Kriging - Wikipedia

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id="toc-Main_principles-sublist" class="vector-toc-list"> <li id="toc-Related_terms_and_techniques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_terms_and_techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Related terms and techniques</span> </div> </a> <ul id="toc-Related_terms_and_techniques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geostatistical_estimator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geostatistical_estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Geostatistical estimator</span> </div> </a> <ul id="toc-Geostatistical_estimator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_estimation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_estimation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Linear estimation</span> </div> </a> <ul id="toc-Linear_estimation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Methods" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Methods</span> </div> </a> <button aria-controls="toc-Methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Methods subsection</span> </button> <ul id="toc-Methods-sublist" class="vector-toc-list"> <li id="toc-Ordinary_kriging" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinary_kriging"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ordinary kriging</span> </div> </a> <ul id="toc-Ordinary_kriging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_kriging" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_kriging"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Simple kriging</span> </div> </a> <ul id="toc-Simple_kriging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bayesian_kriging" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bayesian_kriging"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Bayesian kriging</span> </div> </a> <ul id="toc-Bayesian_kriging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kriging</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 21 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Krigatge" title="Krigatge – Catalan" lang="ca" hreflang="ca" data-title="Krigatge" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kriging" title="Kriging – Czech" lang="cs" hreflang="cs" data-title="Kriging" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kriging" title="Kriging – German" lang="de" hreflang="de" data-title="Kriging" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Krigeaje" title="Krigeaje – Spanish" lang="es" hreflang="es" data-title="Krigeaje" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Krigekuntza" title="Krigekuntza – Basque" lang="eu" hreflang="eu" data-title="Krigekuntza" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Krigeage" title="Krigeage – French" lang="fr" hreflang="fr" data-title="Krigeage" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kriging" title="Kriging – Croatian" lang="hr" hreflang="hr" data-title="Kriging" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kriging" title="Kriging – Indonesian" lang="id" hreflang="id" data-title="Kriging" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Kriging" title="Kriging – Italian" lang="it" hreflang="it" data-title="Kriging" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Krigel%C3%A9s" title="Krigelés – Hungarian" lang="hu" hreflang="hu" data-title="Krigelés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AF%E3%83%AA%E3%82%AE%E3%83%B3%E3%82%B0" title="クリギング – Japanese" lang="ja" hreflang="ja" data-title="クリギング" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kriging" title="Kriging – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kriging" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kriging" title="Kriging – Polish" lang="pl" hreflang="pl" data-title="Kriging" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Krigagem" title="Krigagem – Portuguese" lang="pt" hreflang="pt" data-title="Krigagem" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Kriging" title="Kriging – Romanian" lang="ro" hreflang="ro" data-title="Kriging" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B3%D0%B8%D0%BD%D0%B3" title="Кригинг – Russian" lang="ru" hreflang="ru" data-title="Кригинг" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Kriging" title="Kriging – Sundanese" lang="su" hreflang="su" data-title="Kriging" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kriging-interpolointi" title="Kriging-interpolointi – Finnish" lang="fi" hreflang="fi" data-title="Kriging-interpolointi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kriging" title="Kriging – Swedish" lang="sv" hreflang="sv" data-title="Kriging" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B3%D1%96%D0%BD%D0%B3" title="Кригінг – Ukrainian" lang="uk" hreflang="uk" data-title="Кригінг" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kriging" title="Kriging – Vietnamese" lang="vi" hreflang="vi" data-title="Kriging" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q225926#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Method of interpolation</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_kriging_interpolation_in_1D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Example_of_kriging_interpolation_in_1D.png/400px-Example_of_kriging_interpolation_in_1D.png" decoding="async" width="400" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Example_of_kriging_interpolation_in_1D.png/600px-Example_of_kriging_interpolation_in_1D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Example_of_kriging_interpolation_in_1D.png/800px-Example_of_kriging_interpolation_in_1D.png 2x" data-file-width="885" data-file-height="524" /></a><figcaption>Example of one-dimensional data interpolation by kriging, with <a href="/wiki/Credible_interval" title="Credible interval">credible intervals</a>. Squares indicate the location of the data. The kriging interpolation, shown in red, runs along the means of the normally distributed credible intervals shown in gray. The dashed curve shows a spline that is smooth, but departs significantly from the expected values given by those means.</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, originally in <a href="/wiki/Geostatistics" title="Geostatistics">geostatistics</a>, <b>kriging</b> or <b>Kriging</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'k' in 'kind'">k</span><span title="'r' in 'rye'">r</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/ɡ/: 'g' in 'guy'">ɡ</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="/ŋ/: 'ng' in 'sing'">ŋ</span></span>/</a></span></span>), also known as <b>Gaussian process regression</b>, is a method of <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> based on <a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a> governed by prior <a href="/wiki/Covariance" title="Covariance">covariances</a>. Under suitable assumptions of the prior, kriging gives the <a href="/wiki/Best_linear_unbiased_prediction" title="Best linear unbiased prediction">best linear unbiased prediction</a> (BLUP) at unsampled locations.<sup id="cite_ref-Chung2019_1-0" class="reference"><a href="#cite_note-Chung2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Interpolating methods based on other criteria such as <a href="/wiki/Smoothness" title="Smoothness">smoothness</a> (e.g., <a href="/wiki/Smoothing_spline" title="Smoothing spline">smoothing spline</a>) may not yield the BLUP. The method is widely used in the domain of <a href="/wiki/Spatial_analysis#Sampling" title="Spatial analysis">spatial analysis</a> and <a href="/wiki/Computer_experiment" title="Computer experiment">computer experiments</a>. The technique is also known as <b>Wiener–Kolmogorov prediction</b>, after <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> and <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a>. </p><p>The theoretical basis for the method was developed by the French mathematician <a href="/wiki/Georges_Matheron" title="Georges Matheron">Georges Matheron</a> in 1960, based on the master's thesis of <a href="/wiki/Danie_G._Krige" title="Danie G. Krige">Danie G. Krige</a>, the pioneering plotter of distance-weighted average gold grades at the <a href="/wiki/Witwatersrand" title="Witwatersrand">Witwatersrand</a> reef complex in <a href="/wiki/South_Africa" title="South Africa">South Africa</a>. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. The English verb is <i>to krige</i>, and the most common noun is <i>kriging</i>. The word is sometimes capitalized as <i>Kriging</i> in the literature. </p><p>Though computationally intensive in its basic formulation, kriging can be scaled to larger problems using various <a href="/wiki/Gaussian_process_approximations" title="Gaussian process approximations">approximation methods</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Main_principles">Main principles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=1" title="Edit section: Main principles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Related_terms_and_techniques">Related terms and techniques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=2" title="Edit section: Related terms and techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kriging predicts the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point. The method is closely related to <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>. Both theories derive a <a href="/wiki/Best_linear_unbiased_estimator" class="mw-redirect" title="Best linear unbiased estimator">best linear unbiased estimator</a> based on assumptions on <a href="/wiki/Covariance" title="Covariance">covariances</a>, make use of <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a> to prove independence of the estimate and error, and use very similar formulae. Even so, they are useful in different frameworks: kriging is made for estimation of a single realization of a random field, while regression models are based on multiple observations of a multivariate data set. </p><p>The kriging estimation may also be seen as a <a href="/wiki/Spline_(mathematics)" title="Spline (mathematics)">spline</a> in a <a href="/wiki/Reproducing_kernel_Hilbert_space" title="Reproducing kernel Hilbert space">reproducing kernel Hilbert space</a>, with the reproducing kernel given by the covariance function.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The difference with the classical kriging approach is provided by the interpretation: while the spline is motivated by a minimum-norm interpolation based on a Hilbert-space structure, kriging is motivated by an expected squared prediction error based on a stochastic model. </p><p>Kriging with <i>polynomial trend surfaces</i> is mathematically identical to <a href="/wiki/Generalized_least_squares" title="Generalized least squares">generalized least squares</a> polynomial <a href="/wiki/Curve_fitting" title="Curve fitting">curve fitting</a>. </p><p>Kriging can also be understood as a form of <a href="/wiki/Bayesian_optimization" title="Bayesian optimization">Bayesian optimization</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Kriging starts with a <a href="/wiki/Prior_probability_distribution" class="mw-redirect" title="Prior probability distribution">prior</a> <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> over <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>. This prior takes the form of a Gaussian process: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> samples from a function will be <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a>, where the <a href="/wiki/Covariance" title="Covariance">covariance</a> between any two samples is the covariance function (or <a href="/wiki/Kernel_(statistics)" title="Kernel (statistics)">kernel</a>) of the Gaussian process evaluated at the spatial location of two points. A <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of values is then observed, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location by combining the Gaussian prior with a Gaussian <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> for each of the observed values. The resulting <a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a> distribution is also Gaussian, with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior. </p> <div class="mw-heading mw-heading3"><h3 id="Geostatistical_estimator">Geostatistical estimator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=3" title="Edit section: Geostatistical estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In geostatistical models, sampled data are interpreted as the result of a random process. The fact that these models incorporate uncertainty in their conceptualization doesn't mean that the phenomenon – the forest, the aquifer, the mineral deposit – has resulted from a random process, but rather it allows one to build a methodological basis for the spatial inference of quantities in unobserved locations and to quantify the uncertainty associated with the estimator. </p><p>A <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> is, in the context of this model, simply a way to approach the set of data collected from the samples. The first step in geostatistical modulation is to create a random process that best describes the set of observed data. </p><p>A value from location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> (generic denomination of a set of <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinates</a>) is interpreted as a realization <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace2e0fcf1b028c83e3f4ed82fff0b29a76bdb02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.281ex; height:2.843ex;" alt="{\displaystyle z(x_{1})}"></span> of the <a href="/wiki/Random_variable" title="Random variable">random variable</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09cce78901243d4f03f2c9e6f44e1fdec4f377e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:2.843ex;" alt="{\displaystyle Z(x_{1})}"></span>. In the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, where the set of samples is dispersed, there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> realizations of the random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{1}),Z(x_{2}),\ldots ,Z(x_{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{1}),Z(x_{2}),\ldots ,Z(x_{N})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b66ab5dcc360046d51662ffc9671924c7a38607e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.47ex; height:2.843ex;" alt="{\displaystyle Z(x_{1}),Z(x_{2}),\ldots ,Z(x_{N})}"></span>, correlated between themselves. </p><p>The set of random variables constitutes a random function, of which only one realization is known – the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c5cfcc9655f10b66ab9ae93a1d61b5859ef281" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle z(x_{i})}"></span> of observed data. With only one realization of each random variable, it's theoretically impossible to determine any <a href="/wiki/Statistical_parameter" title="Statistical parameter">statistical parameter</a> of the individual variables or the function. The proposed solution in the geostatistical formalism consists in <i>assuming</i> various degrees of <i>stationarity</i> in the random function, in order to make the inference of some statistic values possible. </p><p>For instance, if one assumes, based on the homogeneity of samples in area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> where the variable is distributed, the hypothesis that the <a href="/wiki/Moment_(mathematics)#Mean" title="Moment (mathematics)">first moment</a> is stationary (i.e. all random variables have the same mean), then one is assuming that the mean can be estimated by the arithmetic mean of sampled values. </p><p>The hypothesis of stationarity related to the <a href="/wiki/Moment_(mathematics)#Variance" title="Moment (mathematics)">second moment</a> is defined in the following way: the correlation between two random variables solely depends on the spatial distance between them and is independent of their location. Thus if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {h} =x_{2}-x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {h} =x_{2}-x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd18a3da8ab4a6ccaeb2b75e7ee0f3ff4f204a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.192ex; height:2.509ex;" alt="{\displaystyle \mathbf {h} =x_{2}-x_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {h} |=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {h} |=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed7a876149d5c6a5c536894fba4d7630f566bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.216ex; height:2.843ex;" alt="{\displaystyle |\mathbf {h} |=h}"></span>, then: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C{\big (}Z(x_{1}),Z(x_{2}){\big )}=C{\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=C(h),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C{\big (}Z(x_{1}),Z(x_{2}){\big )}=C{\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=C(h),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384cae49e9c78b18c6a95b4b9d83d4d2baae014a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.93ex; height:3.176ex;" alt="{\displaystyle C{\big (}Z(x_{1}),Z(x_{2}){\big )}=C{\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=C(h),}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma {\big (}Z(x_{1}),Z(x_{2}){\big )}=\gamma {\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=\gamma (h).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma {\big (}Z(x_{1}),Z(x_{2}){\big )}=\gamma {\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=\gamma (h).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0de2596ffd247af144cb7b03f1b22df99c61485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.418ex; height:3.176ex;" alt="{\displaystyle \gamma {\big (}Z(x_{1}),Z(x_{2}){\big )}=\gamma {\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=\gamma (h).}"></span></dd></dl> <p>For simplicity, we define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x_{i},x_{j})=C{\big (}Z(x_{i}),Z(x_{j}){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x_{i},x_{j})=C{\big (}Z(x_{i}),Z(x_{j}){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a99272f226c631a711a558eced817eb72c2c8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.355ex; height:3.176ex;" alt="{\displaystyle C(x_{i},x_{j})=C{\big (}Z(x_{i}),Z(x_{j}){\big )}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (x_{i},x_{j})=\gamma {\big (}Z(x_{i}),Z(x_{j}){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (x_{i},x_{j})=\gamma {\big (}Z(x_{i}),Z(x_{j}){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcb8aa5037bff4b930c129415ec0f86bc8def57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.347ex; height:3.176ex;" alt="{\displaystyle \gamma (x_{i},x_{j})=\gamma {\big (}Z(x_{i}),Z(x_{j}){\big )}}"></span>. </p><p>This hypothesis allows one to infer those two measures – the <a href="/wiki/Variogram" title="Variogram">variogram</a> and the <a href="/w/index.php?title=Covariogram&action=edit&redlink=1" class="new" title="Covariogram (page does not exist)">covariogram</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-Z(x_{j}){\big )}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-Z(x_{j}){\big )}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313305fa5cac42373a8e5242b78c44b1c7b3536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:42.57ex; height:6.843ex;" alt="{\displaystyle \gamma (h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-Z(x_{j}){\big )}^{2},}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(h)={\frac {1}{|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-m(h){\big )}{\big (}Z(x_{j})-m(h){\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(h)={\frac {1}{|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-m(h){\big )}{\big (}Z(x_{j})-m(h){\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eac910ebb946a6f7d54169edd53852f02e4c203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:56.205ex; height:6.843ex;" alt="{\displaystyle C(h)={\frac {1}{|N(h)|}}\sum _{(i,j)\in N(h)}{\big (}Z(x_{i})-m(h){\big )}{\big (}Z(x_{j})-m(h){\big )},}"></span></dd></dl> <p>where: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}Z(x_{i})+Z(x_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}Z(x_{i})+Z(x_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97684212b657e05180b0af33278f910d53b26e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:39.517ex; height:6.843ex;" alt="{\displaystyle m(h)={\frac {1}{2|N(h)|}}\sum _{(i,j)\in N(h)}Z(x_{i})+Z(x_{j})}"></span>;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c9991d137fd259390c49f41499894e31b04468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.212ex; height:2.843ex;" alt="{\displaystyle N(h)}"></span> denotes the set of pairs of observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,\;j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,\;j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbecf1d9810f61affaff4163bb31a65d0e94070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.44ex; height:2.509ex;" alt="{\displaystyle i,\;j}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{i}-x_{j}|=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{i}-x_{j}|=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b848bdebdd2c0823918635119c6ba25a71eaad4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.94ex; height:3.009ex;" alt="{\displaystyle |x_{i}-x_{j}|=h}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |N(h)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |N(h)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded46e0e3a10c429bb323d16cbe490c5f79663f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.506ex; height:2.843ex;" alt="{\displaystyle |N(h)|}"></span> is the number of pairs in the set.</dd></dl> <p>In this set, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,\;j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,\;j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072f239857e5911de7c40285a1eea3fa35829030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.249ex; height:2.843ex;" alt="{\displaystyle (i,\;j)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (j,\;i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (j,\;i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926780e49565a3864c5e643817011d586a48d04c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.249ex; height:2.843ex;" alt="{\displaystyle (j,\;i)}"></span> denote the same element. Generally an "approximate distance" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is used, implemented using a certain tolerance. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_estimation">Linear estimation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=4" title="Edit section: Linear estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial inference, or estimation, of a quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\colon \mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\colon \mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1d865566ae59f83f99cb118dbe8187c7c6a2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.903ex; height:2.343ex;" alt="{\displaystyle Z\colon \mathbb {R} ^{n}\to \mathbb {R} }"></span>, at an unobserved location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, is calculated from a linear combination of the observed values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}=Z(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}=Z(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad0167cff1d37360f419b3032970e59b4991431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.598ex; height:2.843ex;" alt="{\displaystyle z_{i}=Z(x_{i})}"></span> and weights <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}(x_{0}),\;i=1,\ldots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}(x_{0}),\;i=1,\ldots ,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0225dcf6c6f814884c691d4cf1cebe18086887f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.641ex; height:2.843ex;" alt="{\displaystyle w_{i}(x_{0}),\;i=1,\ldots ,N}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x_{0})={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{N}\end{bmatrix}}{\begin{bmatrix}z_{1}\\z_{2}\\\vdots \\z_{N}\end{bmatrix}}=\sum _{i=1}^{N}w_{i}(x_{0})Z(x_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x_{0})={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{N}\end{bmatrix}}{\begin{bmatrix}z_{1}\\z_{2}\\\vdots \\z_{N}\end{bmatrix}}=\sum _{i=1}^{N}w_{i}(x_{0})Z(x_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d56ce5934898b13b3f057b8add1c7287db42b67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:55.889ex; height:13.843ex;" alt="{\displaystyle {\hat {Z}}(x_{0})={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{N}\end{bmatrix}}{\begin{bmatrix}z_{1}\\z_{2}\\\vdots \\z_{N}\end{bmatrix}}=\sum _{i=1}^{N}w_{i}(x_{0})Z(x_{i}).}"></span></dd></dl> <p>The weights <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"></span> are intended to summarize two extremely important procedures in a spatial inference process: </p> <ul><li>reflect the structural "proximity" of samples to the estimation location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>;</li> <li>at the same time, they should have a desegregation effect, in order to avoid bias caused by eventual sample <i>clusters</i>.</li></ul> <p>When calculating the weights <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"></span>, there are two objectives in the geostatistical formalism: <i>unbias</i> and <i>minimal variance of estimation</i>. </p><p>If the cloud of real values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0641760c6094d43d86ed5e0e13fc8917df8c5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:2.843ex;" alt="{\displaystyle Z(x_{0})}"></span> is plotted against the estimated values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9a4fa0080f228b73e4e27de3e5dbc3c8b5f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:3.343ex;" alt="{\displaystyle {\hat {Z}}(x_{0})}"></span>, the criterion for global unbias, <i>intrinsic stationarity</i> or <a href="/wiki/Stationary_process" title="Stationary process">wide sense stationarity</a> of the field, implies that the mean of the estimations must be equal to mean of the real values. </p><p>The second criterion says that the mean of the squared deviations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}{\hat {Z}}(x)-Z(x){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}{\hat {Z}}(x)-Z(x){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c903b1d3528a8d21e3e49d25206623f51710d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.609ex; height:3.509ex;" alt="{\displaystyle {\big (}{\hat {Z}}(x)-Z(x){\big )}}"></span> must be minimal, which means that when the cloud of estimated values <i>versus</i> the cloud real values is more disperse, the estimator is more imprecise. </p> <div class="mw-heading mw-heading2"><h2 id="Methods">Methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=5" title="Edit section: Methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deduced, i.e. different types of kriging apply. Classical methods are: </p> <ul><li><i>Ordinary kriging</i> assumes constant unknown mean only over the search neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>.</li> <li><i>Simple kriging</i> assumes stationarity of the <a href="/wiki/Moment_(mathematics)#Mean" title="Moment (mathematics)">first moment</a> over the entire domain with a known mean: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\{Z(x)\}=E\{Z(x_{0})\}=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\{Z(x)\}=E\{Z(x_{0})\}=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9725c1c640c1dc650bc3dfe84604a3fc1a2b840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.131ex; height:2.843ex;" alt="{\displaystyle E\{Z(x)\}=E\{Z(x_{0})\}=m}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the known mean.</li> <li><i><a href="/wiki/Universal_kriging" class="mw-redirect" title="Universal kriging">Universal kriging</a></i><span class="anchor" id="Universal"></span> assumes a general polynomial trend model, such as linear trend model <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle E\{Z(x)\}=\sum _{k=0}^{p}\beta _{k}f_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle E\{Z(x)\}=\sum _{k=0}^{p}\beta _{k}f_{k}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d9199a52d56df54b3e683631de770302b41e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.82ex; height:3.176ex;" alt="{\displaystyle \textstyle E\{Z(x)\}=\sum _{k=0}^{p}\beta _{k}f_{k}(x)}"></span>.</li> <li><i>IRFk-kriging</i><span class="anchor" id="IRFk"></span> assumes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\{Z(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\{Z(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78141e66f2de5ce28496dc099548e552dfd49821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.92ex; height:2.843ex;" alt="{\displaystyle E\{Z(x)\}}"></span> to be an unknown <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</li> <li><i>Indicator kriging</i><span class="anchor" id="Indicator"></span> uses <a href="/wiki/Indicator_function" title="Indicator function">indicator functions</a> instead of the process itself, in order to estimate transition probabilities. <ul><li><i>Multiple-indicator kriging</i><span class="anchor" id="Multiple-indicator"></span> is a version of indicator kriging working with a family of indicators. Initially, MIK showed considerable promise as a new method that could more accurately estimate overall global mineral deposit concentrations or grades. However, these benefits have been outweighed by other inherent problems of practicality in modelling due to the inherently large block sizes used and also the lack of mining scale resolution. Conditional simulation is fast, becoming the accepted replacement technique in this case.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2016)">citation needed</span></a></i>]</sup></li></ul></li> <li><i>Disjunctive kriging</i><span class="anchor" id="Disjunctive"></span> is a nonlinear generalisation of kriging.</li> <li><i><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a> kriging</i><span class="anchor" id="Lognormal"></span> interpolates positive data by means of <a href="/wiki/Logarithm" title="Logarithm">logarithms</a>.</li> <li><i>Latent kriging</i> assumes the various krigings on the latent level (second stage) of the <a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">nonlinear mixed-effects model</a> to produce a spatial functional prediction.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This technique is useful when analyzing a spatial functional data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(y_{i},x_{i},s_{i})\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(y_{i},x_{i},s_{i})\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab3b7453f0d1dc73a3173b5686cd65861ebed15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.061ex; height:3.009ex;" alt="{\displaystyle \{(y_{i},x_{i},s_{i})\}_{i=1}^{n}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=(y_{i1},y_{i2},\cdots ,y_{iT_{i}})^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=(y_{i1},y_{i2},\cdots ,y_{iT_{i}})^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f585f78b69312aa54226839d731dac445db566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.615ex; height:3.343ex;" alt="{\displaystyle y_{i}=(y_{i1},y_{i2},\cdots ,y_{iT_{i}})^{\top }}"></span> is a time series data over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}"></span> period, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=(x_{i1},x_{i2},\cdots ,x_{ip})^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=(x_{i1},x_{i2},\cdots ,x_{ip})^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794bf205983e1209c94648d4779b12375a53460a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.619ex; height:3.343ex;" alt="{\displaystyle x_{i}=(x_{i1},x_{i2},\cdots ,x_{ip})^{\top }}"></span> is a vector of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> covariates, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=(s_{i1},s_{i2})^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}=(s_{i1},s_{i2})^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2a5fdfb0babd6663110b869c31b36883ebbade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.767ex; height:3.176ex;" alt="{\displaystyle s_{i}=(s_{i1},s_{i2})^{\top }}"></span> is a spatial location (longitude, latitude) of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th subject.</li> <li><i>Co-kriging</i> denotes the joint kriging of data from multiple sources with a relationship between the different data sources.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Co-kriging is also possible in a <a href="/wiki/List_of_things_named_after_Thomas_Bayes" title="List of things named after Thomas Bayes">Bayesian</a> approach.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><i>Bayesian kriging</i> departs from the optimization of unknown coefficients and hyperparameters, which is understood as a <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimate</a> from the Bayesian perspective. Instead, the coefficients and hyperparameters are estimated from their <a href="/wiki/Expectation_value" class="mw-redirect" title="Expectation value">expectation values</a>. An advantage of Bayesian kriging is, that it allows to quantify the evidence for and the uncertainty of the kriging <a href="/wiki/Surrogate_model" title="Surrogate model">emulator</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> If the emulator is employed to propagate uncertainties, the quality of the kriging emulator can be assessed by comparing the emulator uncertainty to the total uncertainty (see also <a href="/wiki/Polynomial_chaos#Bayesian_polynomial_chaos" title="Polynomial chaos">Bayesian Polynomial Chaos</a>). Bayesian kriging can also be mixed with co-kriging.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-1" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ordinary_kriging">Ordinary kriging<span class="anchor" id="Ordinary"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=6" title="Edit section: Ordinary kriging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The unknown value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0641760c6094d43d86ed5e0e13fc8917df8c5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:2.843ex;" alt="{\displaystyle Z(x_{0})}"></span> is interpreted as a random variable located in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, as well as the values of neighbors samples <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{i}),\ i=1,\ldots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{i}),\ i=1,\ldots ,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3275e60ae5607e805412d07d0d7d116b5dcaded6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.539ex; height:2.843ex;" alt="{\displaystyle Z(x_{i}),\ i=1,\ldots ,N}"></span>. The estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9a4fa0080f228b73e4e27de3e5dbc3c8b5f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:3.343ex;" alt="{\displaystyle {\hat {Z}}(x_{0})}"></span> is also interpreted as a random variable located in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, a result of the linear combination of variables. </p><p>Kriging seeks to minimize the mean square value of the following error in estimating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0641760c6094d43d86ed5e0e13fc8917df8c5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:2.843ex;" alt="{\displaystyle Z(x_{0})}"></span>, subject to lack of bias: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (x_{0})={\hat {Z}}(x_{0})-Z(x_{0})={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}=\sum _{i=1}^{N}w_{i}(x_{0})\times Z(x_{i})-Z(x_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon (x_{0})={\hat {Z}}(x_{0})-Z(x_{0})={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}=\sum _{i=1}^{N}w_{i}(x_{0})\times Z(x_{i})-Z(x_{0}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb75d95b3031ce14c2d4982c3e7c646d7024faec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:101.538ex; height:7.343ex;" alt="{\displaystyle \epsilon (x_{0})={\hat {Z}}(x_{0})-Z(x_{0})={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}=\sum _{i=1}^{N}w_{i}(x_{0})\times Z(x_{i})-Z(x_{0}).}"></span></dd></dl> <p>The two quality criteria referred to previously can now be expressed in terms of the mean and variance of the new random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon (x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af16b76418b91bfc3fa02b1daab87659b712ecd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.137ex; height:2.843ex;" alt="{\displaystyle \epsilon (x_{0})}"></span>: </p> <dl><dt>Lack of bias</dt></dl> <p>Since the random function is stationary, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[Z(x_{i})]=E[Z(x_{0})]=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[Z(x_{i})]=E[Z(x_{0})]=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d171760402e744c733cb7f95cd10701451abe5a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.869ex; height:2.843ex;" alt="{\displaystyle E[Z(x_{i})]=E[Z(x_{0})]=m}"></span>, the weights must sum to 1 in order to ensure that the model is unbiased. This can be seen as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[\epsilon (x_{0})]=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})\times E[Z(x_{i})]-E[Z(x_{0})]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\epsilon (x_{0})]=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})\times E[Z(x_{i})]-E[Z(x_{0})]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6847052de0f719fe0b1fb529ff19f045c6ffb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.054ex; height:7.343ex;" alt="{\displaystyle E[\epsilon (x_{0})]=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})\times E[Z(x_{i})]-E[Z(x_{0})]=0}"></span></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow m\sum _{i=1}^{N}w_{i}(x_{0})-m=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})=1\Leftrightarrow \mathbf {1} ^{T}\cdot W=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> <mi>m</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇔</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <mi>W</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow m\sum _{i=1}^{N}w_{i}(x_{0})-m=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})=1\Leftrightarrow \mathbf {1} ^{T}\cdot W=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c310b7652fab95ce055e6c9d7e65ec1128230f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.573ex; height:7.343ex;" alt="{\displaystyle \Leftrightarrow m\sum _{i=1}^{N}w_{i}(x_{0})-m=0\Leftrightarrow \sum _{i=1}^{N}w_{i}(x_{0})=1\Leftrightarrow \mathbf {1} ^{T}\cdot W=1.}"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dt>Minimum variance</dt></dl> <p>Two estimators can have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[\epsilon (x_{0})]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\epsilon (x_{0})]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca37f3d1521cb193994eafd1d1b6b63593a65c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.468ex; height:2.843ex;" alt="{\displaystyle E[\epsilon (x_{0})]=0}"></span>, but the dispersion around their mean determines the difference between the quality of estimators. To find an estimator with minimum variance, we need to minimize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[\epsilon (x_{0})^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\epsilon (x_{0})^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d934d5584e666a31eab363e7b73982560f13c918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.261ex; height:3.176ex;" alt="{\displaystyle E[\epsilon (x_{0})^{2}]}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {Var} (\epsilon (x_{0}))&=\operatorname {Var} \left({\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\\&={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot \operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\cdot {\begin{bmatrix}W\\-1\end{bmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>W</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Var} (\epsilon (x_{0}))&=\operatorname {Var} \left({\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\\&={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot \operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\cdot {\begin{bmatrix}W\\-1\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982f94df7c80ec44f196c85dccf728177f1d52f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:74.007ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Var} (\epsilon (x_{0}))&=\operatorname {Var} \left({\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\\&={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot \operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})&Z(x_{0})\end{bmatrix}}^{T}\right)\cdot {\begin{bmatrix}W\\-1\end{bmatrix}}.\end{aligned}}}"></span></dd></dl> <p>See <a href="/wiki/Covariance_matrix#As_a_linear_operator" title="Covariance matrix">covariance matrix</a> for a detailed explanation. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (\epsilon (x_{0}))={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Var} _{x_{i}}&\operatorname {Cov} _{x_{i}x_{0}}\\\operatorname {Cov} _{x_{i}x_{0}}^{T}&\operatorname {Var} _{x_{0}}\end{bmatrix}}\cdot {\begin{bmatrix}W\\-1\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>W</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (\epsilon (x_{0}))={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Var} _{x_{i}}&\operatorname {Cov} _{x_{i}x_{0}}\\\operatorname {Cov} _{x_{i}x_{0}}^{T}&\operatorname {Var} _{x_{0}}\end{bmatrix}}\cdot {\begin{bmatrix}W\\-1\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7adf0a9677ebf2efb55aa7322b455c9b454098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.339ex; height:7.509ex;" alt="{\displaystyle \operatorname {Var} (\epsilon (x_{0}))={\begin{bmatrix}W^{T}&-1\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Var} _{x_{i}}&\operatorname {Cov} _{x_{i}x_{0}}\\\operatorname {Cov} _{x_{i}x_{0}}^{T}&\operatorname {Var} _{x_{0}}\end{bmatrix}}\cdot {\begin{bmatrix}W\\-1\end{bmatrix}},}"></span></dd></dl> <p>where the literals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\operatorname {Var} _{x_{i}},\operatorname {Var} _{x_{0}},\operatorname {Cov} _{x_{i}x_{0}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\operatorname {Var} _{x_{i}},\operatorname {Var} _{x_{0}},\operatorname {Cov} _{x_{i}x_{0}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ec5d3ef1a78638083d046c1866f4cd55611ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.466ex; height:3.009ex;" alt="{\displaystyle \left\{\operatorname {Var} _{x_{i}},\operatorname {Var} _{x_{0}},\operatorname {Cov} _{x_{i}x_{0}}\right\}}"></span> stand for </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T}\right),\operatorname {Var} {\big (}Z(x_{0}){\big )},\operatorname {Cov} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T},Z(x_{0})\right)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T}\right),\operatorname {Var} {\big (}Z(x_{0}){\big )},\operatorname {Cov} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T},Z(x_{0})\right)\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d819547ff026804426f158acf2a7feb7b7bfc4fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:85.128ex; height:4.843ex;" alt="{\displaystyle \left\{\operatorname {Var} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T}\right),\operatorname {Var} {\big (}Z(x_{0}){\big )},\operatorname {Cov} \left({\begin{bmatrix}Z(x_{1})&\cdots &Z(x_{N})\end{bmatrix}}^{T},Z(x_{0})\right)\right\}.}"></span></dd></dl> <p>Once defined the covariance model or <a href="/wiki/Variogram" title="Variogram">variogram</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathbf {h} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbf {h} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/defc7480b5baa3509514ee3865e9ea05c76f92ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.061ex; height:2.843ex;" alt="{\displaystyle C(\mathbf {h} )}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (\mathbf {h} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (\mathbf {h} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcbabd347c0b96bc7777581caf720061fb1b101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.557ex; height:2.843ex;" alt="{\displaystyle \gamma (\mathbf {h} )}"></span>, valid in all field of analysis of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2ffb6f5d6e9eeab2867ef3c37ea13f9c294ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle Z(x)}"></span>, then we can write an expression for the estimation variance of any estimator in function of the covariance between the samples and the covariances between the samples and the point to estimate: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}},\\\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=\operatorname {Cov} (0)+\sum _{i}\sum _{j}w_{i}w_{j}\operatorname {Cov} (x_{i},x_{j})-2\sum _{i}w_{i}C(x_{i},x_{0}).\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <mi>W</mi> <mo>−</mo> <msubsup> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo>⋅</mo> <mi>W</mi> <mo>−</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}},\\\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=\operatorname {Cov} (0)+\sum _{i}\sum _{j}w_{i}w_{j}\operatorname {Cov} (x_{i},x_{j})-2\sum _{i}w_{i}C(x_{i},x_{0}).\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f500dec5b391e55f494e9eea6c7929237394d148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.435ex; height:7.509ex;" alt="{\displaystyle {\begin{cases}\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}},\\\operatorname {Var} {\big (}\epsilon (x_{0}){\big )}=\operatorname {Cov} (0)+\sum _{i}\sum _{j}w_{i}w_{j}\operatorname {Cov} (x_{i},x_{j})-2\sum _{i}w_{i}C(x_{i},x_{0}).\end{cases}}}"></span></dd></dl> <p>Some conclusions can be asserted from this expression. The variance of estimation: </p> <ul><li>is not quantifiable to any linear estimator, once the stationarity of the mean and of the spatial covariances, or variograms, are assumed;</li> <li>grows when the covariance between the samples and the point to estimate decreases. This means that, when the samples are farther away from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, the estimation becomes worse;</li> <li>grows with the a priori variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5b9ed75ee078feac0dce20803e26b81f83f844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.738ex; height:2.843ex;" alt="{\displaystyle C(0)}"></span> of the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2ffb6f5d6e9eeab2867ef3c37ea13f9c294ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle Z(x)}"></span>; when the variable is less disperse, the variance is lower in any point of the area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>;</li> <li>does not depend on the values of the samples, which means that the same spatial configuration (with the same geometrical relations between samples and the point to estimate) always reproduces the same estimation variance in any part of the area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>; this way, the variance does not measure the uncertainty of estimation produced by the local variable.</li></ul> <dl><dt>System of equations</dt></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\underset {\mathbf {1} ^{T}\cdot W=1}{\operatorname {arg\,min} }}\left(W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <mi>W</mi> <mo>=</mo> <mn>1</mn> </mrow> </munder> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <mi>W</mi> <mo>−</mo> <msubsup> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo>⋅</mo> <mi>W</mi> <mo>−</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\underset {\mathbf {1} ^{T}\cdot W=1}{\operatorname {arg\,min} }}\left(W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ebd5d3629bffec40a4b3f786aff287e7bfc6e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:69.647ex; height:5.176ex;" alt="{\displaystyle W={\underset {\mathbf {1} ^{T}\cdot W=1}{\operatorname {arg\,min} }}\left(W^{T}\cdot \operatorname {Var} _{x_{i}}\cdot W-\operatorname {Cov} _{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot \operatorname {Cov} _{x_{i}x_{0}}+\operatorname {Var} _{x_{0}}\right).}"></span></dd></dl> <p>Solving this optimization problem (see <a href="/wiki/Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a>) results in the <i>kriging system</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\hat {W}}\\\mu \end{bmatrix}}={\begin{bmatrix}\operatorname {Var} _{x_{i}}&\mathbf {1} \\\mathbf {1} ^{T}&0\end{bmatrix}}^{-1}\cdot {\begin{bmatrix}\operatorname {Cov} _{x_{i}x_{0}}\\1\end{bmatrix}}={\begin{bmatrix}\gamma (x_{1},x_{1})&\cdots &\gamma (x_{1},x_{n})&1\\\vdots &\ddots &\vdots &\vdots \\\gamma (x_{n},x_{1})&\cdots &\gamma (x_{n},x_{n})&1\\1&\cdots &1&0\end{bmatrix}}^{-1}{\begin{bmatrix}\gamma (x_{1},x^{*})\\\vdots \\\gamma (x_{n},x^{*})\\1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>W</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>μ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Var</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\hat {W}}\\\mu \end{bmatrix}}={\begin{bmatrix}\operatorname {Var} _{x_{i}}&\mathbf {1} \\\mathbf {1} ^{T}&0\end{bmatrix}}^{-1}\cdot {\begin{bmatrix}\operatorname {Cov} _{x_{i}x_{0}}\\1\end{bmatrix}}={\begin{bmatrix}\gamma (x_{1},x_{1})&\cdots &\gamma (x_{1},x_{n})&1\\\vdots &\ddots &\vdots &\vdots \\\gamma (x_{n},x_{1})&\cdots &\gamma (x_{n},x_{n})&1\\1&\cdots &1&0\end{bmatrix}}^{-1}{\begin{bmatrix}\gamma (x_{1},x^{*})\\\vdots \\\gamma (x_{n},x^{*})\\1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2713d7bfdac0086a46524d8ae90dc8d95389fba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:88.244ex; height:15.009ex;" alt="{\displaystyle {\begin{bmatrix}{\hat {W}}\\\mu \end{bmatrix}}={\begin{bmatrix}\operatorname {Var} _{x_{i}}&\mathbf {1} \\\mathbf {1} ^{T}&0\end{bmatrix}}^{-1}\cdot {\begin{bmatrix}\operatorname {Cov} _{x_{i}x_{0}}\\1\end{bmatrix}}={\begin{bmatrix}\gamma (x_{1},x_{1})&\cdots &\gamma (x_{1},x_{n})&1\\\vdots &\ddots &\vdots &\vdots \\\gamma (x_{n},x_{1})&\cdots &\gamma (x_{n},x_{n})&1\\1&\cdots &1&0\end{bmatrix}}^{-1}{\begin{bmatrix}\gamma (x_{1},x^{*})\\\vdots \\\gamma (x_{n},x^{*})\\1\end{bmatrix}}.}"></span></dd></dl> <p>The additional parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is a <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> used in the minimization of the kriging error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a4bf6d470a05f84260af55724fa769c8caeb75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.555ex; height:3.176ex;" alt="{\displaystyle \sigma _{k}^{2}(x)}"></span> to honor the unbiasedness condition. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_kriging">Simple kriging<span class="anchor" id="Simpler"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=7" title="Edit section: Simple kriging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>this section is very poor and needs to be improved.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Kriging" title="Special:EditPage/Kriging">improve this section</a> if you can.</span> <span class="date-container"><i>(<span class="date">January 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gaussianprocess.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Gaussianprocess.gif/400px-Gaussianprocess.gif" decoding="async" width="400" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Gaussianprocess.gif/600px-Gaussianprocess.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/d/da/Gaussianprocess.gif 2x" data-file-width="667" data-file-height="292" /></a><figcaption>Simple kriging can be seen as the mean and envelope of Brownian <a href="/wiki/Random_walk" title="Random walk">random walks</a> passing through the data points.</figcaption></figure> <p>Simple kriging is mathematically the simplest, but the least general.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> It assumes the <a href="/wiki/Expected_value" title="Expected value">expectation</a> of the <a href="/wiki/Random_field" title="Random field">random field</a> is known and relies on a <a href="/wiki/Covariance_function" title="Covariance function">covariance function</a>. However, in most applications neither the expectation nor the covariance are known beforehand. </p><p>The practical assumptions for the application of <i>simple kriging</i> are: </p> <ul><li><a href="/wiki/Stationary_process" title="Stationary process">Wide-sense stationarity</a> of the field (variance stationary).</li> <li>The expectation is zero everywhere: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ce1f0410632782365b7d11e010d17b8b64c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.802ex; height:2.843ex;" alt="{\displaystyle \mu (x)=0}"></span>.</li> <li>Known <a href="/wiki/Covariance_function" title="Covariance function">covariance function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)=\operatorname {Cov} {\big (}Z(x),Z(y){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Cov</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(x,y)=\operatorname {Cov} {\big (}Z(x),Z(y){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdb345bd01d570851104c2e8b58cbb492cb6b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.517ex; height:3.176ex;" alt="{\displaystyle c(x,y)=\operatorname {Cov} {\big (}Z(x),Z(y){\big )}}"></span>.</li></ul> <p>The covariance function is a crucial design choice, since it stipulates the properties of the Gaussian process and thereby the behaviour of the model. The covariance function encodes information about, for instance, smoothness and periodicity, which is reflected in the estimate produced. A very common covariance function is the squared exponential, which heavily favours smooth function estimates.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For this reason, it can produce poor estimates in many real-world applications, especially when the true underlying function contains discontinuities and rapid changes. </p> <dl><dt>System of equations</dt></dl> <p>The <i>kriging weights</i> of <i>simple kriging</i> have no unbiasedness condition and are given by the <i>simple kriging equation system</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}={\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}={\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39d6fbeb0f3d91d3ac62a710bc02beb1917ed09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:57.55ex; height:11.676ex;" alt="{\displaystyle {\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}={\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}"></span></dd></dl> <p>This is analogous to a linear regression of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0641760c6094d43d86ed5e0e13fc8917df8c5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.874ex; height:2.843ex;" alt="{\displaystyle Z(x_{0})}"></span> on the other <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1},\ldots ,z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1},\ldots ,z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399be1c816346daeabf844df4597179a1c8bbdf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.613ex; height:2.009ex;" alt="{\displaystyle z_{1},\ldots ,z_{n}}"></span>. </p> <dl><dt>Estimation</dt></dl> <p>The interpolation by simple kriging is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x_{0})={\begin{pmatrix}z_{1}\\\vdots \\z_{n}\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x_{0})={\begin{pmatrix}z_{1}\\\vdots \\z_{n}\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df000cf66c68409d12d516b0b526195f75cb21e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:63.526ex; height:11.676ex;" alt="{\displaystyle {\hat {Z}}(x_{0})={\begin{pmatrix}z_{1}\\\vdots \\z_{n}\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}.}"></span></dd></dl> <p>The kriging error is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}=\underbrace {c(x_{0},x_{0})} _{\operatorname {Var} {\big (}Z(x_{0}){\big )}}-\underbrace {{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}} _{\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </munder> <mo>−</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </munder> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}=\underbrace {c(x_{0},x_{0})} _{\operatorname {Var} {\big (}Z(x_{0}){\big )}}-\underbrace {{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}} _{\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee931e860717c992f7be9cd2365c38548412742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:97.185ex; height:17.009ex;" alt="{\displaystyle \operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}=\underbrace {c(x_{0},x_{0})} _{\operatorname {Var} {\big (}Z(x_{0}){\big )}}-\underbrace {{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}'{\begin{pmatrix}c(x_{1},x_{1})&\cdots &c(x_{1},x_{n})\\\vdots &\ddots &\vdots \\c(x_{n},x_{1})&\cdots &c(x_{n},x_{n})\end{pmatrix}}^{-1}{\begin{pmatrix}c(x_{1},x_{0})\\\vdots \\c(x_{n},x_{0})\end{pmatrix}}} _{\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}},}"></span></dd></dl> <p>which leads to the generalised least-squares version of the <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a> (Chiles & Delfiner 1999, p. 159): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} {\big (}Z(x_{0}){\big )}=\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}+\operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mi>Var</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} {\big (}Z(x_{0}){\big )}=\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}+\operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a904d9154e96626bc088cbc17ea678cee42b5000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.923ex; height:3.509ex;" alt="{\displaystyle \operatorname {Var} {\big (}Z(x_{0}){\big )}=\operatorname {Var} {\big (}{\hat {Z}}(x_{0}){\big )}+\operatorname {Var} {\big (}{\hat {Z}}(x_{0})-Z(x_{0}){\big )}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bayesian_kriging">Bayesian kriging</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=8" title="Edit section: Bayesian kriging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>See also <a href="/wiki/Polynomial_chaos#Bayesian_polynomial_chaos" title="Polynomial chaos">Bayesian Polynomial Chaos</a></i> </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=9" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>this section needs revision. Incorrect or confusing text should be removed.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Kriging" title="Special:EditPage/Kriging">improve this section</a> if you can.</span> <span class="date-container"><i>(<span class="date">January 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li>The kriging estimation is unbiased: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[{\hat {Z}}(x_{i})]=E[Z(x_{i})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[{\hat {Z}}(x_{i})]=E[Z(x_{i})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af483a2d7cd5784c37bd1a3d88ed8a2e5e8b9ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.475ex; height:3.343ex;" alt="{\displaystyle E[{\hat {Z}}(x_{i})]=E[Z(x_{i})]}"></span>.</li> <li>The kriging estimation honors the actually observed value: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x_{i})=Z(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x_{i})=Z(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1230bf07f9217d4080aff17dddee70391aecd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.337ex; height:3.343ex;" alt="{\displaystyle {\hat {Z}}(x_{i})=Z(x_{i})}"></span> (assuming no measurement error is incurred).</li> <li>The kriging estimation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Z}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Z}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef307674ef5abe26c319b0b241a4d2be53dbaacf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:3.343ex;" alt="{\displaystyle {\hat {Z}}(x)}"></span> is the <a href="/wiki/Best_linear_unbiased_estimator" class="mw-redirect" title="Best linear unbiased estimator">best linear unbiased estimator</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2ffb6f5d6e9eeab2867ef3c37ea13f9c294ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle Z(x)}"></span> if the assumptions hold. However (e.g. Cressie 1993):<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <ul><li>As with any method, if the assumptions do not hold, kriging might be bad.</li> <li>There might be better nonlinear and/or biased methods.</li> <li>No properties are guaranteed when the wrong variogram is used. However, typically still a "good" interpolation is achieved.</li> <li>Best is not necessarily good: e.g. in case of no spatial dependence the kriging interpolation is only as good as the arithmetic mean.</li></ul></li> <li>Kriging provides <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065426e4746772f367e2476d16fa02f11460a70d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.416ex; height:3.176ex;" alt="{\displaystyle \sigma _{k}^{2}}"></span> as a measure of precision. However, this measure relies on the correctness of the variogram.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=10" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>this section is very poor and needs to be improved.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Kriging" title="Special:EditPage/Kriging">improve this section</a> if you can.</span> <span class="date-container"><i>(<span class="date">January 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Although kriging was developed originally for applications in geostatistics, it is a general method of statistical interpolation and can be applied within any discipline to sampled data from random fields that satisfy the appropriate mathematical assumptions. It can be used where spatially related data has been collected (in 2-D or 3-D) and estimates of "fill-in" data are desired in the locations (spatial gaps) between the actual measurements. </p><p>To date kriging has been used in a variety of disciplines, including the following: </p> <ul><li><a href="/wiki/Environmental_science" title="Environmental science">Environmental science</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Hydrogeology" title="Hydrogeology">Hydrogeology</a><sup id="cite_ref-Chiles_13-0" class="reference"><a href="#cite_note-Chiles-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Mining" title="Mining">Mining</a><sup id="cite_ref-Journel_16-0" class="reference"><a href="#cite_note-Journel-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Natural_resource" title="Natural resource">Natural resources</a><sup id="cite_ref-multiple_18-0" class="reference"><a href="#cite_note-multiple-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Remote_sensing" title="Remote sensing">Remote sensing</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Real_estate_appraisal" title="Real estate appraisal">Real estate appraisal</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li>Integrated circuit analysis and optimization<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>Modelling of microwave devices<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>Prediction of oil production curve of shale oil wells<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Design_and_analysis_of_computer_experiments">Design and analysis of computer experiments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=11" title="Edit section: Design and analysis of computer experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another very important and rapidly growing field of application, in <a href="/wiki/Engineering" title="Engineering">engineering</a>, is the interpolation of data coming out as response variables of deterministic computer simulations,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> e.g. <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a> (FEM) simulations. In this case, kriging is used as a <a href="/wiki/Metamodeling" title="Metamodeling">metamodeling</a> tool, i.e. a black-box model built over a designed set of <a href="/wiki/Computer_experiment" title="Computer experiment">computer experiments</a>. In many practical engineering problems, such as the design of a <a href="/wiki/Metal_forming" class="mw-redirect" title="Metal forming">metal forming</a> process, a single FEM simulation might be several hours or even a few days long. It is therefore more efficient to design and run a limited number of computer simulations, and then use a kriging interpolator to rapidly predict the response in any other design point. Kriging is therefore used very often as a so-called <a href="/wiki/Surrogate_model" title="Surrogate model">surrogate model</a>, implemented inside <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimization</a> routines.<sup id="cite_ref-sheetforming_29-0" class="reference"><a href="#cite_note-sheetforming-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Kriging-based surrogate models may also be used in the case of mixed integer inputs.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bayes_linear_statistics" title="Bayes linear statistics">Bayes linear statistics</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Multivariate_interpolation" title="Multivariate interpolation">Multivariate interpolation</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric regression</a></li> <li><a href="/wiki/Radial_basis_function_interpolation" title="Radial basis function interpolation">Radial basis function interpolation</a></li> <li><a href="/wiki/Space_mapping" title="Space mapping">Space mapping</a></li> <li><a href="/wiki/Spatial_dependence" class="mw-redirect" title="Spatial dependence">Spatial dependence</a></li> <li><a href="/wiki/Variogram" title="Variogram">Variogram</a></li> <li><a href="/wiki/Gradient-enhanced_kriging" title="Gradient-enhanced kriging">Gradient-enhanced kriging</a> (GEK)</li> <li><a href="/wiki/Surrogate_model" title="Surrogate model">Surrogate model</a></li> <li><a href="/wiki/Information_field_theory" title="Information field theory">Information field theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Chung2019-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chung2019_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChungVenkatramananElzainSelvam2019" class="citation book cs1">Chung, Sang Yong; Venkatramanan, S.; 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"A technique for FEM optimization under reliability constraint of process variables in sheet metal forming". <i>International Journal of Material Forming</i>. <b>1</b> (1): 13–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12289-008-0001-8">10.1007/s12289-008-0001-8</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:136682565">136682565</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Material+Forming&rft.atitle=A+technique+for+FEM+optimization+under+reliability+constraint+of+process+variables+in+sheet+metal+forming&rft.volume=1&rft.issue=1&rft.pages=13-20&rft.date=2008-03&rft_id=info%3Adoi%2F10.1007%2Fs12289-008-0001-8&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A136682565%23id-name%3DS2CID&rft.aulast=Strano&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKriging" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSavesDiouaneBartoliLefebvre2023" class="citation journal cs1">Saves, Paul; Diouane, Youssef; Bartoli, Nathalie; Lefebvre, Thierry; Morlier, Joseph (2023). "A mixed-categorical correlation kernel for Gaussian process". <i>Neurocomputing</i>. <b>550</b>: 126472. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2211.08262">2211.08262</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neucom.2023.126472">10.1016/j.neucom.2023.126472</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neurocomputing&rft.atitle=A+mixed-categorical+correlation+kernel+for+Gaussian+process&rft.volume=550&rft.pages=126472&rft.date=2023&rft_id=info%3Aarxiv%2F2211.08262&rft_id=info%3Adoi%2F10.1016%2Fj.neucom.2023.126472&rft.aulast=Saves&rft.aufirst=Paul&rft.au=Diouane%2C+Youssef&rft.au=Bartoli%2C+Nathalie&rft.au=Lefebvre%2C+Thierry&rft.au=Morlier%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKriging" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Kriging" class="extiw" title="commons:Category:Kriging">Kriging</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Further_reading_cleanup plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This "<a href="/wiki/Wikipedia:Manual_of_Style/Layout#Further_reading" title="Wikipedia:Manual of Style/Layout">Further reading</a>" section <b>may need cleanup</b>.<span class="hide-when-compact"> Please read the <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">editing guide</a> and help improve the section.</span> <span class="date-container"><i>(<span class="date">November 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=15" title="Edit section: Historical references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChilèsDesassis2018" class="citation book cs1">Chilès, Jean-Paul; Desassis, Nicolas (2018). "Fifty Years of Kriging". <i>Handbook of Mathematical Geosciences</i>. Cham: Springer International Publishing. pp. 589–612. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-78999-6_29">10.1007/978-3-319-78999-6_29</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-78998-9" title="Special:BookSources/978-3-319-78998-9"><bdi>978-3-319-78998-9</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125362741">125362741</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fifty+Years+of+Kriging&rft.btitle=Handbook+of+Mathematical+Geosciences&rft.place=Cham&rft.pages=589-612&rft.pub=Springer+International+Publishing&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125362741%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-319-78999-6_29&rft.isbn=978-3-319-78998-9&rft.aulast=Chil%C3%A8s&rft.aufirst=Jean-Paul&rft.au=Desassis%2C+Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKriging" class="Z3988"></span></li> <li>Agterberg, F. P., <i>Geomathematics, Mathematical Background and Geo-Science Applications</i>, Elsevier Scientific Publishing Company, Amsterdam, 1974.</li> <li>Cressie, N. A. C., <i>The origins of kriging, Mathematical Geology</i>, v. 22, pp. 239–252, 1990.</li> <li>Krige, D. G., <i>A statistical approach to some mine valuations and allied problems at the Witwatersrand</i>, Master's thesis of the University of Witwatersrand, 1951.</li> <li>Link, R. F. and Koch, G. S., <i>Experimental Designs and Trend-Surface Analsysis, Geostatistics</i>, A colloquium, Plenum Press, New York, 1970.</li> <li>Matheron, G., "Principles of geostatistics", <i>Economic Geology</i>, 58, pp. 1246–1266, 1963.</li> <li>Matheron, G., "The intrinsic random functions, and their applications", <i>Adv. Appl. Prob.</i>, 5, pp. 439–468, 1973.</li> <li>Merriam, D. F. (editor), <i>Geostatistics</i>, a colloquium, Plenum Press, New York, 1970.</li> <li>Mockus, J., "On Bayesian methods for seeking the extremum." Proceedings of the IFIP Technical Conference. 1974.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kriging&action=edit&section=16" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Abramowitz, M., and Stegun, I. (1972), Handbook of Mathematical Functions, Dover Publications, New York.</li> <li>Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC Press, Taylor and Francis Group.</li> <li>Chiles, J.-P. and P. Delfiner (1999) <i>Geostatistics, Modeling Spatial uncertainty</i>, Wiley Series in Probability and statistics.</li> <li>Clark, I., and Harper, W. V., (2000) <i>Practical Geostatistics 2000</i>, Ecosse North America, USA.</li> <li>Cressie, N. (1993) <i>Statistics for spatial data</i>, Wiley, New York.</li> <li>David, M. (1988) <i>Handbook of Applied Advanced Geostatistical Ore Reserve Estimation</i>, Elsevier Scientific Publishing</li> <li>Deutsch, C. V., and Journel, A. G. (1992), GSLIB – Geostatistical Software Library and User's Guide, Oxford University Press, New York, 338 pp.</li> <li>Goovaerts, P. (1997) <i>Geostatistics for Natural Resources Evaluation</i>, Oxford University Press, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-511538-4" title="Special:BookSources/0-19-511538-4">0-19-511538-4</a>.</li> <li>Isaaks, E. H., and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 pp.</li> <li>Journel, A. G. and C. J. Huijbregts (1978) <i>Mining Geostatistics</i>, Academic Press London.</li> <li>Journel, A. G. (1989), Fundamentals of Geostatistics in Five Lessons, American Geophysical Union, Washington D.C.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=144">"Section 3.7.4. Interpolation by Kriging"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), New York: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3.7.4.+Interpolation+by+Kriging&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=W.+H.&rft.au=Teukolsky%2C+S.+A.&rft.au=Vetterling%2C+W.+T.&rft.au=Flannery%2C+B.+P.&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%3Fpg%3D144&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKriging" class="Z3988"></span>. Also, <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=836">"Section 15.9. Gaussian Process Regression"</a>.</li> <li>Stein, M. L. (1999), <i>Statistical Interpolation of Spatial Data: Some Theory for Kriging</i>, Springer, New York.</li> <li>Wackernagel, H. (1995) <i>Multivariate Geostatistics - An Introduction with Applications</i>, Springer Berlin</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist 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Kriging - Wikipedia