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Literature review
<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Literature review</title> <!--Generated on Thu Apr 3 18:25:41 2025 by LaTeXML (version 0.8.8) http://dlmf.nist.gov/LaTeXML/.--> <!--Document created on January 2024.--> <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2504.02960v1/"/></head> <body> <nav class="ltx_page_navbar"> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Literature review</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Vaidehi Dixit, Scott Holan, Christopher Wikle </span></span> </div> <div class="ltx_dates">(January 2024)</div> <div class="ltx_para" id="p1"> <ul class="ltx_itemize" id="S0.I1"> <li class="ltx_item" id="S0.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S0.I1.i1.p1"> <p class="ltx_p" id="S0.I1.i1.p1.1">Complete spatial randomness (CSR) and homogeneity of Poisson processes are equivalent. Find below testing for homogeneity of a Poisson process <cite class="ltx_cite ltx_citemacro_citep">(<span class="ltx_ref ltx_missing_citation ltx_ref_self">chiu2009homogeneity</span>)</cite>.</p> </div> <div class="ltx_para" id="S0.I1.i1.p2"> <p class="ltx_p" id="S0.I1.i1.p2.11">“Suppose we observe <math alttext="K" class="ltx_Math" display="inline" id="S0.I1.i1.p2.1.m1.1"><semantics id="S0.I1.i1.p2.1.m1.1a"><mi id="S0.I1.i1.p2.1.m1.1.1" xref="S0.I1.i1.p2.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.1.m1.1b"><ci id="S0.I1.i1.p2.1.m1.1.1.cmml" xref="S0.I1.i1.p2.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.1.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.1.m1.1d">italic_K</annotation></semantics></math> independent random variables <math alttext="X_{1},\ldots,X_{K}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.2.m2.3"><semantics id="S0.I1.i1.p2.2.m2.3a"><mrow id="S0.I1.i1.p2.2.m2.3.3.2" xref="S0.I1.i1.p2.2.m2.3.3.3.cmml"><msub id="S0.I1.i1.p2.2.m2.2.2.1.1" xref="S0.I1.i1.p2.2.m2.2.2.1.1.cmml"><mi id="S0.I1.i1.p2.2.m2.2.2.1.1.2" xref="S0.I1.i1.p2.2.m2.2.2.1.1.2.cmml">X</mi><mn id="S0.I1.i1.p2.2.m2.2.2.1.1.3" xref="S0.I1.i1.p2.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S0.I1.i1.p2.2.m2.3.3.2.3" xref="S0.I1.i1.p2.2.m2.3.3.3.cmml">,</mo><mi id="S0.I1.i1.p2.2.m2.1.1" mathvariant="normal" xref="S0.I1.i1.p2.2.m2.1.1.cmml">…</mi><mo id="S0.I1.i1.p2.2.m2.3.3.2.4" xref="S0.I1.i1.p2.2.m2.3.3.3.cmml">,</mo><msub id="S0.I1.i1.p2.2.m2.3.3.2.2" xref="S0.I1.i1.p2.2.m2.3.3.2.2.cmml"><mi id="S0.I1.i1.p2.2.m2.3.3.2.2.2" xref="S0.I1.i1.p2.2.m2.3.3.2.2.2.cmml">X</mi><mi id="S0.I1.i1.p2.2.m2.3.3.2.2.3" xref="S0.I1.i1.p2.2.m2.3.3.2.2.3.cmml">K</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.2.m2.3b"><list id="S0.I1.i1.p2.2.m2.3.3.3.cmml" xref="S0.I1.i1.p2.2.m2.3.3.2"><apply id="S0.I1.i1.p2.2.m2.2.2.1.1.cmml" xref="S0.I1.i1.p2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.2.m2.2.2.1.1.1.cmml" xref="S0.I1.i1.p2.2.m2.2.2.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.2.m2.2.2.1.1.2.cmml" xref="S0.I1.i1.p2.2.m2.2.2.1.1.2">𝑋</ci><cn id="S0.I1.i1.p2.2.m2.2.2.1.1.3.cmml" type="integer" xref="S0.I1.i1.p2.2.m2.2.2.1.1.3">1</cn></apply><ci id="S0.I1.i1.p2.2.m2.1.1.cmml" xref="S0.I1.i1.p2.2.m2.1.1">…</ci><apply id="S0.I1.i1.p2.2.m2.3.3.2.2.cmml" xref="S0.I1.i1.p2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S0.I1.i1.p2.2.m2.3.3.2.2.1.cmml" xref="S0.I1.i1.p2.2.m2.3.3.2.2">subscript</csymbol><ci id="S0.I1.i1.p2.2.m2.3.3.2.2.2.cmml" xref="S0.I1.i1.p2.2.m2.3.3.2.2.2">𝑋</ci><ci id="S0.I1.i1.p2.2.m2.3.3.2.2.3.cmml" xref="S0.I1.i1.p2.2.m2.3.3.2.2.3">𝐾</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.2.m2.3c">X_{1},\ldots,X_{K}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.2.m2.3d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_X start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT</annotation></semantics></math> and each <math alttext="X_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.3.m3.1"><semantics id="S0.I1.i1.p2.3.m3.1a"><msub id="S0.I1.i1.p2.3.m3.1.1" xref="S0.I1.i1.p2.3.m3.1.1.cmml"><mi id="S0.I1.i1.p2.3.m3.1.1.2" xref="S0.I1.i1.p2.3.m3.1.1.2.cmml">X</mi><mi id="S0.I1.i1.p2.3.m3.1.1.3" xref="S0.I1.i1.p2.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.3.m3.1b"><apply id="S0.I1.i1.p2.3.m3.1.1.cmml" xref="S0.I1.i1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.3.m3.1.1.1.cmml" xref="S0.I1.i1.p2.3.m3.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.3.m3.1.1.2.cmml" xref="S0.I1.i1.p2.3.m3.1.1.2">𝑋</ci><ci id="S0.I1.i1.p2.3.m3.1.1.3.cmml" xref="S0.I1.i1.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.3.m3.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> follows a Poisson distribution with mean <math alttext="\lambda_{i}t_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.4.m4.1"><semantics id="S0.I1.i1.p2.4.m4.1a"><mrow id="S0.I1.i1.p2.4.m4.1.1" xref="S0.I1.i1.p2.4.m4.1.1.cmml"><msub id="S0.I1.i1.p2.4.m4.1.1.2" xref="S0.I1.i1.p2.4.m4.1.1.2.cmml"><mi id="S0.I1.i1.p2.4.m4.1.1.2.2" xref="S0.I1.i1.p2.4.m4.1.1.2.2.cmml">λ</mi><mi id="S0.I1.i1.p2.4.m4.1.1.2.3" xref="S0.I1.i1.p2.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S0.I1.i1.p2.4.m4.1.1.1" xref="S0.I1.i1.p2.4.m4.1.1.1.cmml"></mo><msub id="S0.I1.i1.p2.4.m4.1.1.3" xref="S0.I1.i1.p2.4.m4.1.1.3.cmml"><mi id="S0.I1.i1.p2.4.m4.1.1.3.2" xref="S0.I1.i1.p2.4.m4.1.1.3.2.cmml">t</mi><mi id="S0.I1.i1.p2.4.m4.1.1.3.3" xref="S0.I1.i1.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.4.m4.1b"><apply id="S0.I1.i1.p2.4.m4.1.1.cmml" xref="S0.I1.i1.p2.4.m4.1.1"><times id="S0.I1.i1.p2.4.m4.1.1.1.cmml" xref="S0.I1.i1.p2.4.m4.1.1.1"></times><apply id="S0.I1.i1.p2.4.m4.1.1.2.cmml" xref="S0.I1.i1.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S0.I1.i1.p2.4.m4.1.1.2.1.cmml" xref="S0.I1.i1.p2.4.m4.1.1.2">subscript</csymbol><ci id="S0.I1.i1.p2.4.m4.1.1.2.2.cmml" xref="S0.I1.i1.p2.4.m4.1.1.2.2">𝜆</ci><ci id="S0.I1.i1.p2.4.m4.1.1.2.3.cmml" xref="S0.I1.i1.p2.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S0.I1.i1.p2.4.m4.1.1.3.cmml" xref="S0.I1.i1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S0.I1.i1.p2.4.m4.1.1.3.1.cmml" xref="S0.I1.i1.p2.4.m4.1.1.3">subscript</csymbol><ci id="S0.I1.i1.p2.4.m4.1.1.3.2.cmml" xref="S0.I1.i1.p2.4.m4.1.1.3.2">𝑡</ci><ci id="S0.I1.i1.p2.4.m4.1.1.3.3.cmml" xref="S0.I1.i1.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.4.m4.1c">\lambda_{i}t_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.4.m4.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, in which <math alttext="t_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.5.m5.1"><semantics id="S0.I1.i1.p2.5.m5.1a"><msub id="S0.I1.i1.p2.5.m5.1.1" xref="S0.I1.i1.p2.5.m5.1.1.cmml"><mi id="S0.I1.i1.p2.5.m5.1.1.2" xref="S0.I1.i1.p2.5.m5.1.1.2.cmml">t</mi><mi id="S0.I1.i1.p2.5.m5.1.1.3" xref="S0.I1.i1.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.5.m5.1b"><apply id="S0.I1.i1.p2.5.m5.1.1.cmml" xref="S0.I1.i1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.5.m5.1.1.1.cmml" xref="S0.I1.i1.p2.5.m5.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.5.m5.1.1.2.cmml" xref="S0.I1.i1.p2.5.m5.1.1.2">𝑡</ci><ci id="S0.I1.i1.p2.5.m5.1.1.3.cmml" xref="S0.I1.i1.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.5.m5.1c">t_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.5.m5.1d">italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is known but <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.6.m6.1"><semantics id="S0.I1.i1.p2.6.m6.1a"><msub id="S0.I1.i1.p2.6.m6.1.1" xref="S0.I1.i1.p2.6.m6.1.1.cmml"><mi id="S0.I1.i1.p2.6.m6.1.1.2" xref="S0.I1.i1.p2.6.m6.1.1.2.cmml">λ</mi><mi id="S0.I1.i1.p2.6.m6.1.1.3" xref="S0.I1.i1.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.6.m6.1b"><apply id="S0.I1.i1.p2.6.m6.1.1.cmml" xref="S0.I1.i1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.6.m6.1.1.1.cmml" xref="S0.I1.i1.p2.6.m6.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.6.m6.1.1.2.cmml" xref="S0.I1.i1.p2.6.m6.1.1.2">𝜆</ci><ci id="S0.I1.i1.p2.6.m6.1.1.3.cmml" xref="S0.I1.i1.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.6.m6.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.6.m6.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is not. That is, <math alttext="X_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.7.m7.1"><semantics id="S0.I1.i1.p2.7.m7.1a"><msub id="S0.I1.i1.p2.7.m7.1.1" xref="S0.I1.i1.p2.7.m7.1.1.cmml"><mi id="S0.I1.i1.p2.7.m7.1.1.2" xref="S0.I1.i1.p2.7.m7.1.1.2.cmml">X</mi><mi id="S0.I1.i1.p2.7.m7.1.1.3" xref="S0.I1.i1.p2.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.7.m7.1b"><apply id="S0.I1.i1.p2.7.m7.1.1.cmml" xref="S0.I1.i1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.7.m7.1.1.1.cmml" xref="S0.I1.i1.p2.7.m7.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.7.m7.1.1.2.cmml" xref="S0.I1.i1.p2.7.m7.1.1.2">𝑋</ci><ci id="S0.I1.i1.p2.7.m7.1.1.3.cmml" xref="S0.I1.i1.p2.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.7.m7.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.7.m7.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the observed number of occurrences of a temporal (or spatial, respectively) Poisson process with rate (or intensity) <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.8.m8.1"><semantics id="S0.I1.i1.p2.8.m8.1a"><msub id="S0.I1.i1.p2.8.m8.1.1" xref="S0.I1.i1.p2.8.m8.1.1.cmml"><mi id="S0.I1.i1.p2.8.m8.1.1.2" xref="S0.I1.i1.p2.8.m8.1.1.2.cmml">λ</mi><mi id="S0.I1.i1.p2.8.m8.1.1.3" xref="S0.I1.i1.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.8.m8.1b"><apply id="S0.I1.i1.p2.8.m8.1.1.cmml" xref="S0.I1.i1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.8.m8.1.1.1.cmml" xref="S0.I1.i1.p2.8.m8.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.8.m8.1.1.2.cmml" xref="S0.I1.i1.p2.8.m8.1.1.2">𝜆</ci><ci id="S0.I1.i1.p2.8.m8.1.1.3.cmml" xref="S0.I1.i1.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.8.m8.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.8.m8.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in a sampling frame of length (or area) <math alttext="t_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.9.m9.1"><semantics id="S0.I1.i1.p2.9.m9.1a"><msub id="S0.I1.i1.p2.9.m9.1.1" xref="S0.I1.i1.p2.9.m9.1.1.cmml"><mi id="S0.I1.i1.p2.9.m9.1.1.2" xref="S0.I1.i1.p2.9.m9.1.1.2.cmml">t</mi><mi id="S0.I1.i1.p2.9.m9.1.1.3" xref="S0.I1.i1.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.9.m9.1b"><apply id="S0.I1.i1.p2.9.m9.1.1.cmml" xref="S0.I1.i1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.9.m9.1.1.1.cmml" xref="S0.I1.i1.p2.9.m9.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.9.m9.1.1.2.cmml" xref="S0.I1.i1.p2.9.m9.1.1.2">𝑡</ci><ci id="S0.I1.i1.p2.9.m9.1.1.3.cmml" xref="S0.I1.i1.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.9.m9.1c">t_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.9.m9.1d">italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="i=1,\ldots,K" class="ltx_Math" display="inline" id="S0.I1.i1.p2.10.m10.3"><semantics id="S0.I1.i1.p2.10.m10.3a"><mrow id="S0.I1.i1.p2.10.m10.3.4" xref="S0.I1.i1.p2.10.m10.3.4.cmml"><mi id="S0.I1.i1.p2.10.m10.3.4.2" xref="S0.I1.i1.p2.10.m10.3.4.2.cmml">i</mi><mo id="S0.I1.i1.p2.10.m10.3.4.1" xref="S0.I1.i1.p2.10.m10.3.4.1.cmml">=</mo><mrow id="S0.I1.i1.p2.10.m10.3.4.3.2" xref="S0.I1.i1.p2.10.m10.3.4.3.1.cmml"><mn id="S0.I1.i1.p2.10.m10.1.1" xref="S0.I1.i1.p2.10.m10.1.1.cmml">1</mn><mo id="S0.I1.i1.p2.10.m10.3.4.3.2.1" xref="S0.I1.i1.p2.10.m10.3.4.3.1.cmml">,</mo><mi id="S0.I1.i1.p2.10.m10.2.2" mathvariant="normal" xref="S0.I1.i1.p2.10.m10.2.2.cmml">…</mi><mo id="S0.I1.i1.p2.10.m10.3.4.3.2.2" xref="S0.I1.i1.p2.10.m10.3.4.3.1.cmml">,</mo><mi id="S0.I1.i1.p2.10.m10.3.3" xref="S0.I1.i1.p2.10.m10.3.3.cmml">K</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.10.m10.3b"><apply id="S0.I1.i1.p2.10.m10.3.4.cmml" xref="S0.I1.i1.p2.10.m10.3.4"><eq id="S0.I1.i1.p2.10.m10.3.4.1.cmml" xref="S0.I1.i1.p2.10.m10.3.4.1"></eq><ci id="S0.I1.i1.p2.10.m10.3.4.2.cmml" xref="S0.I1.i1.p2.10.m10.3.4.2">𝑖</ci><list id="S0.I1.i1.p2.10.m10.3.4.3.1.cmml" xref="S0.I1.i1.p2.10.m10.3.4.3.2"><cn id="S0.I1.i1.p2.10.m10.1.1.cmml" type="integer" xref="S0.I1.i1.p2.10.m10.1.1">1</cn><ci id="S0.I1.i1.p2.10.m10.2.2.cmml" xref="S0.I1.i1.p2.10.m10.2.2">…</ci><ci id="S0.I1.i1.p2.10.m10.3.3.cmml" xref="S0.I1.i1.p2.10.m10.3.3">𝐾</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.10.m10.3c">i=1,\ldots,K</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.10.m10.3d">italic_i = 1 , … , italic_K</annotation></semantics></math> . Hence, the density function of <math alttext="X_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.11.m11.1"><semantics id="S0.I1.i1.p2.11.m11.1a"><msub id="S0.I1.i1.p2.11.m11.1.1" xref="S0.I1.i1.p2.11.m11.1.1.cmml"><mi id="S0.I1.i1.p2.11.m11.1.1.2" xref="S0.I1.i1.p2.11.m11.1.1.2.cmml">X</mi><mi id="S0.I1.i1.p2.11.m11.1.1.3" xref="S0.I1.i1.p2.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.11.m11.1b"><apply id="S0.I1.i1.p2.11.m11.1.1.cmml" xref="S0.I1.i1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.11.m11.1.1.1.cmml" xref="S0.I1.i1.p2.11.m11.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.11.m11.1.1.2.cmml" xref="S0.I1.i1.p2.11.m11.1.1.2">𝑋</ci><ci id="S0.I1.i1.p2.11.m11.1.1.3.cmml" xref="S0.I1.i1.p2.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.11.m11.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.11.m11.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is,</p> <table class="ltx_equation ltx_eqn_table" id="S0.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(x_{i})=\frac{\exp^{-\lambda_{i}t_{i}}(\lambda_{i}t_{i})^{x_{i}}}{x_{i}!},% \quad i=1,\ldots,K" class="ltx_math_unparsed" display="block" id="S0.Ex1.m1.5"><semantics id="S0.Ex1.m1.5a"><mrow id="S0.Ex1.m1.5.5.2"><mrow id="S0.Ex1.m1.4.4.1.1"><mrow id="S0.Ex1.m1.4.4.1.1.1"><mi id="S0.Ex1.m1.4.4.1.1.1.3">f</mi><mo id="S0.Ex1.m1.4.4.1.1.1.2"></mo><mrow id="S0.Ex1.m1.4.4.1.1.1.1.1"><mo id="S0.Ex1.m1.4.4.1.1.1.1.1.2" stretchy="false">(</mo><msub id="S0.Ex1.m1.4.4.1.1.1.1.1.1"><mi id="S0.Ex1.m1.4.4.1.1.1.1.1.1.2">x</mi><mi id="S0.Ex1.m1.4.4.1.1.1.1.1.1.3">i</mi></msub><mo id="S0.Ex1.m1.4.4.1.1.1.1.1.3" stretchy="false">)</mo></mrow></mrow><mo id="S0.Ex1.m1.4.4.1.1.2">=</mo><mfrac id="S0.Ex1.m1.4.4.1.1.3"><mrow id="S0.Ex1.m1.4.4.1.1.3.2"><msup id="S0.Ex1.m1.4.4.1.1.3.2.1"><mi id="S0.Ex1.m1.4.4.1.1.3.2.1.2">exp</mi><mrow id="S0.Ex1.m1.4.4.1.1.3.2.1.3"><mo id="S0.Ex1.m1.4.4.1.1.3.2.1.3a">−</mo><mrow id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2"><msub id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.2"><mi id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.2.2">λ</mi><mi id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.2.3">i</mi></msub><mo id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.1"></mo><msub id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.3"><mi id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.3.2">t</mi><mi id="S0.Ex1.m1.4.4.1.1.3.2.1.3.2.3.3">i</mi></msub></mrow></mrow></msup><msup id="S0.Ex1.m1.4.4.1.1.3.2.2"><mrow id="S0.Ex1.m1.4.4.1.1.3.2.2.2"><mo id="S0.Ex1.m1.4.4.1.1.3.2.2.2.1" stretchy="false">(</mo><msub id="S0.Ex1.m1.4.4.1.1.3.2.2.2.2"><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.2.2.2">λ</mi><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.2.2.3">i</mi></msub><msub id="S0.Ex1.m1.4.4.1.1.3.2.2.2.3"><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.2.3.2">t</mi><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.2.3.3">i</mi></msub><mo id="S0.Ex1.m1.4.4.1.1.3.2.2.2.4" stretchy="false">)</mo></mrow><msub id="S0.Ex1.m1.4.4.1.1.3.2.2.3"><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.3.2">x</mi><mi id="S0.Ex1.m1.4.4.1.1.3.2.2.3.3">i</mi></msub></msup></mrow><mrow id="S0.Ex1.m1.4.4.1.1.3.3"><msub id="S0.Ex1.m1.4.4.1.1.3.3.2"><mi id="S0.Ex1.m1.4.4.1.1.3.3.2.2">x</mi><mi id="S0.Ex1.m1.4.4.1.1.3.3.2.3">i</mi></msub><mo id="S0.Ex1.m1.4.4.1.1.3.3.1">!</mo></mrow></mfrac></mrow><mo id="S0.Ex1.m1.5.5.2.3" rspace="1.167em">,</mo><mrow id="S0.Ex1.m1.5.5.2.2"><mi id="S0.Ex1.m1.5.5.2.2.2">i</mi><mo id="S0.Ex1.m1.5.5.2.2.1">=</mo><mrow id="S0.Ex1.m1.5.5.2.2.3.2"><mn id="S0.Ex1.m1.1.1">1</mn><mo id="S0.Ex1.m1.5.5.2.2.3.2.1">,</mo><mi id="S0.Ex1.m1.2.2" mathvariant="normal">…</mi><mo id="S0.Ex1.m1.5.5.2.2.3.2.2">,</mo><mi id="S0.Ex1.m1.3.3">K</mi></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S0.Ex1.m1.5b">f(x_{i})=\frac{\exp^{-\lambda_{i}t_{i}}(\lambda_{i}t_{i})^{x_{i}}}{x_{i}!},% \quad i=1,\ldots,K</annotation><annotation encoding="application/x-llamapun" id="S0.Ex1.m1.5c">italic_f ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = divide start_ARG roman_exp start_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ! end_ARG , italic_i = 1 , … , italic_K</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S0.I1.i1.p2.13">The maximum likelihood estimate (MLE) of <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.12.m1.1"><semantics id="S0.I1.i1.p2.12.m1.1a"><msub id="S0.I1.i1.p2.12.m1.1.1" xref="S0.I1.i1.p2.12.m1.1.1.cmml"><mi id="S0.I1.i1.p2.12.m1.1.1.2" xref="S0.I1.i1.p2.12.m1.1.1.2.cmml">λ</mi><mi id="S0.I1.i1.p2.12.m1.1.1.3" xref="S0.I1.i1.p2.12.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.12.m1.1b"><apply id="S0.I1.i1.p2.12.m1.1.1.cmml" xref="S0.I1.i1.p2.12.m1.1.1"><csymbol cd="ambiguous" id="S0.I1.i1.p2.12.m1.1.1.1.cmml" xref="S0.I1.i1.p2.12.m1.1.1">subscript</csymbol><ci id="S0.I1.i1.p2.12.m1.1.1.2.cmml" xref="S0.I1.i1.p2.12.m1.1.1.2">𝜆</ci><ci id="S0.I1.i1.p2.12.m1.1.1.3.cmml" xref="S0.I1.i1.p2.12.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.12.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.12.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is simply the empirical rate <math alttext="X_{i}/t_{i}" class="ltx_Math" display="inline" id="S0.I1.i1.p2.13.m2.1"><semantics id="S0.I1.i1.p2.13.m2.1a"><mrow id="S0.I1.i1.p2.13.m2.1.1" xref="S0.I1.i1.p2.13.m2.1.1.cmml"><msub id="S0.I1.i1.p2.13.m2.1.1.2" xref="S0.I1.i1.p2.13.m2.1.1.2.cmml"><mi id="S0.I1.i1.p2.13.m2.1.1.2.2" xref="S0.I1.i1.p2.13.m2.1.1.2.2.cmml">X</mi><mi id="S0.I1.i1.p2.13.m2.1.1.2.3" xref="S0.I1.i1.p2.13.m2.1.1.2.3.cmml">i</mi></msub><mo id="S0.I1.i1.p2.13.m2.1.1.1" xref="S0.I1.i1.p2.13.m2.1.1.1.cmml">/</mo><msub id="S0.I1.i1.p2.13.m2.1.1.3" xref="S0.I1.i1.p2.13.m2.1.1.3.cmml"><mi id="S0.I1.i1.p2.13.m2.1.1.3.2" xref="S0.I1.i1.p2.13.m2.1.1.3.2.cmml">t</mi><mi id="S0.I1.i1.p2.13.m2.1.1.3.3" xref="S0.I1.i1.p2.13.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p2.13.m2.1b"><apply id="S0.I1.i1.p2.13.m2.1.1.cmml" xref="S0.I1.i1.p2.13.m2.1.1"><divide id="S0.I1.i1.p2.13.m2.1.1.1.cmml" xref="S0.I1.i1.p2.13.m2.1.1.1"></divide><apply id="S0.I1.i1.p2.13.m2.1.1.2.cmml" xref="S0.I1.i1.p2.13.m2.1.1.2"><csymbol cd="ambiguous" id="S0.I1.i1.p2.13.m2.1.1.2.1.cmml" xref="S0.I1.i1.p2.13.m2.1.1.2">subscript</csymbol><ci id="S0.I1.i1.p2.13.m2.1.1.2.2.cmml" xref="S0.I1.i1.p2.13.m2.1.1.2.2">𝑋</ci><ci id="S0.I1.i1.p2.13.m2.1.1.2.3.cmml" xref="S0.I1.i1.p2.13.m2.1.1.2.3">𝑖</ci></apply><apply id="S0.I1.i1.p2.13.m2.1.1.3.cmml" xref="S0.I1.i1.p2.13.m2.1.1.3"><csymbol cd="ambiguous" id="S0.I1.i1.p2.13.m2.1.1.3.1.cmml" xref="S0.I1.i1.p2.13.m2.1.1.3">subscript</csymbol><ci id="S0.I1.i1.p2.13.m2.1.1.3.2.cmml" xref="S0.I1.i1.p2.13.m2.1.1.3.2">𝑡</ci><ci id="S0.I1.i1.p2.13.m2.1.1.3.3.cmml" xref="S0.I1.i1.p2.13.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p2.13.m2.1c">X_{i}/t_{i}</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p2.13.m2.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT / italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. By the homogeneity test of K independent Poisson variates, we mean a test for”</p> <table class="ltx_equation ltx_eqn_table" id="S0.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H_{0}:\lambda_{1}=\ldots=\lambda_{K}\quad\text{against}\quad H_{A}:\lambda_{i}% \neq\lambda_{j}\quad\text{for some}\quad i,j\in\{1,\dots,K\}" class="ltx_math_unparsed" display="block" id="S0.Ex2.m1.1"><semantics id="S0.Ex2.m1.1a"><mrow id="S0.Ex2.m1.1b"><msub id="S0.Ex2.m1.1.1"><mi id="S0.Ex2.m1.1.1.2">H</mi><mn id="S0.Ex2.m1.1.1.3">0</mn></msub><mo id="S0.Ex2.m1.1.2" lspace="0.278em" rspace="0.278em">:</mo><msub id="S0.Ex2.m1.1.3"><mi id="S0.Ex2.m1.1.3.2">λ</mi><mn id="S0.Ex2.m1.1.3.3">1</mn></msub><mo id="S0.Ex2.m1.1.4">=</mo><mi id="S0.Ex2.m1.1.5" mathvariant="normal">…</mi><mo id="S0.Ex2.m1.1.6">=</mo><msub id="S0.Ex2.m1.1.7"><mi id="S0.Ex2.m1.1.7.2">λ</mi><mi id="S0.Ex2.m1.1.7.3">K</mi></msub><mspace id="S0.Ex2.m1.1.8" width="1em"></mspace><mtext id="S0.Ex2.m1.1.9">against</mtext><mspace id="S0.Ex2.m1.1.10" width="1em"></mspace><msub id="S0.Ex2.m1.1.11"><mi id="S0.Ex2.m1.1.11.2">H</mi><mi id="S0.Ex2.m1.1.11.3">A</mi></msub><mo id="S0.Ex2.m1.1.12" lspace="0.278em" rspace="0.278em">:</mo><msub id="S0.Ex2.m1.1.13"><mi id="S0.Ex2.m1.1.13.2">λ</mi><mi id="S0.Ex2.m1.1.13.3">i</mi></msub><mo id="S0.Ex2.m1.1.14">≠</mo><msub id="S0.Ex2.m1.1.15"><mi id="S0.Ex2.m1.1.15.2">λ</mi><mi id="S0.Ex2.m1.1.15.3">j</mi></msub><mspace id="S0.Ex2.m1.1.16" width="1em"></mspace><mtext id="S0.Ex2.m1.1.17">for some</mtext><mspace id="S0.Ex2.m1.1.18" width="1em"></mspace><mi id="S0.Ex2.m1.1.19">i</mi><mo id="S0.Ex2.m1.1.20">,</mo><mi id="S0.Ex2.m1.1.21">j</mi><mo id="S0.Ex2.m1.1.22">∈</mo><mrow id="S0.Ex2.m1.1.23"><mo id="S0.Ex2.m1.1.23.1" stretchy="false">{</mo><mn id="S0.Ex2.m1.1.23.2">1</mn><mo id="S0.Ex2.m1.1.23.3">,</mo><mi id="S0.Ex2.m1.1.23.4" mathvariant="normal">…</mi><mo id="S0.Ex2.m1.1.23.5">,</mo><mi id="S0.Ex2.m1.1.23.6">K</mi><mo id="S0.Ex2.m1.1.23.7" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S0.Ex2.m1.1c">H_{0}:\lambda_{1}=\ldots=\lambda_{K}\quad\text{against}\quad H_{A}:\lambda_{i}% \neq\lambda_{j}\quad\text{for some}\quad i,j\in\{1,\dots,K\}</annotation><annotation encoding="application/x-llamapun" id="S0.Ex2.m1.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = … = italic_λ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT against italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT : italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for some italic_i , italic_j ∈ { 1 , … , italic_K }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S0.I1.i1.p2.14"><span class="ltx_text" id="S0.I1.i1.p2.14.1" style="color:#FF0000;">But this means that we start with K populations. The test is designed for that kind of scenario. For a dataset that does not define K populations, one can define K quadrats?</span></p> </div> </li> <li class="ltx_item" id="S0.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S0.I1.i2.p1"> <p class="ltx_p" id="S0.I1.i2.p1.1">First and second order stationarity. What are we testing? See below snippet from <cite class="ltx_cite ltx_citemacro_citet"><span class="ltx_ref ltx_missing_citation ltx_ref_self">kraamwinkel2018bootstrap</span></cite>.</p> </div> <div class="ltx_para" id="S0.I1.i2.p2"> <p class="ltx_p" id="S0.I1.i2.p2.1">“A first-order stationary process has the same intensity at different locations on a map. This implies that the expected number of observations is proportional to the size of the observation area. Stationarity is related to the expected number of pairs of points in those sections. To study second-order non-stationarity, we consider interactions between the points of the process. Complete spatial randomness (CSR) describes a stationary point pattern where the points are randomly distributed (Diggle, 2014; Gatrell et al., 1996). Three possibilities can occur: points are spread randomly, i.e. without interactions, they are clustered, i.e. with positive interactions or they are regular, i.e. with negative interactions (Baddeley, 2010; Baddeley et al., 2015; Diggle, 2014). Regularity implies that points tend to avoid each other, leading to a pattern within the data with points distributed in a regular fashion. A clustered pattern, also referred to as an ‘aggregated’ point pattern, implies that points are grouped or clustered together. A random pattern point implies that there is no pattern to the points, neither regular nor clustered. In this paper we focus on first-order stationarity.”</p> </div> </li> </ul> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Apr 3 18:25:41 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>