.43.35.3.m1.1.1" xref="S4.T5.43.35.3.m1.1.1.cmml"><mn id="S4.T5.43.35.3.m1.1.1.2" xref="S4.T5.43.35.3.m1.1.1.2.cmml">0.44</mn><mo id="S4.T5.43.35.3.m1.1.1.1" xref="S4.T5.43.35.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.43.35.3.m1.1.1.3" xref="S4.T5.43.35.3.m1.1.1.3.cmml">0.14</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.43.35.3.m1.1b"><apply id="S4.T5.43.35.3.m1.1.1.cmml" xref="S4.T5.43.35.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.43.35.3.m1.1.1.1.cmml" xref="S4.T5.43.35.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.43.35.3.m1.1.1.2.cmml" type="float" xref="S4.T5.43.35.3.m1.1.1.2">0.44</cn><cn id="S4.T5.43.35.3.m1.1.1.3.cmml" type="float" xref="S4.T5.43.35.3.m1.1.1.3">0.14</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.43.35.3.m1.1c">0.44\pm 0.14</annotation><annotation encoding="application/x-llamapun" id="S4.T5.43.35.3.m1.1d">0.44 ± 0.14</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.44.36.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}726.1\pm 93.7\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.44.36.4.m1.1"><semantics id="S4.T5.44.36.4.m1.1a"><mrow id="S4.T5.44.36.4.m1.1.1" xref="S4.T5.44.36.4.m1.1.1.cmml"><mn id="S4.T5.44.36.4.m1.1.1.2" xref="S4.T5.44.36.4.m1.1.1.2.cmml">726.1</mn><mo id="S4.T5.44.36.4.m1.1.1.1" xref="S4.T5.44.36.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.44.36.4.m1.1.1.3" xref="S4.T5.44.36.4.m1.1.1.3.cmml">93.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.44.36.4.m1.1b"><apply id="S4.T5.44.36.4.m1.1.1.cmml" xref="S4.T5.44.36.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.44.36.4.m1.1.1.1.cmml" xref="S4.T5.44.36.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.44.36.4.m1.1.1.2.cmml" type="float" xref="S4.T5.44.36.4.m1.1.1.2">726.1</cn><cn id="S4.T5.44.36.4.m1.1.1.3.cmml" type="float" xref="S4.T5.44.36.4.m1.1.1.3">93.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.44.36.4.m1.1c">\phantom{0}726.1\pm 93.7\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.44.36.4.m1.1d">726.1 ± 93.7</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.48.40"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.48.40.5" style="padding-left:3.0pt;padding-right:3.0pt;">SISVAE</th> <td class="ltx_td ltx_align_center" id="S4.T5.45.37.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.11\pm 0.02" class="ltx_Math" display="inline" id="S4.T5.45.37.1.m1.1"><semantics id="S4.T5.45.37.1.m1.1a"><mrow id="S4.T5.45.37.1.m1.1.1" xref="S4.T5.45.37.1.m1.1.1.cmml"><mn id="S4.T5.45.37.1.m1.1.1.2" xref="S4.T5.45.37.1.m1.1.1.2.cmml">0.11</mn><mo id="S4.T5.45.37.1.m1.1.1.1" xref="S4.T5.45.37.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.45.37.1.m1.1.1.3" xref="S4.T5.45.37.1.m1.1.1.3.cmml">0.02</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.45.37.1.m1.1b"><apply id="S4.T5.45.37.1.m1.1.1.cmml" xref="S4.T5.45.37.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.45.37.1.m1.1.1.1.cmml" xref="S4.T5.45.37.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.45.37.1.m1.1.1.2.cmml" type="float" xref="S4.T5.45.37.1.m1.1.1.2">0.11</cn><cn id="S4.T5.45.37.1.m1.1.1.3.cmml" type="float" xref="S4.T5.45.37.1.m1.1.1.3">0.02</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.45.37.1.m1.1c">0.11\pm 0.02</annotation><annotation encoding="application/x-llamapun" id="S4.T5.45.37.1.m1.1d">0.11 ± 0.02</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.46.38.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.08\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.46.38.2.m1.1"><semantics id="S4.T5.46.38.2.m1.1a"><mrow id="S4.T5.46.38.2.m1.1.1" xref="S4.T5.46.38.2.m1.1.1.cmml"><mn id="S4.T5.46.38.2.m1.1.1.2" xref="S4.T5.46.38.2.m1.1.1.2.cmml">0.08</mn><mo id="S4.T5.46.38.2.m1.1.1.1" xref="S4.T5.46.38.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.46.38.2.m1.1.1.3" xref="S4.T5.46.38.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.46.38.2.m1.1b"><apply id="S4.T5.46.38.2.m1.1.1.cmml" xref="S4.T5.46.38.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.46.38.2.m1.1.1.1.cmml" xref="S4.T5.46.38.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.46.38.2.m1.1.1.2.cmml" type="float" xref="S4.T5.46.38.2.m1.1.1.2">0.08</cn><cn id="S4.T5.46.38.2.m1.1.1.3.cmml" type="float" xref="S4.T5.46.38.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.46.38.2.m1.1c">0.08\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.46.38.2.m1.1d">0.08 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.47.39.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.20\pm 0.09" class="ltx_Math" display="inline" id="S4.T5.47.39.3.m1.1"><semantics id="S4.T5.47.39.3.m1.1a"><mrow id="S4.T5.47.39.3.m1.1.1" xref="S4.T5.47.39.3.m1.1.1.cmml"><mn id="S4.T5.47.39.3.m1.1.1.2" xref="S4.T5.47.39.3.m1.1.1.2.cmml">0.20</mn><mo id="S4.T5.47.39.3.m1.1.1.1" xref="S4.T5.47.39.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.47.39.3.m1.1.1.3" xref="S4.T5.47.39.3.m1.1.1.3.cmml">0.09</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.47.39.3.m1.1b"><apply id="S4.T5.47.39.3.m1.1.1.cmml" xref="S4.T5.47.39.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.47.39.3.m1.1.1.1.cmml" xref="S4.T5.47.39.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.47.39.3.m1.1.1.2.cmml" type="float" xref="S4.T5.47.39.3.m1.1.1.2">0.20</cn><cn id="S4.T5.47.39.3.m1.1.1.3.cmml" type="float" xref="S4.T5.47.39.3.m1.1.1.3">0.09</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.47.39.3.m1.1c">0.20\pm 0.09</annotation><annotation encoding="application/x-llamapun" id="S4.T5.47.39.3.m1.1d">0.20 ± 0.09</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.48.40.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}863.2\pm 76.6\phantom{0}" class="ltx_Math" display="inline" id="S4.T5.48.40.4.m1.1"><semantics id="S4.T5.48.40.4.m1.1a"><mrow id="S4.T5.48.40.4.m1.1.1" xref="S4.T5.48.40.4.m1.1.1.cmml"><mn id="S4.T5.48.40.4.m1.1.1.2" xref="S4.T5.48.40.4.m1.1.1.2.cmml">863.2</mn><mo id="S4.T5.48.40.4.m1.1.1.1" xref="S4.T5.48.40.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.48.40.4.m1.1.1.3" xref="S4.T5.48.40.4.m1.1.1.3.cmml">76.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.48.40.4.m1.1b"><apply id="S4.T5.48.40.4.m1.1.1.cmml" xref="S4.T5.48.40.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.48.40.4.m1.1.1.1.cmml" xref="S4.T5.48.40.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.48.40.4.m1.1.1.2.cmml" type="float" xref="S4.T5.48.40.4.m1.1.1.2">863.2</cn><cn id="S4.T5.48.40.4.m1.1.1.3.cmml" type="float" xref="S4.T5.48.40.4.m1.1.1.3">76.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.48.40.4.m1.1c">\phantom{0}863.2\pm 76.6\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.48.40.4.m1.1d">863.2 ± 76.6</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.52.44"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.52.44.5" style="padding-left:3.0pt;padding-right:3.0pt;">LW-VAE</th> <td class="ltx_td ltx_align_center" id="S4.T5.49.41.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.11\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.49.41.1.m1.1"><semantics id="S4.T5.49.41.1.m1.1a"><mrow id="S4.T5.49.41.1.m1.1.1" xref="S4.T5.49.41.1.m1.1.1.cmml"><mn id="S4.T5.49.41.1.m1.1.1.2" xref="S4.T5.49.41.1.m1.1.1.2.cmml">0.11</mn><mo id="S4.T5.49.41.1.m1.1.1.1" xref="S4.T5.49.41.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.49.41.1.m1.1.1.3" xref="S4.T5.49.41.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.49.41.1.m1.1b"><apply id="S4.T5.49.41.1.m1.1.1.cmml" xref="S4.T5.49.41.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.49.41.1.m1.1.1.1.cmml" xref="S4.T5.49.41.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.49.41.1.m1.1.1.2.cmml" type="float" xref="S4.T5.49.41.1.m1.1.1.2">0.11</cn><cn id="S4.T5.49.41.1.m1.1.1.3.cmml" type="float" xref="S4.T5.49.41.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.49.41.1.m1.1c">0.11\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.49.41.1.m1.1d">0.11 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.50.42.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.07\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.50.42.2.m1.1"><semantics id="S4.T5.50.42.2.m1.1a"><mrow id="S4.T5.50.42.2.m1.1.1" xref="S4.T5.50.42.2.m1.1.1.cmml"><mn id="S4.T5.50.42.2.m1.1.1.2" xref="S4.T5.50.42.2.m1.1.1.2.cmml">0.07</mn><mo id="S4.T5.50.42.2.m1.1.1.1" xref="S4.T5.50.42.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.50.42.2.m1.1.1.3" xref="S4.T5.50.42.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.50.42.2.m1.1b"><apply id="S4.T5.50.42.2.m1.1.1.cmml" xref="S4.T5.50.42.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.50.42.2.m1.1.1.1.cmml" xref="S4.T5.50.42.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.50.42.2.m1.1.1.2.cmml" type="float" xref="S4.T5.50.42.2.m1.1.1.2">0.07</cn><cn id="S4.T5.50.42.2.m1.1.1.3.cmml" type="float" xref="S4.T5.50.42.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.50.42.2.m1.1c">0.07\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.50.42.2.m1.1d">0.07 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.51.43.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.41\pm 0.15" class="ltx_Math" display="inline" id="S4.T5.51.43.3.m1.1"><semantics id="S4.T5.51.43.3.m1.1a"><mrow id="S4.T5.51.43.3.m1.1.1" xref="S4.T5.51.43.3.m1.1.1.cmml"><mn id="S4.T5.51.43.3.m1.1.1.2" xref="S4.T5.51.43.3.m1.1.1.2.cmml">0.41</mn><mo id="S4.T5.51.43.3.m1.1.1.1" xref="S4.T5.51.43.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.51.43.3.m1.1.1.3" xref="S4.T5.51.43.3.m1.1.1.3.cmml">0.15</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.51.43.3.m1.1b"><apply id="S4.T5.51.43.3.m1.1.1.cmml" xref="S4.T5.51.43.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.51.43.3.m1.1.1.1.cmml" xref="S4.T5.51.43.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.51.43.3.m1.1.1.2.cmml" type="float" xref="S4.T5.51.43.3.m1.1.1.2">0.41</cn><cn id="S4.T5.51.43.3.m1.1.1.3.cmml" type="float" xref="S4.T5.51.43.3.m1.1.1.3">0.15</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.51.43.3.m1.1c">0.41\pm 0.15</annotation><annotation encoding="application/x-llamapun" id="S4.T5.51.43.3.m1.1d">0.41 ± 0.15</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.52.44.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}692.2\pm 114.1" class="ltx_Math" display="inline" id="S4.T5.52.44.4.m1.1"><semantics id="S4.T5.52.44.4.m1.1a"><mrow id="S4.T5.52.44.4.m1.1.1" xref="S4.T5.52.44.4.m1.1.1.cmml"><mn id="S4.T5.52.44.4.m1.1.1.2" xref="S4.T5.52.44.4.m1.1.1.2.cmml">692.2</mn><mo id="S4.T5.52.44.4.m1.1.1.1" xref="S4.T5.52.44.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.52.44.4.m1.1.1.3" xref="S4.T5.52.44.4.m1.1.1.3.cmml">114.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.52.44.4.m1.1b"><apply id="S4.T5.52.44.4.m1.1.1.cmml" xref="S4.T5.52.44.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.52.44.4.m1.1.1.1.cmml" xref="S4.T5.52.44.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.52.44.4.m1.1.1.2.cmml" type="float" xref="S4.T5.52.44.4.m1.1.1.2">692.2</cn><cn id="S4.T5.52.44.4.m1.1.1.3.cmml" type="float" xref="S4.T5.52.44.4.m1.1.1.3">114.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.52.44.4.m1.1c">\phantom{0}692.2\pm 114.1</annotation><annotation encoding="application/x-llamapun" id="S4.T5.52.44.4.m1.1d">692.2 ± 114.1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.56.48"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S4.T5.56.48.5" style="padding-left:3.0pt;padding-right:3.0pt;">TSADIS</th> <td class="ltx_td ltx_align_center" id="S4.T5.53.45.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.11\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.53.45.1.m1.1"><semantics id="S4.T5.53.45.1.m1.1a"><mrow id="S4.T5.53.45.1.m1.1.1" xref="S4.T5.53.45.1.m1.1.1.cmml"><mn id="S4.T5.53.45.1.m1.1.1.2" xref="S4.T5.53.45.1.m1.1.1.2.cmml">0.11</mn><mo id="S4.T5.53.45.1.m1.1.1.1" xref="S4.T5.53.45.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.53.45.1.m1.1.1.3" xref="S4.T5.53.45.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.53.45.1.m1.1b"><apply id="S4.T5.53.45.1.m1.1.1.cmml" xref="S4.T5.53.45.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.53.45.1.m1.1.1.1.cmml" xref="S4.T5.53.45.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.53.45.1.m1.1.1.2.cmml" type="float" xref="S4.T5.53.45.1.m1.1.1.2">0.11</cn><cn id="S4.T5.53.45.1.m1.1.1.3.cmml" type="float" xref="S4.T5.53.45.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.53.45.1.m1.1c">0.11\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.53.45.1.m1.1d">0.11 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.54.46.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.08\pm 0.01" class="ltx_Math" display="inline" id="S4.T5.54.46.2.m1.1"><semantics id="S4.T5.54.46.2.m1.1a"><mrow id="S4.T5.54.46.2.m1.1.1" xref="S4.T5.54.46.2.m1.1.1.cmml"><mn id="S4.T5.54.46.2.m1.1.1.2" xref="S4.T5.54.46.2.m1.1.1.2.cmml">0.08</mn><mo id="S4.T5.54.46.2.m1.1.1.1" xref="S4.T5.54.46.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.54.46.2.m1.1.1.3" xref="S4.T5.54.46.2.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.54.46.2.m1.1b"><apply id="S4.T5.54.46.2.m1.1.1.cmml" xref="S4.T5.54.46.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.54.46.2.m1.1.1.1.cmml" xref="S4.T5.54.46.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.54.46.2.m1.1.1.2.cmml" type="float" xref="S4.T5.54.46.2.m1.1.1.2">0.08</cn><cn id="S4.T5.54.46.2.m1.1.1.3.cmml" type="float" xref="S4.T5.54.46.2.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.54.46.2.m1.1c">0.08\pm 0.01</annotation><annotation encoding="application/x-llamapun" id="S4.T5.54.46.2.m1.1d">0.08 ± 0.01</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.55.47.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.35\pm 0.19" class="ltx_Math" display="inline" id="S4.T5.55.47.3.m1.1"><semantics id="S4.T5.55.47.3.m1.1a"><mrow id="S4.T5.55.47.3.m1.1.1" xref="S4.T5.55.47.3.m1.1.1.cmml"><mn id="S4.T5.55.47.3.m1.1.1.2" xref="S4.T5.55.47.3.m1.1.1.2.cmml">0.35</mn><mo id="S4.T5.55.47.3.m1.1.1.1" xref="S4.T5.55.47.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.55.47.3.m1.1.1.3" xref="S4.T5.55.47.3.m1.1.1.3.cmml">0.19</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.55.47.3.m1.1b"><apply id="S4.T5.55.47.3.m1.1.1.cmml" xref="S4.T5.55.47.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.55.47.3.m1.1.1.1.cmml" xref="S4.T5.55.47.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.55.47.3.m1.1.1.2.cmml" type="float" xref="S4.T5.55.47.3.m1.1.1.2">0.35</cn><cn id="S4.T5.55.47.3.m1.1.1.3.cmml" type="float" xref="S4.T5.55.47.3.m1.1.1.3">0.19</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.55.47.3.m1.1c">0.35\pm 0.19</annotation><annotation encoding="application/x-llamapun" id="S4.T5.55.47.3.m1.1d">0.35 ± 0.19</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T5.56.48.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}808.4\pm 143.3" class="ltx_Math" display="inline" id="S4.T5.56.48.4.m1.1"><semantics id="S4.T5.56.48.4.m1.1a"><mrow id="S4.T5.56.48.4.m1.1.1" xref="S4.T5.56.48.4.m1.1.1.cmml"><mn id="S4.T5.56.48.4.m1.1.1.2" xref="S4.T5.56.48.4.m1.1.1.2.cmml">808.4</mn><mo id="S4.T5.56.48.4.m1.1.1.1" xref="S4.T5.56.48.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.56.48.4.m1.1.1.3" xref="S4.T5.56.48.4.m1.1.1.3.cmml">143.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.56.48.4.m1.1b"><apply id="S4.T5.56.48.4.m1.1.1.cmml" xref="S4.T5.56.48.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.56.48.4.m1.1.1.1.cmml" xref="S4.T5.56.48.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.56.48.4.m1.1.1.2.cmml" type="float" xref="S4.T5.56.48.4.m1.1.1.2">808.4</cn><cn id="S4.T5.56.48.4.m1.1.1.3.cmml" type="float" xref="S4.T5.56.48.4.m1.1.1.3">143.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.56.48.4.m1.1c">\phantom{0}808.4\pm 143.3</annotation><annotation encoding="application/x-llamapun" id="S4.T5.56.48.4.m1.1d">808.4 ± 143.3</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T5.60.52"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b" id="S4.T5.60.52.5" style="padding-left:3.0pt;padding-right:3.0pt;">TeVAE</th> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.57.49.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.56}\pm 0.06" class="ltx_Math" display="inline" id="S4.T5.57.49.1.m1.1"><semantics id="S4.T5.57.49.1.m1.1a"><mrow id="S4.T5.57.49.1.m1.1.1" xref="S4.T5.57.49.1.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.57.49.1.m1.1.1.2" mathvariant="bold" xref="S4.T5.57.49.1.m1.1.1.2.cmml">0.56</mn><mo id="S4.T5.57.49.1.m1.1.1.1" xref="S4.T5.57.49.1.m1.1.1.1.cmml">±</mo><mn id="S4.T5.57.49.1.m1.1.1.3" xref="S4.T5.57.49.1.m1.1.1.3.cmml">0.06</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.57.49.1.m1.1b"><apply id="S4.T5.57.49.1.m1.1.1.cmml" xref="S4.T5.57.49.1.m1.1.1"><csymbol cd="latexml" id="S4.T5.57.49.1.m1.1.1.1.cmml" xref="S4.T5.57.49.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.57.49.1.m1.1.1.2.cmml" type="float" xref="S4.T5.57.49.1.m1.1.1.2">0.56</cn><cn id="S4.T5.57.49.1.m1.1.1.3.cmml" type="float" xref="S4.T5.57.49.1.m1.1.1.3">0.06</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.57.49.1.m1.1c">\mathbf{0.56}\pm 0.06</annotation><annotation encoding="application/x-llamapun" id="S4.T5.57.49.1.m1.1d">bold_0.56 ± 0.06</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.58.50.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.64}\pm 0.09" class="ltx_Math" display="inline" id="S4.T5.58.50.2.m1.1"><semantics id="S4.T5.58.50.2.m1.1a"><mrow id="S4.T5.58.50.2.m1.1.1" xref="S4.T5.58.50.2.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.58.50.2.m1.1.1.2" mathvariant="bold" xref="S4.T5.58.50.2.m1.1.1.2.cmml">0.64</mn><mo id="S4.T5.58.50.2.m1.1.1.1" xref="S4.T5.58.50.2.m1.1.1.1.cmml">±</mo><mn id="S4.T5.58.50.2.m1.1.1.3" xref="S4.T5.58.50.2.m1.1.1.3.cmml">0.09</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.58.50.2.m1.1b"><apply id="S4.T5.58.50.2.m1.1.1.cmml" xref="S4.T5.58.50.2.m1.1.1"><csymbol cd="latexml" id="S4.T5.58.50.2.m1.1.1.1.cmml" xref="S4.T5.58.50.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.58.50.2.m1.1.1.2.cmml" type="float" xref="S4.T5.58.50.2.m1.1.1.2">0.64</cn><cn id="S4.T5.58.50.2.m1.1.1.3.cmml" type="float" xref="S4.T5.58.50.2.m1.1.1.3">0.09</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.58.50.2.m1.1c">\mathbf{0.64}\pm 0.09</annotation><annotation encoding="application/x-llamapun" id="S4.T5.58.50.2.m1.1d">bold_0.64 ± 0.09</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.59.51.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{0.51}\pm 0.08" class="ltx_Math" display="inline" id="S4.T5.59.51.3.m1.1"><semantics id="S4.T5.59.51.3.m1.1a"><mrow id="S4.T5.59.51.3.m1.1.1" xref="S4.T5.59.51.3.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.59.51.3.m1.1.1.2" mathvariant="bold" xref="S4.T5.59.51.3.m1.1.1.2.cmml">0.51</mn><mo id="S4.T5.59.51.3.m1.1.1.1" xref="S4.T5.59.51.3.m1.1.1.1.cmml">±</mo><mn id="S4.T5.59.51.3.m1.1.1.3" xref="S4.T5.59.51.3.m1.1.1.3.cmml">0.08</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.59.51.3.m1.1b"><apply id="S4.T5.59.51.3.m1.1.1.cmml" xref="S4.T5.59.51.3.m1.1.1"><csymbol cd="latexml" id="S4.T5.59.51.3.m1.1.1.1.cmml" xref="S4.T5.59.51.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.59.51.3.m1.1.1.2.cmml" type="float" xref="S4.T5.59.51.3.m1.1.1.2">0.51</cn><cn id="S4.T5.59.51.3.m1.1.1.3.cmml" type="float" xref="S4.T5.59.51.3.m1.1.1.3">0.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.59.51.3.m1.1c">\mathbf{0.51}\pm 0.08</annotation><annotation encoding="application/x-llamapun" id="S4.T5.59.51.3.m1.1d">bold_0.51 ± 0.08</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.60.52.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\mathbf{\phantom{0}651.0}\pm 83.0\phantom{7}" class="ltx_Math" display="inline" id="S4.T5.60.52.4.m1.1"><semantics id="S4.T5.60.52.4.m1.1a"><mrow id="S4.T5.60.52.4.m1.1.1" xref="S4.T5.60.52.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S4.T5.60.52.4.m1.1.1.2" mathvariant="bold" xref="S4.T5.60.52.4.m1.1.1.2.cmml">651.0</mn><mo id="S4.T5.60.52.4.m1.1.1.1" xref="S4.T5.60.52.4.m1.1.1.1.cmml">±</mo><mn id="S4.T5.60.52.4.m1.1.1.3" xref="S4.T5.60.52.4.m1.1.1.3.cmml">83.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.60.52.4.m1.1b"><apply id="S4.T5.60.52.4.m1.1.1.cmml" xref="S4.T5.60.52.4.m1.1.1"><csymbol cd="latexml" id="S4.T5.60.52.4.m1.1.1.1.cmml" xref="S4.T5.60.52.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T5.60.52.4.m1.1.1.2.cmml" type="float" xref="S4.T5.60.52.4.m1.1.1.2">651.0</cn><cn id="S4.T5.60.52.4.m1.1.1.3.cmml" type="float" xref="S4.T5.60.52.4.m1.1.1.3">83.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.60.52.4.m1.1c">\mathbf{\phantom{0}651.0}\pm 83.0\phantom{7}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.60.52.4.m1.1d">bold_651.0 ± 83.0</annotation></semantics></math></td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.3">First, it is evident that there is a large gap between the unsupervised and theoretical best threshold results. The unsupervised threshold is a rudimentary estimation based on the unlabelled validation subset <math alttext="\mathcal{D}^{\text{val}}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><msup id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.1.m1.1.1.2" xref="S4.SS3.p2.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S4.SS3.p2.1.m1.1.1.3" xref="S4.SS3.p2.1.m1.1.1.3a.cmml">val</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><apply id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS3.p2.1.m1.1.1.2.cmml" xref="S4.SS3.p2.1.m1.1.1.2">𝒟</ci><ci id="S4.SS3.p2.1.m1.1.1.3a.cmml" xref="S4.SS3.p2.1.m1.1.1.3"><mtext id="S4.SS3.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p2.1.m1.1.1.3">val</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\mathcal{D}^{\text{val}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT val end_POSTSUPERSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> and tends to be set higher than the theoretical best. This is because in the unsupervised version of the dataset, there are anomalous sequences within <math alttext="\mathcal{D}^{\text{val}}" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><msup id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.2.m2.1.1.2" xref="S4.SS3.p2.2.m2.1.1.2.cmml">𝒟</mi><mtext id="S4.SS3.p2.2.m2.1.1.3" xref="S4.SS3.p2.2.m2.1.1.3a.cmml">val</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1.2">𝒟</ci><ci id="S4.SS3.p2.2.m2.1.1.3a.cmml" xref="S4.SS3.p2.2.m2.1.1.3"><mtext id="S4.SS3.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p2.2.m2.1.1.3">val</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">\mathcal{D}^{\text{val}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">caligraphic_D start_POSTSUPERSCRIPT val end_POSTSUPERSCRIPT</annotation></semantics></math>, which are associated with a higher maximum anomaly score and therefore threshold. It is clear, however, that the results are far from good, which sets a foundation for future work. The threshold choice can be taken out of the equation by performing a grid search on different thresholds, which allows us to find the threshold which yields the theoretical best <math alttext="F_{1}" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><msub id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml"><mi id="S4.SS3.p2.3.m3.1.1.2" xref="S4.SS3.p2.3.m3.1.1.2.cmml">F</mi><mn id="S4.SS3.p2.3.m3.1.1.3" xref="S4.SS3.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><apply id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m3.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p2.3.m3.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.2">𝐹</ci><cn id="S4.SS3.p2.3.m3.1.1.3.cmml" type="integer" xref="S4.SS3.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> score. At this threshold, we observe the best possible anomaly detection performance the model can achieve on the test set, though it is not observable in the real-world. Here, TeVAE performs best in all metrics, though with a high average detection delay due to the number of high number of false negatives. While these results are better than with the unsupervised threshold, they still leave room for improvement. Regardless of the results, it cannot be denied how much less computationally intensive TSADIS is compared to methods based on deep learning. Even on commodity hardware, more specifically a laptop with an Intel Core i7-1185G7, it can evaluate test data faster than deep learning models. It should be noted that this is mainly because no reverse-windowing is needed with TSADIS, a process that, unlike inference in deep learning models, runs on the CPU, not the GPU. Additionally, TSADIS <em class="ltx_emph ltx_font_italic" id="S4.SS3.p2.3.1">does not</em> require a training procedure. At first glance, this property is a benefit, as the implementation hurdle is much lower than with deep learning models, which essentially require GPU-acceleration. However, without training data, TSADIS cannot know what is a nominal time series and what is not, therefore the <em class="ltx_emph ltx_font_italic" id="S4.SS3.p2.3.2">nominal behaviour is not modelled</em>. It can solely rely on the information present within a sequence to calculate an anomaly score, which is part of the reason it cannot outperform deep learning-based methods.</p> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.1">We also performed limited testing on the version of the dataset for <em class="ltx_emph ltx_font_italic" id="S4.SS3.p3.1.1">semi-supervised</em> anomaly detection. It involves the same testing procedure, except that the clean version, i.e. anomaly-free, of the training subset is used.</p> </div> <figure class="ltx_table" id="S4.T6"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 6: </span><math alttext="F_{1}" class="ltx_Math" display="inline" id="S4.T6.5.m1.1"><semantics id="S4.T6.5.m1.1b"><msub id="S4.T6.5.m1.1.1" xref="S4.T6.5.m1.1.1.cmml"><mi id="S4.T6.5.m1.1.1.2" xref="S4.T6.5.m1.1.1.2.cmml">F</mi><mn id="S4.T6.5.m1.1.1.3" xref="S4.T6.5.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T6.5.m1.1c"><apply id="S4.T6.5.m1.1.1.cmml" xref="S4.T6.5.m1.1.1"><csymbol cd="ambiguous" id="S4.T6.5.m1.1.1.1.cmml" xref="S4.T6.5.m1.1.1">subscript</csymbol><ci id="S4.T6.5.m1.1.1.2.cmml" xref="S4.T6.5.m1.1.1.2">𝐹</ci><cn id="S4.T6.5.m1.1.1.3.cmml" type="integer" xref="S4.T6.5.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.5.m1.1d">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.T6.5.m1.1e">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> score, precision <math alttext="P" class="ltx_Math" display="inline" id="S4.T6.6.m2.1"><semantics id="S4.T6.6.m2.1b"><mi id="S4.T6.6.m2.1.1" xref="S4.T6.6.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.T6.6.m2.1c"><ci id="S4.T6.6.m2.1.1.cmml" xref="S4.T6.6.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.6.m2.1d">P</annotation><annotation encoding="application/x-llamapun" id="S4.T6.6.m2.1e">italic_P</annotation></semantics></math>, recall <math alttext="R" class="ltx_Math" display="inline" id="S4.T6.7.m3.1"><semantics id="S4.T6.7.m3.1b"><mi id="S4.T6.7.m3.1.1" xref="S4.T6.7.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.T6.7.m3.1c"><ci id="S4.T6.7.m3.1.1.cmml" xref="S4.T6.7.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.7.m3.1d">R</annotation><annotation encoding="application/x-llamapun" id="S4.T6.7.m3.1e">italic_R</annotation></semantics></math>, and average detection delay <math alttext="\bar{\delta}" class="ltx_Math" display="inline" id="S4.T6.8.m4.1"><semantics id="S4.T6.8.m4.1b"><mover accent="true" id="S4.T6.8.m4.1.1" xref="S4.T6.8.m4.1.1.cmml"><mi id="S4.T6.8.m4.1.1.2" xref="S4.T6.8.m4.1.1.2.cmml">δ</mi><mo id="S4.T6.8.m4.1.1.1" xref="S4.T6.8.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.T6.8.m4.1c"><apply id="S4.T6.8.m4.1.1.cmml" xref="S4.T6.8.m4.1.1"><ci id="S4.T6.8.m4.1.1.1.cmml" xref="S4.T6.8.m4.1.1.1">¯</ci><ci id="S4.T6.8.m4.1.1.2.cmml" xref="S4.T6.8.m4.1.1.2">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.8.m4.1d">\bar{\delta}</annotation><annotation encoding="application/x-llamapun" id="S4.T6.8.m4.1e">over¯ start_ARG italic_δ end_ARG</annotation></semantics></math> using the <em class="ltx_emph ltx_font_italic" id="S4.T6.23.1">unsupervised</em> threshold (top half) and <em class="ltx_emph ltx_font_italic" id="S4.T6.24.2">theoretical best</em> threshold (bottom half) for TeVAE applied to the semi-supervised anomaly detection version of the PATH dataset.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T6.20"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T6.12.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S4.T6.12.4.5" style="padding-left:3.0pt;padding-right:3.0pt;">Model</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T6.9.1.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="F_{1}" class="ltx_Math" display="inline" id="S4.T6.9.1.1.m1.1"><semantics id="S4.T6.9.1.1.m1.1a"><msub id="S4.T6.9.1.1.m1.1.1" xref="S4.T6.9.1.1.m1.1.1.cmml"><mi id="S4.T6.9.1.1.m1.1.1.2" xref="S4.T6.9.1.1.m1.1.1.2.cmml">F</mi><mn id="S4.T6.9.1.1.m1.1.1.3" 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id="S4.T6.11.3.3.m1.1d">italic_R</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T6.12.4.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\bar{\delta}\ [s]" class="ltx_Math" display="inline" id="S4.T6.12.4.4.m1.1"><semantics id="S4.T6.12.4.4.m1.1a"><mrow id="S4.T6.12.4.4.m1.1.2" xref="S4.T6.12.4.4.m1.1.2.cmml"><mover accent="true" id="S4.T6.12.4.4.m1.1.2.2" xref="S4.T6.12.4.4.m1.1.2.2.cmml"><mi id="S4.T6.12.4.4.m1.1.2.2.2" xref="S4.T6.12.4.4.m1.1.2.2.2.cmml">δ</mi><mo id="S4.T6.12.4.4.m1.1.2.2.1" xref="S4.T6.12.4.4.m1.1.2.2.1.cmml">¯</mo></mover><mo id="S4.T6.12.4.4.m1.1.2.1" lspace="0.500em" xref="S4.T6.12.4.4.m1.1.2.1.cmml">⁢</mo><mrow id="S4.T6.12.4.4.m1.1.2.3.2" xref="S4.T6.12.4.4.m1.1.2.3.1.cmml"><mo id="S4.T6.12.4.4.m1.1.2.3.2.1" stretchy="false" xref="S4.T6.12.4.4.m1.1.2.3.1.1.cmml">[</mo><mi id="S4.T6.12.4.4.m1.1.1" xref="S4.T6.12.4.4.m1.1.1.cmml">s</mi><mo id="S4.T6.12.4.4.m1.1.2.3.2.2" stretchy="false" 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]</annotation></semantics></math></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T6.16.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S4.T6.16.8.5" style="padding-left:3.0pt;padding-right:3.0pt;">TeVAE</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T6.13.5.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.67\pm 0.16" class="ltx_Math" display="inline" id="S4.T6.13.5.1.m1.1"><semantics id="S4.T6.13.5.1.m1.1a"><mrow id="S4.T6.13.5.1.m1.1.1" xref="S4.T6.13.5.1.m1.1.1.cmml"><mn id="S4.T6.13.5.1.m1.1.1.2" xref="S4.T6.13.5.1.m1.1.1.2.cmml">0.67</mn><mo id="S4.T6.13.5.1.m1.1.1.1" xref="S4.T6.13.5.1.m1.1.1.1.cmml">±</mo><mn id="S4.T6.13.5.1.m1.1.1.3" xref="S4.T6.13.5.1.m1.1.1.3.cmml">0.16</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.13.5.1.m1.1b"><apply id="S4.T6.13.5.1.m1.1.1.cmml" xref="S4.T6.13.5.1.m1.1.1"><csymbol cd="latexml" id="S4.T6.13.5.1.m1.1.1.1.cmml" xref="S4.T6.13.5.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.13.5.1.m1.1.1.2.cmml" type="float" xref="S4.T6.13.5.1.m1.1.1.2">0.67</cn><cn id="S4.T6.13.5.1.m1.1.1.3.cmml" type="float" xref="S4.T6.13.5.1.m1.1.1.3">0.16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.13.5.1.m1.1c">0.67\pm 0.16</annotation><annotation encoding="application/x-llamapun" id="S4.T6.13.5.1.m1.1d">0.67 ± 0.16</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T6.14.6.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.97\pm 0.03" class="ltx_Math" display="inline" id="S4.T6.14.6.2.m1.1"><semantics id="S4.T6.14.6.2.m1.1a"><mrow id="S4.T6.14.6.2.m1.1.1" xref="S4.T6.14.6.2.m1.1.1.cmml"><mn id="S4.T6.14.6.2.m1.1.1.2" xref="S4.T6.14.6.2.m1.1.1.2.cmml">0.97</mn><mo id="S4.T6.14.6.2.m1.1.1.1" xref="S4.T6.14.6.2.m1.1.1.1.cmml">±</mo><mn id="S4.T6.14.6.2.m1.1.1.3" xref="S4.T6.14.6.2.m1.1.1.3.cmml">0.03</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.14.6.2.m1.1b"><apply id="S4.T6.14.6.2.m1.1.1.cmml" xref="S4.T6.14.6.2.m1.1.1"><csymbol cd="latexml" id="S4.T6.14.6.2.m1.1.1.1.cmml" xref="S4.T6.14.6.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.14.6.2.m1.1.1.2.cmml" type="float" xref="S4.T6.14.6.2.m1.1.1.2">0.97</cn><cn id="S4.T6.14.6.2.m1.1.1.3.cmml" type="float" xref="S4.T6.14.6.2.m1.1.1.3">0.03</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.14.6.2.m1.1c">0.97\pm 0.03</annotation><annotation encoding="application/x-llamapun" id="S4.T6.14.6.2.m1.1d">0.97 ± 0.03</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T6.15.7.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.54\pm 0.20" class="ltx_Math" display="inline" id="S4.T6.15.7.3.m1.1"><semantics id="S4.T6.15.7.3.m1.1a"><mrow id="S4.T6.15.7.3.m1.1.1" xref="S4.T6.15.7.3.m1.1.1.cmml"><mn id="S4.T6.15.7.3.m1.1.1.2" xref="S4.T6.15.7.3.m1.1.1.2.cmml">0.54</mn><mo id="S4.T6.15.7.3.m1.1.1.1" xref="S4.T6.15.7.3.m1.1.1.1.cmml">±</mo><mn id="S4.T6.15.7.3.m1.1.1.3" xref="S4.T6.15.7.3.m1.1.1.3.cmml">0.20</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.15.7.3.m1.1b"><apply id="S4.T6.15.7.3.m1.1.1.cmml" xref="S4.T6.15.7.3.m1.1.1"><csymbol cd="latexml" id="S4.T6.15.7.3.m1.1.1.1.cmml" xref="S4.T6.15.7.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.15.7.3.m1.1.1.2.cmml" type="float" xref="S4.T6.15.7.3.m1.1.1.2">0.54</cn><cn id="S4.T6.15.7.3.m1.1.1.3.cmml" type="float" xref="S4.T6.15.7.3.m1.1.1.3">0.20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.15.7.3.m1.1c">0.54\pm 0.20</annotation><annotation encoding="application/x-llamapun" id="S4.T6.15.7.3.m1.1d">0.54 ± 0.20</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S4.T6.16.8.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}576.3\pm 168.2" class="ltx_Math" display="inline" id="S4.T6.16.8.4.m1.1"><semantics id="S4.T6.16.8.4.m1.1a"><mrow id="S4.T6.16.8.4.m1.1.1" xref="S4.T6.16.8.4.m1.1.1.cmml"><mn id="S4.T6.16.8.4.m1.1.1.2" xref="S4.T6.16.8.4.m1.1.1.2.cmml">576.3</mn><mo id="S4.T6.16.8.4.m1.1.1.1" xref="S4.T6.16.8.4.m1.1.1.1.cmml">±</mo><mn id="S4.T6.16.8.4.m1.1.1.3" xref="S4.T6.16.8.4.m1.1.1.3.cmml">168.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.16.8.4.m1.1b"><apply id="S4.T6.16.8.4.m1.1.1.cmml" xref="S4.T6.16.8.4.m1.1.1"><csymbol cd="latexml" id="S4.T6.16.8.4.m1.1.1.1.cmml" xref="S4.T6.16.8.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.16.8.4.m1.1.1.2.cmml" type="float" xref="S4.T6.16.8.4.m1.1.1.2">576.3</cn><cn id="S4.T6.16.8.4.m1.1.1.3.cmml" type="float" xref="S4.T6.16.8.4.m1.1.1.3">168.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.16.8.4.m1.1c">\phantom{0}576.3\pm 168.2</annotation><annotation encoding="application/x-llamapun" id="S4.T6.16.8.4.m1.1d">576.3 ± 168.2</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T6.20.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_t" id="S4.T6.20.12.5" style="padding-left:3.0pt;padding-right:3.0pt;">TeVAE</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_t" id="S4.T6.17.9.1" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.83\pm 0.10" class="ltx_Math" display="inline" id="S4.T6.17.9.1.m1.1"><semantics id="S4.T6.17.9.1.m1.1a"><mrow id="S4.T6.17.9.1.m1.1.1" xref="S4.T6.17.9.1.m1.1.1.cmml"><mn id="S4.T6.17.9.1.m1.1.1.2" xref="S4.T6.17.9.1.m1.1.1.2.cmml">0.83</mn><mo id="S4.T6.17.9.1.m1.1.1.1" xref="S4.T6.17.9.1.m1.1.1.1.cmml">±</mo><mn id="S4.T6.17.9.1.m1.1.1.3" xref="S4.T6.17.9.1.m1.1.1.3.cmml">0.10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.17.9.1.m1.1b"><apply id="S4.T6.17.9.1.m1.1.1.cmml" xref="S4.T6.17.9.1.m1.1.1"><csymbol cd="latexml" id="S4.T6.17.9.1.m1.1.1.1.cmml" xref="S4.T6.17.9.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.17.9.1.m1.1.1.2.cmml" type="float" xref="S4.T6.17.9.1.m1.1.1.2">0.83</cn><cn id="S4.T6.17.9.1.m1.1.1.3.cmml" type="float" xref="S4.T6.17.9.1.m1.1.1.3">0.10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.17.9.1.m1.1c">0.83\pm 0.10</annotation><annotation encoding="application/x-llamapun" id="S4.T6.17.9.1.m1.1d">0.83 ± 0.10</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_t" id="S4.T6.18.10.2" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.94\pm 0.05" class="ltx_Math" display="inline" id="S4.T6.18.10.2.m1.1"><semantics id="S4.T6.18.10.2.m1.1a"><mrow id="S4.T6.18.10.2.m1.1.1" xref="S4.T6.18.10.2.m1.1.1.cmml"><mn id="S4.T6.18.10.2.m1.1.1.2" xref="S4.T6.18.10.2.m1.1.1.2.cmml">0.94</mn><mo id="S4.T6.18.10.2.m1.1.1.1" xref="S4.T6.18.10.2.m1.1.1.1.cmml">±</mo><mn id="S4.T6.18.10.2.m1.1.1.3" xref="S4.T6.18.10.2.m1.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.18.10.2.m1.1b"><apply id="S4.T6.18.10.2.m1.1.1.cmml" xref="S4.T6.18.10.2.m1.1.1"><csymbol cd="latexml" id="S4.T6.18.10.2.m1.1.1.1.cmml" xref="S4.T6.18.10.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.18.10.2.m1.1.1.2.cmml" type="float" xref="S4.T6.18.10.2.m1.1.1.2">0.94</cn><cn id="S4.T6.18.10.2.m1.1.1.3.cmml" type="float" xref="S4.T6.18.10.2.m1.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.18.10.2.m1.1c">0.94\pm 0.05</annotation><annotation encoding="application/x-llamapun" id="S4.T6.18.10.2.m1.1d">0.94 ± 0.05</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_t" id="S4.T6.19.11.3" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="0.76\pm 0.13" class="ltx_Math" display="inline" id="S4.T6.19.11.3.m1.1"><semantics id="S4.T6.19.11.3.m1.1a"><mrow id="S4.T6.19.11.3.m1.1.1" xref="S4.T6.19.11.3.m1.1.1.cmml"><mn id="S4.T6.19.11.3.m1.1.1.2" xref="S4.T6.19.11.3.m1.1.1.2.cmml">0.76</mn><mo id="S4.T6.19.11.3.m1.1.1.1" xref="S4.T6.19.11.3.m1.1.1.1.cmml">±</mo><mn id="S4.T6.19.11.3.m1.1.1.3" xref="S4.T6.19.11.3.m1.1.1.3.cmml">0.13</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.19.11.3.m1.1b"><apply id="S4.T6.19.11.3.m1.1.1.cmml" xref="S4.T6.19.11.3.m1.1.1"><csymbol cd="latexml" id="S4.T6.19.11.3.m1.1.1.1.cmml" xref="S4.T6.19.11.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.19.11.3.m1.1.1.2.cmml" type="float" xref="S4.T6.19.11.3.m1.1.1.2">0.76</cn><cn id="S4.T6.19.11.3.m1.1.1.3.cmml" type="float" xref="S4.T6.19.11.3.m1.1.1.3">0.13</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.19.11.3.m1.1c">0.76\pm 0.13</annotation><annotation encoding="application/x-llamapun" id="S4.T6.19.11.3.m1.1d">0.76 ± 0.13</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_t" id="S4.T6.20.12.4" style="padding-left:3.0pt;padding-right:3.0pt;"><math alttext="\phantom{0}383.9\pm 92.1\phantom{0}" class="ltx_Math" display="inline" id="S4.T6.20.12.4.m1.1"><semantics id="S4.T6.20.12.4.m1.1a"><mrow id="S4.T6.20.12.4.m1.1.1" xref="S4.T6.20.12.4.m1.1.1.cmml"><mn id="S4.T6.20.12.4.m1.1.1.2" xref="S4.T6.20.12.4.m1.1.1.2.cmml">383.9</mn><mo id="S4.T6.20.12.4.m1.1.1.1" xref="S4.T6.20.12.4.m1.1.1.1.cmml">±</mo><mn id="S4.T6.20.12.4.m1.1.1.3" xref="S4.T6.20.12.4.m1.1.1.3.cmml">92.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.20.12.4.m1.1b"><apply id="S4.T6.20.12.4.m1.1.1.cmml" xref="S4.T6.20.12.4.m1.1.1"><csymbol cd="latexml" id="S4.T6.20.12.4.m1.1.1.1.cmml" xref="S4.T6.20.12.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S4.T6.20.12.4.m1.1.1.2.cmml" type="float" xref="S4.T6.20.12.4.m1.1.1.2">383.9</cn><cn id="S4.T6.20.12.4.m1.1.1.3.cmml" type="float" xref="S4.T6.20.12.4.m1.1.1.3">92.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.20.12.4.m1.1c">\phantom{0}383.9\pm 92.1\phantom{0}</annotation><annotation encoding="application/x-llamapun" id="S4.T6.20.12.4.m1.1d">383.9 ± 92.1</annotation></semantics></math></td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS3.p4"> <p class="ltx_p" id="S4.SS3.p4.2">The corresponding results for TeVAE are shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.T6" title="Table 6 ‣ 4.3 Results and Discussion ‣ 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">6</span></a>. The gap between results obtained using the unsupervised threshold and the theoretical best is now much smaller than observed in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.13951v1#S4.T5" title="Table 5 ‣ 4.3 Results and Discussion ‣ 4 Baseline Results on the Dataset ‣ A Discrete-sequence Dataset for Evaluating Online Unsupervised Anomaly Detection Approaches for Multivariate Time Series"><span class="ltx_text ltx_ref_tag">5</span></a>, which can be attributed to the lack of anomalous data in <math alttext="\mathcal{D}^{\text{val}}" class="ltx_Math" display="inline" id="S4.SS3.p4.1.m1.1"><semantics id="S4.SS3.p4.1.m1.1a"><msup id="S4.SS3.p4.1.m1.1.1" xref="S4.SS3.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p4.1.m1.1.1.2" xref="S4.SS3.p4.1.m1.1.1.2.cmml">𝒟</mi><mtext id="S4.SS3.p4.1.m1.1.1.3" xref="S4.SS3.p4.1.m1.1.1.3a.cmml">val</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.1.m1.1b"><apply id="S4.SS3.p4.1.m1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.1.m1.1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1">superscript</csymbol><ci id="S4.SS3.p4.1.m1.1.1.2.cmml" xref="S4.SS3.p4.1.m1.1.1.2">𝒟</ci><ci id="S4.SS3.p4.1.m1.1.1.3a.cmml" xref="S4.SS3.p4.1.m1.1.1.3"><mtext id="S4.SS3.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p4.1.m1.1.1.3">val</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.1.m1.1c">\mathcal{D}^{\text{val}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.1.m1.1d">caligraphic_D start_POSTSUPERSCRIPT val end_POSTSUPERSCRIPT</annotation></semantics></math>. Additionally, there is a large gap between the theoretical best results between the unsupervised and semi-supervised versions, indicating the need for more robust future work when anomalous data is present in <math alttext="\mathcal{D}^{\text{train}}" class="ltx_Math" display="inline" id="S4.SS3.p4.2.m2.1"><semantics id="S4.SS3.p4.2.m2.1a"><msup id="S4.SS3.p4.2.m2.1.1" xref="S4.SS3.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p4.2.m2.1.1.2" xref="S4.SS3.p4.2.m2.1.1.2.cmml">𝒟</mi><mtext id="S4.SS3.p4.2.m2.1.1.3" xref="S4.SS3.p4.2.m2.1.1.3a.cmml">train</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.2.m2.1b"><apply id="S4.SS3.p4.2.m2.1.1.cmml" xref="S4.SS3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.2.m2.1.1.1.cmml" xref="S4.SS3.p4.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.p4.2.m2.1.1.2.cmml" xref="S4.SS3.p4.2.m2.1.1.2">𝒟</ci><ci id="S4.SS3.p4.2.m2.1.1.3a.cmml" xref="S4.SS3.p4.2.m2.1.1.3"><mtext id="S4.SS3.p4.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p4.2.m2.1.1.3">train</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.2.m2.1c">\mathcal{D}^{\text{train}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.2.m2.1d">caligraphic_D start_POSTSUPERSCRIPT train end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusion and Outlook</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We propose a novel multivariate time series dataset for online anomaly detection, called the Powertrain Anomaly Time series bencHmark (PATH) dataset. The PATH dataset is generated using simulation, where the model it is based on resembles a real-world dynamic system. In addition to that, simulation is done in a variety of different initial states to further add to the diversity of the dataset. To increase the complexity of the dataset, noise is applied and the beginning of time series are randomly trimmed. The anomalies in the PATH dataset arise from changing parameters prior to simulation, as opposed to manual data manipulation, resulting in anomalies that are non-trivial and realistic. We offer the dataset in three different versions: one for unsupervised anomaly detection, where the training subset consists of both anomalous and nominal sequences, another for semi-supervised anomaly detection, where the training subset consists of nominal sequences only, and one for time series generation or forecasting, where both the training and test subsets are nominal. Lastly, for each of the versions, we offer three different folds with a pre-defined train and test split to ensure generalised and comparable results.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.4">The experiments conducted in this work further support the claim of non-triviality because, even when the threshold choice is removed as a factor, the best approach in an unsupervised setting only manages to achieve an <math alttext="F_{1}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><msub id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi id="S5.p2.1.m1.1.1.2" xref="S5.p2.1.m1.1.1.2.cmml">F</mi><mn id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1">subscript</csymbol><ci id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1.2">𝐹</ci><cn id="S5.p2.1.m1.1.1.3.cmml" type="integer" xref="S5.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> score of <math alttext="0.56" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mn id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">0.56</mn><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><cn id="S5.p2.2.m2.1.1.cmml" type="float" xref="S5.p2.2.m2.1.1">0.56</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">0.56</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">0.56</annotation></semantics></math> and an average detection delay of <span class="ltx_ERROR undefined" id="S5.p2.4.1">\qty</span>651.0. In contrast, however, the results significantly improve when the clean version of the test subset is used. Here, the average theoretical best <math alttext="F_{1}" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><msub id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml"><mi id="S5.p2.3.m3.1.1.2" xref="S5.p2.3.m3.1.1.2.cmml">F</mi><mn id="S5.p2.3.m3.1.1.3" xref="S5.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><apply id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p2.3.m3.1.1.1.cmml" xref="S5.p2.3.m3.1.1">subscript</csymbol><ci id="S5.p2.3.m3.1.1.2.cmml" xref="S5.p2.3.m3.1.1.2">𝐹</ci><cn id="S5.p2.3.m3.1.1.3.cmml" type="integer" xref="S5.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">F_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> score reaches <math alttext="0.83" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mn id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml">0.83</mn><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><cn id="S5.p2.4.m4.1.1.cmml" type="float" xref="S5.p2.4.m4.1.1">0.83</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">0.83</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">0.83</annotation></semantics></math> and an average detection delay of <span class="ltx_ERROR undefined" id="S5.p2.4.2">\qty</span>383.9, highlighting the need for methods more robust to anomalous data in the unlabelled and contaminated training subset.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">In the future, the PATH dataset should be extended to a standardised benchmark consisting of not only a dataset based on the longitudinal electric vehicle dynamics, but also on simulation models from other domains. Additionally, the simulation model can be adapted to take battery ageing into account. Battery ageing can be characterised by the charge capacity, which, fixed in this dataset, can be changed dynamically to simulate battery ageing, which will have an impact on the entire system. This property can be especially useful for research in the area of unsupervised predictive maintenance, where an explicit health signal is not present. Furthermore, there is a need for more sophisticated evolution of the online evaluation metrics <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">correia_tevae_2024</span>]</cite> that do not assume a singular anomalous sub-sequence per test time series. When said metrics are available, the dataset could be extended to such anomalies types.</p> </div> <div class="ltx_para" id="S5.p4"> <span class="ltx_ERROR undefined" id="S5.p4.1">\printbibliography</span> </div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Nov 21 08:59:27 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>