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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.11250">arXiv:2503.11250</a> <span> [<a href="https://arxiv.org/pdf/2503.11250">pdf</a>, <a href="https://arxiv.org/format/2503.11250">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> </div> </div> <p class="title is-5 mathjax"> CRPS-Based Targeted Sequential Design with Application in Chemical Space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Friedli%2C+L">Lea Friedli</a>, <a href="/search/stat?searchtype=author&query=Gautier%2C+A">Ath茅na茂s Gautier</a>, <a href="/search/stat?searchtype=author&query=Broccard%2C+A">Anna Broccard</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.11250v1-abstract-short" style="display: inline;"> Sequential design of real and computer experiments via Gaussian Process (GP) models has proven useful for parsimonious, goal-oriented data acquisition purposes. In this work, we focus on acquisition strategies for a GP model that needs to be accurate within a predefined range of the response of interest. Such an approach is useful in various fields including synthetic chemistry, where finding mole… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.11250v1-abstract-full').style.display = 'inline'; document.getElementById('2503.11250v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.11250v1-abstract-full" style="display: none;"> Sequential design of real and computer experiments via Gaussian Process (GP) models has proven useful for parsimonious, goal-oriented data acquisition purposes. In this work, we focus on acquisition strategies for a GP model that needs to be accurate within a predefined range of the response of interest. Such an approach is useful in various fields including synthetic chemistry, where finding molecules with particular properties is essential for developing useful materials and effective medications. GP modeling and sequential design of experiments have been successfully applied to a plethora of domains, including molecule research. Our main contribution here is to use the threshold-weighted Continuous Ranked Probability Score (CRPS) as a basic building block for acquisition functions employed within sequential design. We study pointwise and integral criteria relying on two different weighting measures and benchmark them against competitors, demonstrating improved performance with respect to considered goals. The resulting acquisition strategies are applicable to a wide range of fields and pave the way to further developing sequential design relying on scoring rules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.11250v1-abstract-full').style.display = 'none'; document.getElementById('2503.11250v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.16246">arXiv:2411.16246</a> <span> [<a href="https://arxiv.org/pdf/2411.16246">pdf</a>, <a href="https://arxiv.org/format/2411.16246">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Efficient pooling of predictions via kernel embeddings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Allen%2C+S">Sam Allen</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Ziegel%2C+J">Johanna Ziegel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.16246v1-abstract-short" style="display: inline;"> Probabilistic predictions are probability distributions over the set of possible outcomes. Such predictions quantify the uncertainty in the outcome, making them essential for effective decision making. By combining multiple predictions, the information sources used to generate the predictions are pooled, often resulting in a more informative forecast. Probabilistic predictions are typically combin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16246v1-abstract-full').style.display = 'inline'; document.getElementById('2411.16246v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.16246v1-abstract-full" style="display: none;"> Probabilistic predictions are probability distributions over the set of possible outcomes. Such predictions quantify the uncertainty in the outcome, making them essential for effective decision making. By combining multiple predictions, the information sources used to generate the predictions are pooled, often resulting in a more informative forecast. Probabilistic predictions are typically combined by linearly pooling the individual predictive distributions; this encompasses several ensemble learning techniques, for example. The weights assigned to each prediction can be estimated based on their past performance, allowing more accurate predictions to receive a higher weight. This can be achieved by finding the weights that optimise a proper scoring rule over some training data. By embedding predictions into a Reproducing Kernel Hilbert Space (RKHS), we illustrate that estimating the linear pool weights that optimise kernel-based scoring rules is a convex quadratic optimisation problem. This permits an efficient implementation of the linear pool when optimally combining predictions on arbitrary outcome domains. This result also holds for other combination strategies, and we additionally study a flexible generalisation of the linear pool that overcomes some of its theoretical limitations, whilst allowing an efficient implementation within the RKHS framework. These approaches are compared in an application to operational wind speed forecasts, where this generalisation is found to offer substantial improvements upon the traditional linear pool. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16246v1-abstract-full').style.display = 'none'; document.getElementById('2411.16246v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.16003">arXiv:2409.16003</a> <span> [<a href="https://arxiv.org/pdf/2409.16003">pdf</a>, <a href="https://arxiv.org/format/2409.16003">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Easy Conditioning far beyond Gaussian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Faul%2C+A">Antoine Faul</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Spycher%2C+B">Ben Spycher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.16003v3-abstract-short" style="display: inline;"> Estimating and sampling from conditional densities plays a critical role in statistics and data science, with a plethora of applications. Numerous methods are available ranging from simple fitting approaches to sophisticated machine learning algorithms. However, selecting from among these often involves a trade-off between conflicting objectives of efficiency, flexibility and interpretability. Sta… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16003v3-abstract-full').style.display = 'inline'; document.getElementById('2409.16003v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16003v3-abstract-full" style="display: none;"> Estimating and sampling from conditional densities plays a critical role in statistics and data science, with a plethora of applications. Numerous methods are available ranging from simple fitting approaches to sophisticated machine learning algorithms. However, selecting from among these often involves a trade-off between conflicting objectives of efficiency, flexibility and interpretability. Starting from well known easy conditioning results in the Gaussian case, we show, thanks to results pertaining to stability by mixing and marginal transformations, that the latter carry over far beyond the Gaussian case. This enables us to flexibly model multivariate data by accommodating broad classes of multi-modal dependence structures and marginal distributions, while enjoying fast conditioning of fitted joint distributions. In applications, we primarily focus on conditioning via Gaussian versus Gaussian mixture copula models, comparing different fitting implementations for the latter. Numerical experiments with simulated and real data demonstrate the relevance of the approach for conditional sampling, evaluated using multivariate scoring rules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16003v3-abstract-full').style.display = 'none'; document.getElementById('2409.16003v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 12 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.12909">arXiv:2311.12909</a> <span> [<a href="https://arxiv.org/pdf/2311.12909">pdf</a>, <a href="https://arxiv.org/format/2311.12909">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Non-Sequential Ensemble Kalman Filtering using Distributed Arrays </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Travelletti%2C+C">C茅dric Travelletti</a>, <a href="/search/stat?searchtype=author&query=Franke%2C+J">J枚rg Franke</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Br%C3%B6nnimann%2C+S">Stefan Br枚nnimann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.12909v1-abstract-short" style="display: inline;"> This work introduces a new, distributed implementation of the Ensemble Kalman Filter (EnKF) that allows for non-sequential assimilation of large datasets in high-dimensional problems. The traditional EnKF algorithm is computationally intensive and exhibits difficulties in applications requiring interaction with the background covariance matrix, prompting the use of methods like sequential assimila… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.12909v1-abstract-full').style.display = 'inline'; document.getElementById('2311.12909v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.12909v1-abstract-full" style="display: none;"> This work introduces a new, distributed implementation of the Ensemble Kalman Filter (EnKF) that allows for non-sequential assimilation of large datasets in high-dimensional problems. The traditional EnKF algorithm is computationally intensive and exhibits difficulties in applications requiring interaction with the background covariance matrix, prompting the use of methods like sequential assimilation which can introduce unwanted consequences, such as dependency on observation ordering. Our implementation leverages recent advancements in distributed computing to enable the construction and use of the full model error covariance matrix in distributed memory, allowing for single-batch assimilation of all observations and eliminating order dependencies. Comparative performance assessments, involving both synthetic and real-world paleoclimatic reconstruction applications, indicate that the new, non-sequential implementation outperforms the traditional, sequential one. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.12909v1-abstract-full').style.display = 'none'; document.getElementById('2311.12909v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.07315">arXiv:2310.07315</a> <span> [<a href="https://arxiv.org/pdf/2310.07315">pdf</a>, <a href="https://arxiv.org/ps/2310.07315">ps</a>, <a href="https://arxiv.org/format/2310.07315">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Consistency of some sequential experimental design strategies for excursion set estimation based on vector-valued Gaussian processes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Stange%2C+P">Philip Stange</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.07315v1-abstract-short" style="display: inline;"> We tackle the extension to the vector-valued case of consistency results for Stepwise Uncertainty Reduction sequential experimental design strategies established in [Bect et al., A supermartingale approach to Gaussian process based sequential design of experiments, Bernoulli 25, 2019]. This lead us in the first place to clarify, assuming a compact index set, how the connection between continuous G… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.07315v1-abstract-full').style.display = 'inline'; document.getElementById('2310.07315v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.07315v1-abstract-full" style="display: none;"> We tackle the extension to the vector-valued case of consistency results for Stepwise Uncertainty Reduction sequential experimental design strategies established in [Bect et al., A supermartingale approach to Gaussian process based sequential design of experiments, Bernoulli 25, 2019]. This lead us in the first place to clarify, assuming a compact index set, how the connection between continuous Gaussian processes and Gaussian measures on the Banach space of continuous functions carries over to vector-valued settings. From there, a number of concepts and properties from the aforementioned paper can be readily extended. However, vector-valued settings do complicate things for some results, mainly due to the lack of continuity for the pseudo-inverse mapping that affects the conditional mean and covariance function given finitely many pointwise observations. We apply obtained results to the Integrated Bernoulli Variance and the Expected Measure Variance uncertainty functionals employed in [Fossum et al., Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling, The Annals of Applied Statistics 15, 2021] for the estimation for excursion sets of vector-valued functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.07315v1-abstract-full').style.display = 'none'; document.getElementById('2310.07315v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.04082">arXiv:2310.04082</a> <span> [<a href="https://arxiv.org/pdf/2310.04082">pdf</a>, <a href="https://arxiv.org/format/2310.04082">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> An energy-based model approach to rare event probability estimation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Friedli%2C+L">Lea Friedli</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Doucet%2C+A">Arnaud Doucet</a>, <a href="/search/stat?searchtype=author&query=Linde%2C+N">Niklas Linde</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.04082v1-abstract-short" style="display: inline;"> The estimation of rare event probabilities plays a pivotal role in diverse fields. Our aim is to determine the probability of a hazard or system failure occurring when a quantity of interest exceeds a critical value. In our approach, the distribution of the quantity of interest is represented by an energy density, characterized by a free energy function. To efficiently estimate the free energy, a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04082v1-abstract-full').style.display = 'inline'; document.getElementById('2310.04082v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.04082v1-abstract-full" style="display: none;"> The estimation of rare event probabilities plays a pivotal role in diverse fields. Our aim is to determine the probability of a hazard or system failure occurring when a quantity of interest exceeds a critical value. In our approach, the distribution of the quantity of interest is represented by an energy density, characterized by a free energy function. To efficiently estimate the free energy, a bias potential is introduced. Using concepts from energy-based models (EBM), this bias potential is optimized such that the corresponding probability density function approximates a pre-defined distribution targeting the failure region of interest. Given the optimal bias potential, the free energy function and the rare event probability of interest can be determined. The approach is applicable not just in traditional rare event settings where the variable upon which the quantity of interest relies has a known distribution, but also in inversion settings where the variable follows a posterior distribution. By combining the EBM approach with a Stein discrepancy-based stopping criterion, we aim for a balanced accuracy-efficiency trade-off. Furthermore, we explore both parametric and non-parametric approaches for the bias potential, with the latter eliminating the need for choosing a particular parameterization, but depending strongly on the accuracy of the kernel density estimate used in the optimization process. Through three illustrative test cases encompassing both traditional and inversion settings, we show that the proposed EBM approach, when properly configured, (i) allows stable and efficient estimation of rare event probabilities and (ii) compares favorably against subset sampling approaches. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04082v1-abstract-full').style.display = 'none'; document.getElementById('2310.04082v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.05846">arXiv:2307.05846</a> <span> [<a href="https://arxiv.org/pdf/2307.05846">pdf</a>, <a href="https://arxiv.org/format/2307.05846">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> </div> <p class="title is-5 mathjax"> Assessing the calibration of multivariate probabilistic forecasts </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Allen%2C+S">Sam Allen</a>, <a href="/search/stat?searchtype=author&query=Ziegel%2C+J">Johanna Ziegel</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.05846v1-abstract-short" style="display: inline;"> Rank and PIT histograms are established tools to assess the calibration of probabilistic forecasts. They not only check whether an ensemble forecast is calibrated, but they also reveal what systematic biases (if any) are present in the forecasts. Several extensions of rank histograms have been proposed to evaluate the calibration of probabilistic forecasts for multivariate outcomes. These extensio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.05846v1-abstract-full').style.display = 'inline'; document.getElementById('2307.05846v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.05846v1-abstract-full" style="display: none;"> Rank and PIT histograms are established tools to assess the calibration of probabilistic forecasts. They not only check whether an ensemble forecast is calibrated, but they also reveal what systematic biases (if any) are present in the forecasts. Several extensions of rank histograms have been proposed to evaluate the calibration of probabilistic forecasts for multivariate outcomes. These extensions introduce a so-called pre-rank function that condenses the multivariate forecasts and observations into univariate objects, from which a standard rank histogram can be produced. Existing pre-rank functions typically aim to preserve as much information as possible when condensing the multivariate forecasts and observations into univariate objects. Although this is sensible when conducting statistical tests for multivariate calibration, it can hinder the interpretation of the resulting histograms. In this paper, we demonstrate that there are few restrictions on the choice of pre-rank function, meaning forecasters can choose a pre-rank function depending on what information they want to extract from their forecasts. We introduce the concept of simple pre-rank functions, and provide examples that can be used to assess the location, scale, and dependence structure of multivariate probabilistic forecasts, as well as pre-rank functions tailored to the evaluation of probabilistic spatial field forecasts. The simple pre-rank functions that we introduce are easy to interpret, easy to implement, and they deliberately provide complementary information, meaning several pre-rank functions can be employed to achieve a more complete understanding of multivariate forecast performance. We then discuss how e-values can be employed to formally test for multivariate calibration over time. This is demonstrated in an application to wind speed forecasting using the EUPPBench post-processing benchmark data set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.05846v1-abstract-full').style.display = 'none'; document.getElementById('2307.05846v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.07588">arXiv:2206.07588</a> <span> [<a href="https://arxiv.org/pdf/2206.07588">pdf</a>, <a href="https://arxiv.org/ps/2206.07588">ps</a>, <a href="https://arxiv.org/format/2206.07588">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Ziegel%2C+J">Johanna Ziegel</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=D%C3%BCmbgen%2C+L">Lutz D眉mbgen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.07588v1-abstract-short" style="display: inline;"> We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.07588v1-abstract-full').style.display = 'inline'; document.getElementById('2206.07588v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.07588v1-abstract-full" style="display: none;"> We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.07588v1-abstract-full').style.display = 'none'; document.getElementById('2206.07588v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.12732">arXiv:2202.12732</a> <span> [<a href="https://arxiv.org/pdf/2202.12732">pdf</a>, <a href="https://arxiv.org/format/2202.12732">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Evaluating forecasts for high-impact events using transformed kernel scores </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Allen%2C+S">Sam Allen</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Ziegel%2C+J">Johanna Ziegel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.12732v1-abstract-short" style="display: inline;"> It is informative to evaluate a forecaster's ability to predict outcomes that have a large impact on the forecast user. Although weighted scoring rules have become a well-established tool to achieve this, such scores have been studied almost exclusively in the univariate case, with interest typically placed on extreme events. However, a large impact may also result from events not considered to be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12732v1-abstract-full').style.display = 'inline'; document.getElementById('2202.12732v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.12732v1-abstract-full" style="display: none;"> It is informative to evaluate a forecaster's ability to predict outcomes that have a large impact on the forecast user. Although weighted scoring rules have become a well-established tool to achieve this, such scores have been studied almost exclusively in the univariate case, with interest typically placed on extreme events. However, a large impact may also result from events not considered to be extreme from a statistical perspective: the interaction of several moderate events could also generate a high impact. Compound weather events provide a good example of this. To assess forecasts made for high-impact events, this work extends existing results on weighted scoring rules by introducing weighted multivariate scores. To do so, we utilise kernel scores. We demonstrate that the threshold-weighted continuous ranked probability score (twCRPS), arguably the most well-known weighted scoring rule, is a kernel score. This result leads to a convenient representation of the twCRPS when the forecast is an ensemble, and also permits a generalisation that can be employed with alternative kernels, allowing us to introduce, for example, a threshold-weighted energy score and threshold-weighted variogram score. To illustrate the additional information that these weighted multivariate scoring rules provide, results are presented for a case study in which the weighted scores are used to evaluate daily precipitation accumulation forecasts, with particular interest on events that could lead to flooding. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12732v1-abstract-full').style.display = 'none'; document.getElementById('2202.12732v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.05210">arXiv:2110.05210</a> <span> [<a href="https://arxiv.org/pdf/2110.05210">pdf</a>, <a href="https://arxiv.org/format/2110.05210">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geophysics">physics.geo-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1093/gji/ggab381">10.1093/gji/ggab381 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lithological Tomography with the Correlated Pseudo-Marginal Method </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Friedli%2C+L">Lea Friedli</a>, <a href="/search/stat?searchtype=author&query=Linde%2C+N">Niklas Linde</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Doucet%2C+A">Arnaud Doucet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.05210v1-abstract-short" style="display: inline;"> We consider lithological tomography in which the posterior distribution of (hydro)geological parameters of interest is inferred from geophysical data by treating the intermediate geophysical properties as latent variables. In such a latent variable model, one needs to estimate the intractable likelihood of the (hydro)geological parameters given the geophysical data. The pseudo-marginal method is a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.05210v1-abstract-full').style.display = 'inline'; document.getElementById('2110.05210v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.05210v1-abstract-full" style="display: none;"> We consider lithological tomography in which the posterior distribution of (hydro)geological parameters of interest is inferred from geophysical data by treating the intermediate geophysical properties as latent variables. In such a latent variable model, one needs to estimate the intractable likelihood of the (hydro)geological parameters given the geophysical data. The pseudo-marginal method is an adaptation of the Metropolis-Hastings algorithm in which an unbiased approximation of this likelihood is obtained by Monte Carlo averaging over samples from, in this setting, the noisy petrophysical relationship linking (hydro)geological and geophysical properties. To make the method practical in data-rich geophysical settings with low noise levels, we demonstrate that the Monte Carlo sampling must rely on importance sampling distributions that well approximate the posterior distribution of petrophysical scatter around the sampled (hydro)geological parameter field. To achieve a suitable acceptance rate, we rely both on (1) the correlated pseudo-marginal method, which correlates the samples used in the proposed and current states of the Markov chain, and (2) a model proposal scheme that preserves the prior distribution. As a synthetic test example, we infer porosity fields using crosshole ground-penetrating radar (GPR) first-arrival travel times. We use a (50x50)-dimensional pixel-based parameterization of the multi-Gaussian porosity field with known statistical parameters, resulting in a parameter space of high dimension. We demonstrate that the correlated pseudo-marginal method with our proposed importance sampling and prior-preserving proposal scheme outperforms current state-of-the-art methods in both linear and non-linear settings by greatly enhancing the posterior exploration. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.05210v1-abstract-full').style.display = 'none'; document.getElementById('2110.05210v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geophysical Journal International, Volume 228, Issue 2, February 2022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.03457">arXiv:2109.03457</a> <span> [<a href="https://arxiv.org/pdf/2109.03457">pdf</a>, <a href="https://arxiv.org/format/2109.03457">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Travelletti%2C+C">C茅dric Travelletti</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Linde%2C+N">Niklas Linde</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.03457v4-abstract-short" style="display: inline;"> We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covarianc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.03457v4-abstract-full').style.display = 'inline'; document.getElementById('2109.03457v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.03457v4-abstract-full" style="display: none;"> We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.03457v4-abstract-full').style.display = 'none'; document.getElementById('2109.03457v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 86A22; 60G15; 62F15; 62L05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.08156">arXiv:2104.08156</a> <span> [<a href="https://arxiv.org/pdf/2104.08156">pdf</a>, <a href="https://arxiv.org/format/2104.08156">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Fast ABC with joint generative modelling and subset simulation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Maalouf%2C+E">Eliane Maalouf</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Linde%2C+N">Niklas Linde</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.08156v1-abstract-short" style="display: inline;"> We propose a novel approach for solving inverse-problems with high-dimensional inputs and an expensive forward mapping. It leverages joint deep generative modelling to transfer the original problem spaces to a lower dimensional latent space. By jointly modelling input and output variables and endowing the latent with a prior distribution, the fitted probabilistic model indirectly gives access to t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.08156v1-abstract-full').style.display = 'inline'; document.getElementById('2104.08156v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.08156v1-abstract-full" style="display: none;"> We propose a novel approach for solving inverse-problems with high-dimensional inputs and an expensive forward mapping. It leverages joint deep generative modelling to transfer the original problem spaces to a lower dimensional latent space. By jointly modelling input and output variables and endowing the latent with a prior distribution, the fitted probabilistic model indirectly gives access to the approximate conditional distributions of interest. Since model error and observational noise with unknown distributions are common in practice, we resort to likelihood-free inference with Approximate Bayesian Computation (ABC). Our method calls on ABC by Subset Simulation to explore the regions of the latent space with dissimilarities between generated and observed outputs below prescribed thresholds. We diagnose the diversity of approximate posterior solutions by monitoring the probability content of these regions as a function of the threshold. We further analyze the curvature of the resulting diagnostic curve to propose an adequate ABC threshold. When applied to a cross-borehole tomography example from geophysics, our approach delivers promising performance without using prior knowledge of the forward nor of the noise distribution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.08156v1-abstract-full').style.display = 'none'; document.getElementById('2104.08156v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 6 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.07612">arXiv:2102.07612</a> <span> [<a href="https://arxiv.org/pdf/2102.07612">pdf</a>, <a href="https://arxiv.org/format/2102.07612">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1051/proc/202171108">10.1051/proc/202171108 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Goal-oriented adaptive sampling under random field modelling of response probability distributions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Gautier%2C+A">Ath茅na茂s Gautier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Pirot%2C+G">Guillaume Pirot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.07612v2-abstract-short" style="display: inline;"> In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features inc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.07612v2-abstract-full').style.display = 'inline'; document.getElementById('2102.07612v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.07612v2-abstract-full" style="display: none;"> In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model. The considered models deliver probabilistic predictions of response distributions at candidate points, allowing for instance to perform (approximate) posterior simulations of probability density functions, to jointly predict multiple moments and other functionals of target distributions, as well as to quantify the impact of collecting new samples on the state of knowledge of the distribution field of interest. In particular, we introduce adaptive sampling strategies leveraging the potential of the considered random distribution field models to guide system evaluations in a goal-oriented way, with a view towards parsimoniously addressing calibration and related problems from non-linear (stochastic) inversion and global optimisation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.07612v2-abstract-full').style.display = 'none'; document.getElementById('2102.07612v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.03108">arXiv:2101.03108</a> <span> [<a href="https://arxiv.org/pdf/2101.03108">pdf</a>, <a href="https://arxiv.org/format/2101.03108">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Fast calculation of Gaussian Process multiple-fold cross-validation residuals and their covariances </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Sch%C3%A4rer%2C+C">Cedric Sch盲rer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.03108v3-abstract-short" style="display: inline;"> We generalize fast Gaussian process leave-one-out formulae to multiple-fold cross-validation, highlighting in turn the covariance structure of cross-validation residuals in both Simple and Universal Kriging frameworks. We illustrate how resulting covariances affect model diagnostics. We further establish in the case of noiseless observations that correcting for covariances between residuals in cro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.03108v3-abstract-full').style.display = 'inline'; document.getElementById('2101.03108v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.03108v3-abstract-full" style="display: none;"> We generalize fast Gaussian process leave-one-out formulae to multiple-fold cross-validation, highlighting in turn the covariance structure of cross-validation residuals in both Simple and Universal Kriging frameworks. We illustrate how resulting covariances affect model diagnostics. We further establish in the case of noiseless observations that correcting for covariances between residuals in cross-validation-based estimation of the scale parameter leads back to MLE. Also, we highlight in broader settings how differences between pseudo-likelihood and likelihood methods boil down to accounting or not for residual covariances. The proposed fast calculation of cross-validation residuals is implemented and benchmarked against a naive implementation. Numerical experiments highlight the accuracy and substantial speed-ups that our approach enables. However, as supported by a discussion on main drivers of computational costs and by a numerical benchmark, speed-ups steeply decline as the number of folds (say, all sharing the same size) decreases. An application to a contaminant localization test case illustrates that grouping clustered observations in folds may help improving model assessment and parameter fitting compared to Leave-One-Out. Overall, our results enable fast multiple-fold cross-validation, have direct consequences in model diagnostics, and pave the way to future work on hyperparameter fitting and on the promising field of goal-oriented fold design. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.03108v3-abstract-full').style.display = 'none'; document.getElementById('2101.03108v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.03722">arXiv:2007.03722</a> <span> [<a href="https://arxiv.org/pdf/2007.03722">pdf</a>, <a href="https://arxiv.org/format/2007.03722">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Fossum%2C+T+O">Trygve Olav Fossum</a>, <a href="/search/stat?searchtype=author&query=Travelletti%2C+C">C茅dric Travelletti</a>, <a href="/search/stat?searchtype=author&query=Eidsvik%2C+J">Jo Eidsvik</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Rajan%2C+K">Kanna Rajan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.03722v2-abstract-short" style="display: inline;"> Improving and optimizing oceanographic sampling is a crucial task for marine science and maritime resource management. Faced with limited resources in understanding processes in the water-column, the combination of statistics and autonomous systems provide new opportunities for experimental design. In this work we develop efficient spatial sampling methods for characterizing regions defined by sim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.03722v2-abstract-full').style.display = 'inline'; document.getElementById('2007.03722v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.03722v2-abstract-full" style="display: none;"> Improving and optimizing oceanographic sampling is a crucial task for marine science and maritime resource management. Faced with limited resources in understanding processes in the water-column, the combination of statistics and autonomous systems provide new opportunities for experimental design. In this work we develop efficient spatial sampling methods for characterizing regions defined by simultaneous exceedances above prescribed thresholds of several responses, with an application focus on mapping coastal ocean phenomena based on temperature and salinity measurements. Specifically, we define a design criterion based on uncertainty in the excursions of vector-valued Gaussian random fields, and derive tractable expressions for the expected integrated Bernoulli variance reduction in such a framework. We demonstrate how this criterion can be used to prioritize sampling efforts at locations that are ambiguous, making exploration more effective. We use simulations to study and compare properties of the considered approaches, followed by results from field deployments with an autonomous underwater vehicle as part of a study mapping the boundary of a river plume. The results demonstrate the potential of combining statistical methods and robotic platforms to effectively inform and execute data-driven environmental sampling. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.03722v2-abstract-full').style.display = 'none'; document.getElementById('2007.03722v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.11827">arXiv:1912.11827</a> <span> [<a href="https://arxiv.org/pdf/1912.11827">pdf</a>, <a href="https://arxiv.org/format/1912.11827">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Area-covering postprocessing of ensemble precipitation forecasts using topographical and seasonal conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Friedli%2C+L">Lea Friedli</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Bhend%2C+J">Jonas Bhend</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.11827v3-abstract-short" style="display: inline;"> Probabilistic weather forecasts from ensemble systems require statistical postprocessing to yield calibrated and sharp predictive distributions. This paper presents an area-covering postprocessing method for ensemble precipitation predictions. We rely on the ensemble model output statistics (EMOS) approach, which generates probabilistic forecasts with a parametric distribution whose parameters dep… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.11827v3-abstract-full').style.display = 'inline'; document.getElementById('1912.11827v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.11827v3-abstract-full" style="display: none;"> Probabilistic weather forecasts from ensemble systems require statistical postprocessing to yield calibrated and sharp predictive distributions. This paper presents an area-covering postprocessing method for ensemble precipitation predictions. We rely on the ensemble model output statistics (EMOS) approach, which generates probabilistic forecasts with a parametric distribution whose parameters depend on (statistics of) the ensemble prediction. A case study with daily precipitation predictions across Switzerland highlights that postprocessing at observation locations indeed improves high-resolution ensemble forecasts, with 4.5% CRPS reduction on average in the case of a lead time of 1 day. Our main aim is to achieve such an improvement without binding the model to stations, by leveraging topographical covariates. Specifically, regression coefficients are estimated by weighting the training data in relation to the topographical similarity between their station of origin and the prediction location. In our case study, this approach is found to reproduce the performance of the local model without using local historical data for calibration. We further identify that one key difficulty is that postprocessing often degrades the performance of the ensemble forecast during summer and early autumn. To mitigate, we additionally estimate on the training set whether postprocessing at a specific location is expected to improve the prediction. If not, the direct model output is used. This extension reduces the CRPS of the topographical model by up to another 1.7% on average at the price of a slight degradation in calibration. In this case, the highest improvement is achieved for a lead time of 4 days. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.11827v3-abstract-full').style.display = 'none'; document.getElementById('1912.11827v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.04086">arXiv:1910.04086</a> <span> [<a href="https://arxiv.org/pdf/1910.04086">pdf</a>, <a href="https://arxiv.org/format/1910.04086">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Buathong%2C+P">Poompol Buathong</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Krityakierne%2C+T">Tipaluck Krityakierne</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.04086v2-abstract-short" style="display: inline;"> We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.04086v2-abstract-full').style.display = 'inline'; document.getElementById('1910.04086v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.04086v2-abstract-full" style="display: none;"> We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels typically suffer from a lack of strict positive definiteness, vast subclasses of DE kernels built upon DS kernels do possess this property, enabling in turn combinatorial optimization without requiring to introduce a jitter parameter. Proofs of theoretical results about considered kernels are complemented by a few practicalities regarding hyperparameter fitting. We furthermore demonstrate the applicability of our approach in prediction and optimization tasks, relying both on toy examples and on two test cases from mechanical engineering and hydrogeology, respectively. Experimental results highlight the applicability and compared merits of the considered approaches while opening new perspectives in prediction and sequential design with set inputs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.04086v2-abstract-full').style.display = 'none'; document.getElementById('1910.04086v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.00753">arXiv:1805.00753</a> <span> [<a href="https://arxiv.org/pdf/1805.00753">pdf</a>, <a href="https://arxiv.org/format/1805.00753">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Bachoc%2C+F">Francois Bachoc</a>, <a href="/search/stat?searchtype=author&query=Suvorikova%2C+A">Alexandra Suvorikova</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Loubes%2C+J">Jean-Michel Loubes</a>, <a href="/search/stat?searchtype=author&query=Spokoiny%2C+V">Vladimir Spokoiny</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.00753v2-abstract-short" style="display: inline;"> In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional case, such constructions do not extend well to the multidimensional case as we illustrate here. To tackle the problem of defining positive definite kernels betwe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.00753v2-abstract-full').style.display = 'inline'; document.getElementById('1805.00753v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.00753v2-abstract-full" style="display: none;"> In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional case, such constructions do not extend well to the multidimensional case as we illustrate here. To tackle the problem of defining positive definite kernels between multivariate distributions based on optimal transport, we appeal instead to Hilbert space embeddings relying on optimal transport maps to a reference distribution, that we suggest to take as a Wasserstein barycenter. We characterize in turn radial positive definite kernels on Hilbert spaces, and show that the covariance parameters of virtually all parametric families of covariance functions are microergodic in the case of (infinite-dimensional) Hilbert spaces. We also investigate statistical properties of our suggested positive definite kernels on multidimensional distributions, with a focus on consistency when a population Wasserstein barycenter is replaced by an empirical barycenter and additional explicit results in the special case of Gaussian distributions. Finally, we study the Gaussian process methodology based on our suggested positive definite kernels in regression problems with multidimensional distribution inputs, on simulation data stemming both from synthetic examples and from a mechanical engineering test case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.00753v2-abstract-full').style.display = 'none'; document.getElementById('1805.00753v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 April, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1711.01878">arXiv:1711.01878</a> <span> [<a href="https://arxiv.org/pdf/1711.01878">pdf</a>, <a href="https://arxiv.org/format/1711.01878">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Modeling non-stationary extreme dependence with stationary max-stable processes and multidimensional scaling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Martius%2C+O">Olivia Martius</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1711.01878v2-abstract-short" style="display: inline;"> Modeling the joint distribution of extreme weather events in multiple locations is a challenging task with important applications. In this study, we use max-stable models to study extreme daily precipitation events in Switzerland. The non-stationarity of the spatial process at hand involves important challenges, which are often dealt with by using a stationary model in a so-called climate space, w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.01878v2-abstract-full').style.display = 'inline'; document.getElementById('1711.01878v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1711.01878v2-abstract-full" style="display: none;"> Modeling the joint distribution of extreme weather events in multiple locations is a challenging task with important applications. In this study, we use max-stable models to study extreme daily precipitation events in Switzerland. The non-stationarity of the spatial process at hand involves important challenges, which are often dealt with by using a stationary model in a so-called climate space, with well-chosen covariates. Here, we instead chose to warp the weather stations under study in a latent space of higher dimension using multidimensional scaling (MDS). The advantage of this approach is its improved flexibility to reproduce highly non-stationary phenomena, while keeping a tractable stationary spatial model in the latent space. Two model fitting approaches, which both use MDS, are presented and compared to a classical approach that relies on composite likelihood maximization in a climate space. Results suggest that the proposed methods better reproduce the observed extremal coefficients and their complex spatial dependence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.01878v2-abstract-full').style.display = 'none'; document.getElementById('1711.01878v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 November, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.00688">arXiv:1710.00688</a> <span> [<a href="https://arxiv.org/pdf/1710.00688">pdf</a>, <a href="https://arxiv.org/format/1710.00688">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/00401706.2018.1562987">10.1080/00401706.2018.1562987 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Profile extrema for visualizing and quantifying uncertainties on excursion regions. Application to coastal flooding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Azzimonti%2C+D">Dario Azzimonti</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Rohmer%2C+J">J茅r茅my Rohmer</a>, <a href="/search/stat?searchtype=author&query=Idier%2C+D">D茅borah Idier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.00688v2-abstract-short" style="display: inline;"> We consider the problem of describing excursion sets of a real-valued function $f$, i.e. the set of inputs where $f$ is above a fixed threshold. Such regions are hard to visualize if the input space dimension, $d$, is higher than 2. For a given projection matrix from the input space to a lower dimensional (usually $1,2$) subspace, we introduce profile sup (inf) functions that associate to each poi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.00688v2-abstract-full').style.display = 'inline'; document.getElementById('1710.00688v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.00688v2-abstract-full" style="display: none;"> We consider the problem of describing excursion sets of a real-valued function $f$, i.e. the set of inputs where $f$ is above a fixed threshold. Such regions are hard to visualize if the input space dimension, $d$, is higher than 2. For a given projection matrix from the input space to a lower dimensional (usually $1,2$) subspace, we introduce profile sup (inf) functions that associate to each point in the projection's image the sup (inf) of the function constrained over the pre-image of this point by the considered projection. Plots of profile extrema functions convey a simple, although intrinsically partial, visualization of the set. We consider expensive to evaluate functions where only a very limited number of evaluations, $n$, is available, e.g. $n<100d$, and we surrogate $f$ with a posterior quantity of a Gaussian process (GP) model. We first compute profile extrema functions for the posterior mean given $n$ evaluations of $f$. We quantify the uncertainty on such estimates by studying the distribution of GP profile extrema with posterior quasi-realizations obtained from an approximating process. We control such approximation with a bound inherited from the Borell-TIS inequality. The technique is applied to analytical functions ($d=2,3$) and to a $5$-dimensional coastal flooding test case for a site located on the Atlantic French coast. Here $f$ is a numerical model returning the area of flooded surface in the coastal region given some offshore conditions. Profile extrema functions allowed us to better understand which offshore conditions impact large flooding events. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.00688v2-abstract-full').style.display = 'none'; document.getElementById('1710.00688v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Technometrics, 2019 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1704.05318">arXiv:1704.05318</a> <span> [<a href="https://arxiv.org/pdf/1704.05318">pdf</a>, <a href="https://arxiv.org/format/1704.05318">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> On the choice of the low-dimensional domain for global optimization via random embeddings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Binois%2C+M">Micka毛l Binois</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1704.05318v3-abstract-short" style="display: inline;"> The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1704.05318v3-abstract-full').style.display = 'inline'; document.getElementById('1704.05318v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1704.05318v3-abstract-full" style="display: none;"> The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budget of evaluations is severely limited, global optimization with random embeddings appears as a sound alternative to random search. Yet, in the case of box constraints on the native variables, defining suitable bounds on a low dimensional domain appears to be complex. Indeed, a small search domain does not guarantee to find a solution even under restrictive hypotheses about the function, while a larger one may slow down convergence dramatically. Here we tackle the issue of low-dimensional domain selection based on a detailed study of the properties of the random embedding, giving insight on the aforementioned difficulties. In particular, we describe a minimal low-dimensional set in correspondence with the embedded search space. We additionally show that an alternative equivalent embedding procedure yields simultaneously a simpler definition of the low-dimensional minimal set and better properties in practice. Finally, the performance and robustness gains of the proposed enhancements for Bayesian optimization are illustrated on numerical examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1704.05318v3-abstract-full').style.display = 'none'; document.getElementById('1704.05318v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 April, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1611.07256">arXiv:1611.07256</a> <span> [<a href="https://arxiv.org/pdf/1611.07256">pdf</a>, <a href="https://arxiv.org/format/1611.07256">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/00401706.2019.1693427">10.1080/00401706.2019.1693427 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Adaptive Design of Experiments for Conservative Estimation of Excursion Sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Azzimonti%2C+D">Dario Azzimonti</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Bect%2C+J">Julien Bect</a>, <a href="/search/stat?searchtype=author&query=Richet%2C+Y">Yann Richet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1611.07256v6-abstract-short" style="display: inline;"> We consider the problem of estimating the set of all inputs that leads a system to some particular behavior. The system is modeled by an expensive-to-evaluate function, such as a computer experiment, and we are interested in its excursion set, i.e. the set of points where the function takes values above or below some prescribed threshold. The objective function is emulated with a Gaussian Process… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.07256v6-abstract-full').style.display = 'inline'; document.getElementById('1611.07256v6-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1611.07256v6-abstract-full" style="display: none;"> We consider the problem of estimating the set of all inputs that leads a system to some particular behavior. The system is modeled by an expensive-to-evaluate function, such as a computer experiment, and we are interested in its excursion set, i.e. the set of points where the function takes values above or below some prescribed threshold. The objective function is emulated with a Gaussian Process (GP) model based on an initial design of experiments enriched with evaluation results at (batch-)sequentially determined input points. The GP model provides conservative estimates for the excursion set, which control false positives while minimizing false negatives. We introduce adaptive strategies that sequentially select new evaluations of the function by reducing the uncertainty on conservative estimates. Following the Stepwise Uncertainty Reduction approach we obtain new evaluations by minimizing adapted criteria. Tractable formulae for the conservative criteria are derived, which allow more convenient optimization. The method is benchmarked on random functions generated under the model assumptions in different scenarios of noise and batch size. We then apply it to a reliability engineering test case. Overall, the proposed strategy of minimizing false negatives in conservative estimation achieves competitive performance both in terms of model-based and model-free indicators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.07256v6-abstract-full').style.display = 'none'; document.getElementById('1611.07256v6-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Technometrics, 63(1):13-26, 2021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.02700">arXiv:1609.02700</a> <span> [<a href="https://arxiv.org/pdf/1609.02700">pdf</a>, <a href="https://arxiv.org/ps/1609.02700">ps</a>, <a href="https://arxiv.org/format/1609.02700">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Efficient batch-sequential Bayesian optimization with moments of truncated Gaussian vectors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Marmin%2C+S">S茅bastien Marmin</a>, <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1609.02700v1-abstract-short" style="display: inline;"> We deal with the efficient parallelization of Bayesian global optimization algorithms, and more specifically of those based on the expected improvement criterion and its variants. A closed form formula relying on multivariate Gaussian cumulative distribution functions is established for a generalized version of the multipoint expected improvement criterion. In turn, the latter relies on intermedia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.02700v1-abstract-full').style.display = 'inline'; document.getElementById('1609.02700v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.02700v1-abstract-full" style="display: none;"> We deal with the efficient parallelization of Bayesian global optimization algorithms, and more specifically of those based on the expected improvement criterion and its variants. A closed form formula relying on multivariate Gaussian cumulative distribution functions is established for a generalized version of the multipoint expected improvement criterion. In turn, the latter relies on intermediate results that could be of independent interest concerning moments of truncated Gaussian vectors. The obtained expansion of the criterion enables studying its differentiability with respect to point batches and calculating the corresponding gradient in closed form. Furthermore , we derive fast numerical approximations of this gradient and propose efficient batch optimization strategies. Numerical experiments illustrate that the proposed approaches enable computational savings of between one and two order of magnitudes, hence enabling derivative-based batch-sequential acquisition function maximization to become a practically implementable and efficient standard. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.02700v1-abstract-full').style.display = 'none'; document.getElementById('1609.02700v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.01118">arXiv:1608.01118</a> <span> [<a href="https://arxiv.org/pdf/1608.01118">pdf</a>, <a href="https://arxiv.org/ps/1608.01118">ps</a>, <a href="https://arxiv.org/format/1608.01118">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> A supermartingale approach to Gaussian process based sequential design of experiments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Bect%2C+J">Julien Bect</a>, <a href="/search/stat?searchtype=author&query=Bachoc%2C+F">Fran莽ois Bachoc</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1608.01118v3-abstract-short" style="display: inline;"> Gaussian process (GP) models have become a well-established frameworkfor the adaptive design of costly experiments, and notably of computerexperiments. GP-based sequential designs have been found practicallyefficient for various objectives, such as global optimization(estimating the global maximum or maximizer(s) of a function),reliability analysis (estimating a probability of failure) or theesti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.01118v3-abstract-full').style.display = 'inline'; document.getElementById('1608.01118v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.01118v3-abstract-full" style="display: none;"> Gaussian process (GP) models have become a well-established frameworkfor the adaptive design of costly experiments, and notably of computerexperiments. GP-based sequential designs have been found practicallyefficient for various objectives, such as global optimization(estimating the global maximum or maximizer(s) of a function),reliability analysis (estimating a probability of failure) or theestimation of level sets and excursion sets. In this paper, we studythe consistency of an important class of sequential designs, known asstepwise uncertainty reduction (SUR) strategies. Our approach relieson the key observation that the sequence of residual uncertaintymeasures, in SUR strategies, is generally a supermartingale withrespect to the filtration generated by the observations. Thisobservation enables us to establish generic consistency results for abroad class of SUR strategies. The consistency of several popularsequential design strategies is then obtained by means of this generalresult. Notably, we establish the consistency of two SUR strategiesproposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comp.,2012)---to the best of our knowledge, these are the first proofs ofconsistency for GP-based sequential design algorithms dedicated to theestimation of excursion sets and their measure. We also establish anew, more general proof of consistency for the expected improvementalgorithm for global optimization which, unlike previous results inthe literature, applies to any GP with continuous sample paths. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.01118v3-abstract-full').style.display = 'none'; document.getElementById('1608.01118v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.05031">arXiv:1603.05031</a> <span> [<a href="https://arxiv.org/pdf/1603.05031">pdf</a>, <a href="https://arxiv.org/format/1603.05031">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/10618600.2017.1360781">10.1080/10618600.2017.1360781 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Azzimonti%2C+D">Dario Azzimonti</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1603.05031v3-abstract-short" style="display: inline;"> The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.05031v3-abstract-full').style.display = 'inline'; document.getElementById('1603.05031v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.05031v3-abstract-full" style="display: none;"> The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.05031v3-abstract-full').style.display = 'none'; document.getElementById('1603.05031v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Computational and Graphical Statistics, Taylor \& Francis, 2018, 27 (2), pp.255-267 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.05509">arXiv:1503.05509</a> <span> [<a href="https://arxiv.org/pdf/1503.05509">pdf</a>, <a href="https://arxiv.org/ps/1503.05509">ps</a>, <a href="https://arxiv.org/format/1503.05509">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Differentiating the multipoint Expected Improvement for optimal batch design </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Marmin%2C+S">S茅bastien Marmin</a>, <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.05509v4-abstract-short" style="display: inline;"> This work deals with parallel optimization of expensive objective functions which are modeled as sample realizations of Gaussian processes. The study is formalized as a Bayesian optimization problem, or continuous multi-armed bandit problem, where a batch of q > 0 arms is pulled in parallel at each iteration. Several algorithms have been developed for choosing batches by trading off exploitation a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.05509v4-abstract-full').style.display = 'inline'; document.getElementById('1503.05509v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.05509v4-abstract-full" style="display: none;"> This work deals with parallel optimization of expensive objective functions which are modeled as sample realizations of Gaussian processes. The study is formalized as a Bayesian optimization problem, or continuous multi-armed bandit problem, where a batch of q > 0 arms is pulled in parallel at each iteration. Several algorithms have been developed for choosing batches by trading off exploitation and exploration. As of today, the maximum Expected Improvement (EI) and Upper Confidence Bound (UCB) selection rules appear as the most prominent approaches for batch selection. Here, we build upon recent work on the multipoint Expected Improvement criterion, for which an analytic expansion relying on Tallis' formula was recently established. The computational burden of this selection rule being still an issue in application, we derive a closed-form expression for the gradient of the multipoint Expected Improvement, which aims at facilitating its maximization using gradient-based ascent algorithms. Substantial computational savings are shown in application. In addition, our algorithms are tested numerically and compared to state-of-the-art UCB-based batch-sequential algorithms. Combining starting designs relying on UCB with gradient-based EI local optimization finally appears as a sound option for batch design in distributed Gaussian Process optimization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.05509v4-abstract-full').style.display = 'none'; document.getElementById('1503.05509v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1501.03659">arXiv:1501.03659</a> <span> [<a href="https://arxiv.org/pdf/1501.03659">pdf</a>, <a href="https://arxiv.org/ps/1501.03659">ps</a>, <a href="https://arxiv.org/format/1501.03659">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1137/141000749">10.1137/141000749 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantifying uncertainties on excursion sets under a Gaussian random field prior </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Azzimonti%2C+D">Dario Azzimonti</a>, <a href="/search/stat?searchtype=author&query=Bect%2C+J">Julien Bect</a>, <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1501.03659v2-abstract-short" style="display: inline;"> We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approache… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1501.03659v2-abstract-full').style.display = 'inline'; document.getElementById('1501.03659v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1501.03659v2-abstract-full" style="display: none;"> We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approaches exist to summarize the distribution of such sets based on random closed set theory. While the recently proposed Vorob'ev approach exploits analytical formulae, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose Monte Carlo simulation points and obtain quasi-realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they minimize the posterior expected distance in measure between the excursion set and its reconstruction. The proposed method reduces the computational costs due to Monte Carlo simulations and enables the computation of quasi-realizations on fine designs in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present a safety engineering test case where the simulation method is employed to compute a Monte Carlo estimate of a contour line. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1501.03659v2-abstract-full').style.display = 'none'; document.getElementById('1501.03659v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIAM/ASA Journal on Uncertainty Quantification, 4(1):850-874, 2016 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1411.3685">arXiv:1411.3685</a> <span> [<a href="https://arxiv.org/pdf/1411.3685">pdf</a>, <a href="https://arxiv.org/format/1411.3685">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> A warped kernel improving robustness in Bayesian optimization via random embeddings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Binois%2C+M">Micka毛l Binois</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1411.3685v3-abstract-short" style="display: inline;"> This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.3685v3-abstract-full').style.display = 'inline'; document.getElementById('1411.3685v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.3685v3-abstract-full" style="display: none;"> This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates constraints on bound selection for the embedded domain, thus improving the robustness, as illustrated with a test case with 25 variables and intrinsic dimension 6. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.3685v3-abstract-full').style.display = 'none'; document.getElementById('1411.3685v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1308.1359">arXiv:1308.1359</a> <span> [<a href="https://arxiv.org/pdf/1308.1359">pdf</a>, <a href="https://arxiv.org/format/1308.1359">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Invariances of random fields paths, with applications in Gaussian Process Regression </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a>, <a href="/search/stat?searchtype=author&query=Durrande%2C+N">Nicolas Durrande</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1308.1359v1-abstract-short" style="display: inline;"> We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Lo猫ve isometry. S… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1308.1359v1-abstract-full').style.display = 'inline'; document.getElementById('1308.1359v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1308.1359v1-abstract-full" style="display: none;"> We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Lo猫ve isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process regression. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1308.1359v1-abstract-full').style.display = 'none'; document.getElementById('1308.1359v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 August, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2013. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1203.6452">arXiv:1203.6452</a> <span> [<a href="https://arxiv.org/pdf/1203.6452">pdf</a>, <a href="https://arxiv.org/ps/1203.6452">ps</a>, <a href="https://arxiv.org/format/1203.6452">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> </div> </div> <p class="title is-5 mathjax"> Corrected Kriging update formulae for batch-sequential data assimilation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Chevalier%2C+C">Cl茅ment Chevalier</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1203.6452v1-abstract-short" style="display: inline;"> Recently, a lot of effort has been paid to the efficient computation of Kriging predictors when observations are assimilated sequentially. In particular, Kriging update formulae enabling significant computational savings were derived in Barnes and Watson (1992), Gao et al. (1996), and Emery (2009). Taking advantage of the previous Kriging mean and variance calculations helps avoiding a costly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.6452v1-abstract-full').style.display = 'inline'; document.getElementById('1203.6452v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1203.6452v1-abstract-full" style="display: none;"> Recently, a lot of effort has been paid to the efficient computation of Kriging predictors when observations are assimilated sequentially. In particular, Kriging update formulae enabling significant computational savings were derived in Barnes and Watson (1992), Gao et al. (1996), and Emery (2009). Taking advantage of the previous Kriging mean and variance calculations helps avoiding a costly $(n+1) \times (n+1)$ matrix inversion when adding one observation to the $n$ already available ones. In addition to traditional update formulae taking into account a single new observation, Emery (2009) also proposed formulae for the batch-sequential case, i.e. when $r > 1$ new observations are simultaneously assimilated. However, the Kriging variance and covariance formulae given without proof in Emery (2009) for the batch-sequential case are not correct. In this paper we fix this issue and establish corrected expressions for updated Kriging variances and covariances when assimilating several observations in parallel. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.6452v1-abstract-full').style.display = 'none'; document.getElementById('1203.6452v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 March, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1111.6233">arXiv:1111.6233</a> <span> [<a href="https://arxiv.org/pdf/1111.6233">pdf</a>, <a href="https://arxiv.org/ps/1111.6233">ps</a>, <a href="https://arxiv.org/format/1111.6233">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Additive Covariance Kernels for High-Dimensional Gaussian Process Modeling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Durrande%2C+N">Nicolas Durrande</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a>, <a href="/search/stat?searchtype=author&query=Carraro%2C+L">Laurent Carraro</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1111.6233v1-abstract-short" style="display: inline;"> Gaussian process models -also called Kriging models- are often used as mathematical approximations of expensive experiments. However, the number of observation required for building an emulator becomes unrealistic when using classical covariance kernels when the dimension of input increases. In oder to get round the curse of dimensionality, a popular approach is to consider simplified models such… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.6233v1-abstract-full').style.display = 'inline'; document.getElementById('1111.6233v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1111.6233v1-abstract-full" style="display: none;"> Gaussian process models -also called Kriging models- are often used as mathematical approximations of expensive experiments. However, the number of observation required for building an emulator becomes unrealistic when using classical covariance kernels when the dimension of input increases. In oder to get round the curse of dimensionality, a popular approach is to consider simplified models such as additive models. The ambition of the present work is to give an insight into covariance kernels that are well suited for building additive Kriging models and to describe some properties of the resulting models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.6233v1-abstract-full').style.display = 'none'; document.getElementById('1111.6233v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:1103.4023</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Annales de la Facult茅 de Sciences de Toulouse Tome 21, num茅ro 3 (2012) p. 481-499 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1106.3571">arXiv:1106.3571</a> <span> [<a href="https://arxiv.org/pdf/1106.3571">pdf</a>, <a href="https://arxiv.org/ps/1106.3571">ps</a>, <a href="https://arxiv.org/format/1106.3571">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Durrande%2C+N">Nicolas Durrande</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a>, <a href="/search/stat?searchtype=author&query=Carraro%2C+L">Laurent Carraro</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1106.3571v2-abstract-short" style="display: inline;"> Given a reproducing kernel Hilbert space H of real-valued functions and a suitable measure mu over the source space D (subset of R), we decompose H as the sum of a subspace of centered functions for mu and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an inter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.3571v2-abstract-full').style.display = 'inline'; document.getElementById('1106.3571v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1106.3571v2-abstract-full" style="display: none;"> Given a reproducing kernel Hilbert space H of real-valued functions and a suitable measure mu over the source space D (subset of R), we decompose H as the sum of a subspace of centered functions for mu and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the e ffect of each (group of) variable(s) and computing sensitivity indices without recursivity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.3571v2-abstract-full').style.display = 'none'; document.getElementById('1106.3571v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 June, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Multivariate Analysis 115 (2013) 57-67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1103.4023">arXiv:1103.4023</a> <span> [<a href="https://arxiv.org/pdf/1103.4023">pdf</a>, <a href="https://arxiv.org/ps/1103.4023">ps</a>, <a href="https://arxiv.org/format/1103.4023">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Additive Kernels for Gaussian Process Modeling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Durrande%2C+N">Nicolas Durrande</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1103.4023v1-abstract-short" style="display: inline;"> Gaussian Process (GP) models are often used as mathematical approximations of computationally expensive experiments. Provided that its kernel is suitably chosen and that enough data is available to obtain a reasonable fit of the simulator, a GP model can beneficially be used for tasks such as prediction, optimization, or Monte-Carlo-based quantification of uncertainty. However, the former conditio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1103.4023v1-abstract-full').style.display = 'inline'; document.getElementById('1103.4023v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1103.4023v1-abstract-full" style="display: none;"> Gaussian Process (GP) models are often used as mathematical approximations of computationally expensive experiments. Provided that its kernel is suitably chosen and that enough data is available to obtain a reasonable fit of the simulator, a GP model can beneficially be used for tasks such as prediction, optimization, or Monte-Carlo-based quantification of uncertainty. However, the former conditions become unrealistic when using classical GPs as the dimension of input increases. One popular alternative is then to turn to Generalized Additive Models (GAMs), relying on the assumption that the simulator's response can approximately be decomposed as a sum of univariate functions. If such an approach has been successfully applied in approximation, it is nevertheless not completely compatible with the GP framework and its versatile applications. The ambition of the present work is to give an insight into the use of GPs for additive models by integrating additivity within the kernel, and proposing a parsimonious numerical method for data-driven parameter estimation. The first part of this article deals with the kernels naturally associated to additive processes and the properties of the GP models based on such kernels. The second part is dedicated to a numerical procedure based on relaxation for additive kernel parameter estimation. Finally, the efficiency of the proposed method is illustrated and compared to other approaches on Sobol's g-function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1103.4023v1-abstract-full').style.display = 'none'; document.getElementById('1103.4023v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 March, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2011. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1009.5177">arXiv:1009.5177</a> <span> [<a href="https://arxiv.org/pdf/1009.5177">pdf</a>, <a href="https://arxiv.org/ps/1009.5177">ps</a>, <a href="https://arxiv.org/format/1009.5177">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11222-011-9241-4">10.1007/s11222-011-9241-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Sequential design of computer experiments for the estimation of a probability of failure </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Bect%2C+J">Julien Bect</a>, <a href="/search/stat?searchtype=author&query=Ginsbourger%2C+D">David Ginsbourger</a>, <a href="/search/stat?searchtype=author&query=Li%2C+L">Ling Li</a>, <a href="/search/stat?searchtype=author&query=Picheny%2C+V">Victor Picheny</a>, <a href="/search/stat?searchtype=author&query=Vazquez%2C+E">Emmanuel Vazquez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1009.5177v2-abstract-short" style="display: inline;"> This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is availab… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.5177v2-abstract-full').style.display = 'inline'; document.getElementById('1009.5177v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1009.5177v2-abstract-full" style="display: none;"> This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of $f$ and aim at performing evaluations of $f$ as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.5177v2-abstract-full').style.display = 'none'; document.getElementById('1009.5177v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 April, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 September, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This is an author-generated postprint version. The published version is available at http://www.springerlink.com</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62L05; 62C10; 62P30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Statistics and Computing, 22(3):773-793, 2012 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" 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