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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Efficient prediction designs for random fields</title><author><name sortKey="Muller, Werner G" sort="Muller, Werner G" uniqKey="Muller W" first="Werner G" last="Müller">Werner G. Müller</name><affiliation><nlm:aff id="au1"><institution>Department of Applied Statistics, Johannes-Kepler-University of Linz</institution><addr-line>Linz, Austria</addr-line></nlm:aff></affiliation></author><author><name sortKey="Pronzato, Luc" sort="Pronzato, Luc" uniqKey="Pronzato L" first="Luc" last="Pronzato">Luc Pronzato</name><affiliation><nlm:aff id="au2"><institution>Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis</institution><addr-line>Nice, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Rendas, Joao" sort="Rendas, Joao" uniqKey="Rendas J" first="Joao" last="Rendas">Joao Rendas</name><affiliation><nlm:aff id="au2"><institution>Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis</institution><addr-line>Nice, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Waldl, Helmut" sort="Waldl, Helmut" uniqKey="Waldl H" first="Helmut" last="Waldl">Helmut Waldl</name><affiliation><nlm:aff id="au1"><institution>Department of Applied Statistics, Johannes-Kepler-University of Linz</institution><addr-line>Linz, Austria</addr-line></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26300698</idno><idno type="pmc">4540167</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4540167</idno><idno type="RBID">PMC:4540167</idno><idno type="doi">10.1002/asmb.2084</idno><date when="2014">2014</date><idno type="wicri:Area/Pmc/Corpus">000006</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Efficient prediction designs for random fields</title><author><name sortKey="Muller, Werner G" sort="Muller, Werner G" uniqKey="Muller W" first="Werner G" last="Müller">Werner G. Müller</name><affiliation><nlm:aff id="au1"><institution>Department of Applied Statistics, Johannes-Kepler-University of Linz</institution><addr-line>Linz, Austria</addr-line></nlm:aff></affiliation></author><author><name sortKey="Pronzato, Luc" sort="Pronzato, Luc" uniqKey="Pronzato L" first="Luc" last="Pronzato">Luc Pronzato</name><affiliation><nlm:aff id="au2"><institution>Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis</institution><addr-line>Nice, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Rendas, Joao" sort="Rendas, Joao" uniqKey="Rendas J" first="Joao" last="Rendas">Joao Rendas</name><affiliation><nlm:aff id="au2"><institution>Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis</institution><addr-line>Nice, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Waldl, Helmut" sort="Waldl, Helmut" uniqKey="Waldl H" first="Helmut" last="Waldl">Helmut Waldl</name><affiliation><nlm:aff id="au1"><institution>Department of Applied Statistics, Johannes-Kepler-University of Linz</institution><addr-line>Linz, Austria</addr-line></nlm:aff></affiliation></author></analytic><series><title level="j">Applied Stochastic Models in Business and Industry</title><idno type="ISSN">1524-1904</idno><idno type="eISSN">1526-4025</idno><imprint><date when="2014">2014</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>For estimation and predictions of random fields, it is increasingly acknowledged that the kriging variance may be a poor representative of true uncertainty. Experimental designs based on more elaborate criteria that are appropriate for empirical kriging (EK) are then often non-space-filling and very costly to determine. In this paper, we investigate the possibility of using a compound criterion inspired by an equivalence theorem type relation to build designs quasi-optimal for the EK variance when space-filling designs become unsuitable. Two algorithms are proposed, one relying on stochastic optimization to explicitly identify the Pareto front, whereas the second uses the surrogate criteria as local heuristic to choose the points at which the (costly) true EK variance is effectively computed. We illustrate the performance of the algorithms presented on both a simple simulated example and a real oceanographic dataset. © 2014 The Authors. <italic>Applied Stochastic Models in Business and Industry</italic> published by John Wiley & Sons, Ltd.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Cressie, N" uniqKey="Cressie N">N Cressie</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Fang, Kt" uniqKey="Fang K">KT Fang</name></author><author><name sortKey="Li, R" uniqKey="Li R">R Li</name></author><author><name sortKey="Sudjianto, A" uniqKey="Sudjianto A">A Sudjianto</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kleijnen, Jpc" uniqKey="Kleijnen J">JPC Kleijnen</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Rasmussen, Ce" uniqKey="Rasmussen C">CE Rasmussen</name></author><author><name sortKey="Williams, Cki" uniqKey="Williams C">CKI Williams</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Santner, Tj" uniqKey="Santner T">TJ Santner</name></author><author><name sortKey="Williams, Bj" uniqKey="Williams B">BJ Williams</name></author><author><name sortKey="Notz, W" uniqKey="Notz W">W Notz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Todini, E" uniqKey="Todini E">E Todini</name></author><author><name sortKey="Ferraresi, M" uniqKey="Ferraresi M">M Ferraresi</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Harville, Da" uniqKey="Harville D">DA Harville</name></author><author><name sortKey="Jeske, Dr" uniqKey="Jeske D">DR Jeske</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Abt, M" uniqKey="Abt M">M Abt</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Zimmerman, Dl" uniqKey="Zimmerman D">DL Zimmerman</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Zhu, Z" uniqKey="Zhu Z">Z Zhu</name></author><author><name sortKey="Stein, Ml" uniqKey="Stein M">ML Stein</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Den Hertog, D" uniqKey="Den Hertog D">D den Hertog</name></author><author><name sortKey="Kleijnen, Jpc" uniqKey="Kleijnen J">JPC Kleijnen</name></author><author><name sortKey="Siem, Ayd" uniqKey="Siem A">AYD Siem</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Sjostedt De Luna, S" uniqKey="Sjostedt De Luna S">S Sjöstedt-De-Luna</name></author><author><name sortKey="Young, A" uniqKey="Young A">A Young</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Benassi, R" uniqKey="Benassi R">R Benassi</name></author><author><name sortKey="Bect, J" uniqKey="Bect J">J Bect</name></author><author><name sortKey="Vazquez, E" uniqKey="Vazquez E">E Vazquez</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Gauthier, B" uniqKey="Gauthier B">B Gauthier</name></author><author><name sortKey="Pronzato, L" uniqKey="Pronzato L">L Pronzato</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Pronzato, L" uniqKey="Pronzato L">L Pronzato</name></author><author><name sortKey="Muller, Wg" uniqKey="Muller W">WG Müller</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Vazquez, E" uniqKey="Vazquez E">E Vazquez</name></author><author><name sortKey="Bect, E" uniqKey="Bect E">E Bect</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kiefer, J" uniqKey="Kiefer J">J Kiefer</name></author><author><name sortKey="Wolfowitz, J" uniqKey="Wolfowitz J">J Wolfowitz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Muller, Wg" uniqKey="Muller W">WG Müller</name></author><author><name sortKey="Stehlik, M" uniqKey="Stehlik M">M Stehlík</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Stein, Ml" uniqKey="Stein M">ML Stein</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Zhang, H" uniqKey="Zhang H">H Zhang</name></author><author><name sortKey="Zimmerman, Dl" uniqKey="Zimmerman D">DL Zimmerman</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Zhu, Z" uniqKey="Zhu Z">Z Zhu</name></author><author><name sortKey="Zhang, H" uniqKey="Zhang H">H Zhang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Mardia, Kv" uniqKey="Mardia K">KV Mardia</name></author><author><name sortKey="Marshall, Rj" uniqKey="Marshall R">RJ Marshall</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Smirnov, Oa" uniqKey="Smirnov O">OA Smirnov</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Baldi Antognini, A" uniqKey="Baldi Antognini A">A Baldi Antognini</name></author><author><name sortKey="Zagoraiou, M" uniqKey="Zagoraiou M">M Zagoraiou</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Muller, Wg" uniqKey="Muller W">WG Müller</name></author><author><name sortKey="Pronzato, L" uniqKey="Pronzato L">L Pronzato</name></author><author><name sortKey="Waldl, H" uniqKey="Waldl H">H Waldl</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Muller, Wg" uniqKey="Muller W">WG Müller</name></author><author><name sortKey="Pronzato, L" uniqKey="Pronzato L">L Pronzato</name></author><author><name sortKey="Waldl, H" uniqKey="Waldl H">H Waldl</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Mckay, Md" uniqKey="Mckay M">MD McKay</name></author><author><name sortKey="Beckman, Rj" uniqKey="Beckman R">RJ Beckman</name></author><author><name sortKey="Conover, Wj" uniqKey="Conover W">WJ Conover</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lu, L" uniqKey="Lu L">L Lu</name></author><author><name sortKey="Anderson Cook, Cm" uniqKey="Anderson Cook C">CM Anderson-Cook</name></author><author><name sortKey="Robinson, Tj" uniqKey="Robinson T">TJ Robinson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lu, L" uniqKey="Lu L">L Lu</name></author><author><name sortKey="Anderson Cook, Cm" uniqKey="Anderson Cook C">CM Anderson-Cook</name></author><author><name sortKey="Robinson, Tj" uniqKey="Robinson T">TJ Robinson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Fang, Kt" uniqKey="Fang K">KT Fang</name></author><author><name sortKey="Wang, Y" uniqKey="Wang Y">Y Wang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Auffray, Y" uniqKey="Auffray Y">Y Auffray</name></author><author><name sortKey="Barbillon, P" uniqKey="Barbillon P">P Barbillon</name></author><author><name sortKey="Marin, Jm" uniqKey="Marin J">JM Marin</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Bohachevsky, Io" uniqKey="Bohachevsky I">IO Bohachevsky</name></author><author><name sortKey="Johnson, Me" uniqKey="Johnson M">ME Johnson</name></author><author><name sortKey="Stein, Ml" uniqKey="Stein M">ML Stein</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Jin, R" uniqKey="Jin R">R Jin</name></author><author><name sortKey="Chen, W" uniqKey="Chen W">W Chen</name></author><author><name sortKey="Sudjianto, A" uniqKey="Sudjianto A">A Sudjianto</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lacroix, G" uniqKey="Lacroix G">G Lacroix</name></author><author><name sortKey="Ruddick, K" uniqKey="Ruddick K">K Ruddick</name></author><author><name sortKey="Park, Y" uniqKey="Park Y">Y Park</name></author><author><name sortKey="Gypens, N" uniqKey="Gypens N">N Gypens</name></author><author><name sortKey="Lancelot, C" uniqKey="Lancelot C">C Lancelot</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ribeiro, Pj" uniqKey="Ribeiro P">PJ Ribeiro</name></author><author><name sortKey="Diggle, Pj" uniqKey="Diggle P">PJ Diggle</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Bivand, Rs" uniqKey="Bivand R">RS Bivand</name></author><author><name sortKey="Pebesma, Ej" uniqKey="Pebesma E">EJ Pebesma</name></author><author><name sortKey="Gmez Rubio, V" uniqKey="Gmez Rubio V">V Gmez-Rubio</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Fields Development Team" uniqKey="Fields Development Team">Fields Development Team</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Abramowitz, M" uniqKey="Abramowitz M">M Abramowitz</name></author><author><name sortKey="Stegun, Ia" uniqKey="Stegun I">IA Stegun</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Weisstein, Ew" uniqKey="Weisstein E">EW Weisstein</name></author></analytic></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Appl Stoch Models Bus Ind</journal-id><journal-id journal-id-type="iso-abbrev">Appl Stoch Models Bus Ind</journal-id><journal-id journal-id-type="publisher-id">asmb</journal-id><journal-title-group><journal-title>Applied Stochastic Models in Business and Industry</journal-title></journal-title-group><issn pub-type="ppub">1524-1904</issn><issn pub-type="epub">1526-4025</issn><publisher><publisher-name>Blackwell Publishing Ltd</publisher-name><publisher-loc>Oxford, UK</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26300698</article-id><article-id pub-id-type="pmc">4540167</article-id><article-id pub-id-type="doi">10.1002/asmb.2084</article-id><article-categories><subj-group subj-group-type="heading"><subject>Discussion Papers</subject></subj-group></article-categories><title-group><article-title>Efficient prediction designs for random fields</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Müller</surname><given-names>Werner G</given-names></name><xref ref-type="aff" rid="au1">a</xref><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="corresp" rid="cor2">†</xref></contrib><contrib contrib-type="author"><name><surname>Pronzato</surname><given-names>Luc</given-names></name><xref ref-type="aff" rid="au2">b</xref></contrib><contrib contrib-type="author"><name><surname>Rendas</surname><given-names>Joao</given-names></name><xref ref-type="aff" rid="au2">b</xref></contrib><contrib contrib-type="author"><name><surname>Waldl</surname><given-names>Helmut</given-names></name><xref ref-type="aff" rid="au1">a</xref></contrib><aff id="au1"><label>a</label><institution>Department of Applied Statistics, Johannes-Kepler-University of Linz</institution><addr-line>Linz, Austria</addr-line></aff><aff id="au2"><label>b</label><institution>Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis</institution><addr-line>Nice, France</addr-line></aff></contrib-group><author-notes><corresp id="cor1">*Correspondence to: Werner G. Müller, Department of Applied Statistics, Johannes-Kepler-University of Linz, Linz, Austria.</corresp><corresp id="cor2">†E-mail: <email>werner.mueller@jku.at</email></corresp></author-notes><pub-date pub-type="ppub"><month>3</month><year>2015</year></pub-date><pub-date pub-type="epub"><day>26</day><month>11</month><year>2014</year></pub-date><volume>31</volume><issue>2</issue><fpage>178</fpage><lpage>194</lpage><history><date date-type="received"><day>04</day><month>12</month><year>2013</year></date><date date-type="rev-recd"><day>20</day><month>10</month><year>2014</year></date><date date-type="accepted"><day>20</day><month>10</month><year>2014</year></date></history><permissions><copyright-statement>Copyright © 2015 John Wiley & Sons, Ltd.</copyright-statement><copyright-year>2015</copyright-year><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0/"><license-p>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</license-p></license></permissions><abstract><p>For estimation and predictions of random fields, it is increasingly acknowledged that the kriging variance may be a poor representative of true uncertainty. Experimental designs based on more elaborate criteria that are appropriate for empirical kriging (EK) are then often non-space-filling and very costly to determine. In this paper, we investigate the possibility of using a compound criterion inspired by an equivalence theorem type relation to build designs quasi-optimal for the EK variance when space-filling designs become unsuitable. Two algorithms are proposed, one relying on stochastic optimization to explicitly identify the Pareto front, whereas the second uses the surrogate criteria as local heuristic to choose the points at which the (costly) true EK variance is effectively computed. We illustrate the performance of the algorithms presented on both a simple simulated example and a real oceanographic dataset. © 2014 The Authors. <italic>Applied Stochastic Models in Business and Industry</italic> published by John Wiley & Sons, Ltd.</p></abstract><kwd-group><kwd>optimal design</kwd><kwd>Pareto front</kwd><kwd>empirical kriging</kwd><kwd>Gaussian process models</kwd></kwd-group></article-meta></front><body><sec><title>1. Introduction</title><p>The model underlying our investigations is the correlated scalar random field given by</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m1.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>Here, <italic>β</italic> is an unknown vector of parameters in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu2.jpg" mimetype="image"></inline-graphic></inline-formula>, a known function and the random term <italic>ε</italic>(<italic>x</italic>) has zero mean, (unknown) variance <italic>σ</italic><sup>2</sup> and a parameterized correlation structure such that <bold>E</bold>[<italic>ε</italic>(<italic>x</italic>)<italic>ε</italic>(<italic>x</italic><sup>′</sup>)] = <italic>σ</italic><sup>2</sup><italic>c</italic>(<italic>x</italic>,<italic>x</italic><sup>′</sup>;<italic>ν</italic>) with <italic>ν</italic> some unknown parameters. It is often assumed that the deterministic term has a linear structure, that is, <italic>η</italic>(<italic>x</italic>,<italic>β</italic>) = <italic>f</italic><sup>⊤</sup>(<italic>x</italic>)<italic>β</italic>, and that the random field <italic>ε</italic>(<italic>x</italic>) is Gaussian, allowing estimation of <italic>β</italic> and <italic>θ</italic>={<italic>σ</italic><sup>2</sup>,<italic>ν</italic>} by Maximum Likelihood. We are interested in making predictions <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu3.jpg" mimetype="image"></inline-graphic></inline-formula> of <italic>Y</italic>(·) at unsampled locations <italic>x</italic> in a compact subset <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu4.jpg" mimetype="image"></inline-graphic></inline-formula> of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu5.jpg" mimetype="image"></inline-graphic></inline-formula> using observations <italic>Y</italic>(<italic>x</italic><sub>1</sub>),…,<italic>Y</italic>(<italic>x</italic><sub><italic>n</italic></sub>) collected at some design points <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu6.jpg" mimetype="image"></inline-graphic></inline-formula>. Our objective is to select <italic>ξ</italic> (of given size <italic>n</italic>) in order to maximize the precision of the predictions <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu7.jpg" mimetype="image"></inline-graphic></inline-formula> over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu8.jpg" mimetype="image"></inline-graphic></inline-formula>. Problems of this type arise in diverse areas of spatial data analysis such as mining, hydrogeology, natural resource monitoring, and environmental sciences; see, for example, <xref rid="b1" ref-type="bibr">1</xref>. This has become the standard modeling paradigm in computer simulation experiments (cf. <xref rid="b2" ref-type="bibr">2</xref>–<xref rid="b5" ref-type="bibr">5</xref>), known under the designations of Gaussian process (GP) modelling and kriging analysis.</p><p>It is conventional practice that all unknown parameters are estimated from the same data set, but clearly, the classic kriging variance <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu9.jpg" mimetype="image"></inline-graphic></inline-formula> (see Appendix A2) does not reflect the additional uncertainty resulting from the estimation of the covariance parameters; for an early discussion of this issue, see <xref rid="b6" ref-type="bibr">6</xref>. A first-order expansion of the kriging variance for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu10.jpg" mimetype="image"></inline-graphic></inline-formula> around its true value is used in <xref rid="b7" ref-type="bibr">7</xref>, see also <xref rid="b8" ref-type="bibr">8</xref> for more precise developments, leading to an explicit additive correction term to the (normalized) kriging variance. This corrected kriging variance, considered in this paper, is given by</p><p><disp-formula id="m11"><graphic xlink:href="asmb0031-0178-m11.jpg" mimetype="image" position="float"></graphic><label>(1)</label></disp-formula></p><p>The design <italic>ξ</italic>* that minimizes criterion <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic>) (<xref ref-type="disp-formula" rid="m11">1</xref>) is called empirical kriging (EK)-optimal in <xref rid="b9" ref-type="bibr">9</xref>; see also <xref rid="b10" ref-type="bibr">10</xref> for another criterion, which is similar in spirit. Mentioned earlier, <italic>V</italic><sub><italic>ν</italic></sub>=<italic>V</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) stands for the covariance matrix of the estimate of the covariance parameters <italic>ν</italic> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu12.jpg" mimetype="image"></inline-graphic></inline-formula> is the posterior mean of <italic>Y</italic>(<italic>x</italic>) given the data at <italic>ξ</italic> = (<italic>x</italic><sub>1</sub>,…,<italic>x</italic><sub><italic>n</italic></sub>). Note that <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu13.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu14.jpg" mimetype="image"></inline-graphic></inline-formula> all depend on <italic>ξ</italic>.</p><p>Bootstrap solutions can be found in <xref rid="b11" ref-type="bibr">11</xref> and <xref rid="b12" ref-type="bibr">12</xref>. The influence of the uncertainty of covariance parameters on the precision of predictions can also be taken into account through a full Bayesian approach, but the computational cost is then much higher than for the standard kriging methodology; see, for example, <xref rid="b13" ref-type="bibr">13</xref>.</p><p>In contrast to designs that simply minimize the kriging variance, EK-optimal designs are typically not space-filling, in particular, for small numbers of observations. Unfortunately, maximization of the EK-criterion is computationally demanding, because each evaluation of (<xref ref-type="disp-formula" rid="m11">1</xref>) requires the evaluation of the target function for all points in the candidate set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu15.jpg" mimetype="image"></inline-graphic></inline-formula>, being unfeasible for high-dimensional design spaces as it is often the case with computer experiments. It would thus be useful to have an alternative criterion that can substitute (<xref ref-type="disp-formula" rid="m11">1</xref>) in the optimization procedure while still closely reflecting the actual prediction uncertainty. Note that although minimizing the traditional kriging variance looks equally demanding, one here can usually resort to efficient methods of generating space-filling designs rather than taking a direct approach. Also, the use of an integrated criterion rather than a minimax only alleviates the burden marginally for the corrected kriging variance (but the computational cost is then much reduced when using a spectral approach for the classic kriging variance; see <xref rid="b14" ref-type="bibr">14</xref>).</p><p>The paper is organized as follows. In Section 2, we motivate our approach, exploiting the intimate link that should exist between the precision of predictions of the values of the field from a given dataset and the accuracy of the estimates of the process parameters based on the same observations. Section 3 presents the actually new contributions of the paper, proposing two algorithms for identification of EK-suboptimal designs using as surrogates two design criteria related to parameter estimation. Two algorithms for Pareto-optimal design are proposed, both based on the idea of constraining the actual evaluation of <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic>), see (<xref ref-type="disp-formula" rid="m11">1</xref>), to points in the Pareto front of the surrogate criteria. Finally, Section 4 considers the identification of Pareto-optimal designs for a spatial oceanographic field produced by a biogeochemical mathematical model for the North Sea, and Section 5 draws conclusions on the efficiency and limitations of the approach and suggests topics for future work.</p><p>Before presenting the contributions of this paper, it is useful to consider the impact of the correction term in Equation (<xref ref-type="disp-formula" rid="m11">1</xref>) earlier, <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu16.jpg" mimetype="image"></inline-graphic></inline-formula>: its influence diminishes as the designs get denser, which happens, for a fixed <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu17.jpg" mimetype="image"></inline-graphic></inline-formula>, when the number of observations <italic>n</italic> increases. Designs that minimize <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu18.jpg" mimetype="image"></inline-graphic></inline-formula> are thus expected to resemble optimal designs for the EK-criterion when <italic>n</italic> is sufficiently large. We illustrate this on a simple example by comparing the behaviors of two greedy strategies for the sequential construction of designs that (<italic>S</italic><sub>1</sub>) place the next design point at the current maximum of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu19.jpg" mimetype="image"></inline-graphic></inline-formula>, or (<italic>S</italic><sub>2</sub>) at the current maximizer of the corrected kriging variance <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu20.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p><italic>Example 1</italic>Let <italic>σ</italic><sup>2</sup>=1,<italic>c</italic>(<italic>x</italic>,<italic>x</italic><sup>′</sup>;<italic>ν</italic>)= exp(−<italic>ν</italic>∥<italic>x</italic> − <italic>x</italic><sup>′</sup>∥), with <italic>ν</italic> = 7. We assume a constant mean <italic>η</italic>(<italic>x</italic>,<italic>β</italic>) = <italic>β</italic>. For this problem, the design</p><p><disp-formula id="m21"><graphic xlink:href="asmb0031-0178-m21.jpg" mimetype="image" position="float"></graphic><label>(2)</label></disp-formula></p><p>plotted in Figure <xref ref-type="fig" rid="fig07">7</xref>, left, is simultaneously maximin and minimax distance optimal in[0,1]<sup>2</sup> in the class of Latin hypercube (Lh) designs with <italic>n</italic> = 7 points; see <xref rid="b15" ref-type="bibr">15</xref>. We consider the sequential augmentation of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu22.jpg" mimetype="image"></inline-graphic></inline-formula> with strategies <italic>S</italic><sub>1</sub> and <italic>S</italic><sub>2</sub> defined earlier. Denote by <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu23.jpg" mimetype="image"></inline-graphic></inline-formula> the prediction at <italic>x</italic> for the design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu24.jpg" mimetype="image"></inline-graphic></inline-formula>. The design obtained by <italic>S</italic><sub>1</sub> is space-filling; see <xref rid="b16" ref-type="bibr">16</xref> for an analysis of its convergence properties in terms of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu25.jpg" mimetype="image"></inline-graphic></inline-formula> as <italic>k</italic>→<italic>∞</italic>. Figure <xref ref-type="fig" rid="fig01">1</xref> shows the sequence of design points generated by the two strategies when the design space is <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu26.jpg" mimetype="image"></inline-graphic></inline-formula>, which was chosen to encompass the Lh-designs. Figure <xref ref-type="fig" rid="fig02">2</xref> shows the evolution of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu27.jpg" mimetype="image"></inline-graphic></inline-formula> (triangles) and <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub><italic>k</italic></sub>) (squares) given by (<xref ref-type="disp-formula" rid="m11">1</xref>) as functions of <italic>k</italic>: the dashed line corresponds to <italic>S</italic><sub>1</sub> and the solid line to <italic>S</italic><sub>2</sub>.</p><fig id="fig01" position="float"><label>Figure 1</label><caption><p>First 15 additional points generated by the greedy strategies <italic>S</italic><sub>1</sub>(left) and <italic>S</italic><sub>2</sub>(right) in Example 1.</p></caption><graphic xlink:href="asmb0031-0178-f1"></graphic></fig><fig id="fig02" position="float"><label>Figure 2</label><caption><p><inline-formula><inline-graphic xlink:href="asmb0031-0178-mu30.jpg" mimetype="image"></inline-graphic></inline-formula> (triangles) and <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub><italic>k</italic></sub>) (squares) as functions of <italic>k</italic> for S<sub>1</sub> (dashed line) and S<sub>2</sub> (solid line).</p></caption><graphic xlink:href="asmb0031-0178-f2"></graphic></fig><p>All design points added by <italic>S</italic><sub>1</sub> tend to fill the design space, whereas the first three points added by <italic>S</italic><sub>2</sub> make a compromise between the precision of the prediction with <italic>ν</italic> supposed to be known and the precision of the estimation of <italic>ν</italic>. However, starting with <italic>k</italic> = 4,<italic>S</italic><sub>2</sub> tends to be space-filling, too. For <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu31.jpg" mimetype="image"></inline-graphic></inline-formula>, both strategies yield similar values for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu32.jpg" mimetype="image"></inline-graphic></inline-formula> and <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub><italic>k</italic></sub>), respectively, indicating that the effect of the correcting term in <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub><italic>k</italic></sub>) becomes negligible as the number of observations increases.</p><p>This illustrates the fact that application of the methods presented in this paper is only justified when improvements over space-filling designs are potentially significant. Then, the impact of the correction term added to the classic kriging variance in criterion (<xref ref-type="disp-formula" rid="m11">1</xref>) becomes important, which is the specific setting addressed by this paper. Note that this may depend upon the size of the designs (smaller), the dimension of the problem (larger), and the parameter values. The problem is of practical importance whenever the cost of each observation is large, as it is the case, for instance, in geophysical applications, where it reflects both installation and maintenance of the sensing equipment.</p></sec><sec><title>2. A relationship inspired by the equivalence theorem</title><p>Intuitively, accurate predictions of a spatial field in non-observed sites require good knowledge of the process parameters, and thus designs that optimize prediction-oriented criteria should perform well under criteria that measure estimation accuracy. Such relationships are commonly exploited in the field of design of experiments and run under the heading ‘equivalence theory’. They go back to the celebrated paper by Kiefer and Wolfowitz <xref rid="b17" ref-type="bibr">17</xref> who, by employing so-called design measures, and for parametric regression models with independent errors <italic>ε</italic>(<italic>x</italic>), established the equivalence of optimal designs for two criteria of optimality, one related to parameter estimation (D-optimality), <italic>i.e</italic>.</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m33.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>to be maximized, the other related to prediction (G-optimality), that is,</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m34.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>to be minimized.</p><p>The analogue to G-optimality for the correlated setup considered here is the EK-criterion (<xref ref-type="disp-formula" rid="m11">1</xref>), which provides a closed-form characterization of prediction uncertainty. Because parameters <italic>β</italic> and <italic>ν</italic>, related to the trend and covariance function, respectively, influence different moments of the process' statistical characterization, they have a remarkably distinct impact on the prediction error. This motivated Müller and Stehlík, see <xref rid="b18" ref-type="bibr">18</xref> to suggest the use of a convex composition of the two corresponding D-optimality criteria as a surrogate for EK:</p><p><disp-formula id="m35"><graphic xlink:href="asmb0031-0178-m35.jpg" mimetype="image" position="float"></graphic><label>(3)</label></disp-formula></p><p>to be maximized, where</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m36.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>with <italic>L</italic>(<italic>β</italic>,<italic>θ</italic>), the likelihood of <italic>β</italic> and <italic>θ</italic> = (<italic>σ</italic><sup>2</sup>,<italic>ν</italic>), and <italic>M</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) in the second term of (<xref ref-type="disp-formula" rid="m35">3</xref>) is the lower diagonal block of <italic>M</italic><sub><italic>θ</italic></sub>(<italic>ξ</italic>,<italic>θ</italic>); see (<xref ref-type="disp-formula" rid="m40">4</xref>).</p><p>For the linear model <italic>η</italic>(<italic>x</italic>,<italic>β</italic>) = <italic>f</italic><sup>⊤</sup>(<italic>x</italic>)<italic>β</italic>, simple computations lead to</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m37.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>and</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m38.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>where we used the notation <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu39.jpg" mimetype="image"></inline-graphic></inline-formula>. One may note that</p><p><disp-formula id="m40"><graphic xlink:href="asmb0031-0178-m40.jpg" mimetype="image" position="float"></graphic><label>(4)</label></disp-formula></p><p>with</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m41.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>The block <italic>V</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) of</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m42.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>which characterizes the precision of the estimation of <italic>ν</italic> and is used in (<xref ref-type="disp-formula" rid="m11">1</xref>), is given by <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu43.jpg" mimetype="image"></inline-graphic></inline-formula> and does not depend on <italic>σ</italic><sup>2</sup>.</p><p>The reason for considering <italic>M</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) in the definition of <italic>J</italic><sub><italic>α</italic></sub>(<italic>ξ</italic>), Equation (<xref ref-type="disp-formula" rid="m35">3</xref>), instead of the entire matrix <italic>M</italic><sub><italic>θ</italic></sub>(<italic>ξ</italic>,<italic>θ</italic>), is that <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu44.jpg" mimetype="image"></inline-graphic></inline-formula> is independent of <italic>σ</italic><sup>2</sup>, which only intervenes as a multiplicative factor in (<xref ref-type="disp-formula" rid="m11">1</xref>), which thus has no influence on the optimality of a given design for the EK criterion.</p><p>The parameter <italic>σ</italic><sup>2</sup> is sometimes assumed to be known, and in that case, <italic>V</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) coincides with <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu45.jpg" mimetype="image"></inline-graphic></inline-formula>. Assumption of knowledge about <italic>σ</italic><sup>2</sup> may be motivated by estimability considerations: under the infill design framework, typically not all components of <italic>θ</italic> = (<italic>σ</italic><sup>2</sup>,<italic>ν</italic>) are estimable, and only some of them, or some suitable functions of them, are micro-ergodic; see <xref rid="b19" ref-type="bibr">19</xref>,<xref rid="b20" ref-type="bibr">20</xref>. A reparametrization can then be used; see, for example, <xref rid="b21" ref-type="bibr">21</xref>, with <italic>σ</italic><sup>2</sup> set to an arbitrary value. When both <italic>σ</italic><sup>2</sup> and <italic>ν</italic> are estimable, there is usually no big difference between <italic>V</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu46.jpg" mimetype="image"></inline-graphic></inline-formula>. One may refer to <xref rid="b22" ref-type="bibr">22</xref> for more details on these information matrices and to <xref rid="b23" ref-type="bibr">23</xref> for computationally efficient implementations for their calculation. We have preferred <italic>M</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>) over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu47.jpg" mimetype="image"></inline-graphic></inline-formula> in the definition (<xref ref-type="disp-formula" rid="m35">3</xref>) as it more strongly sharpens the desired balance between space-filling and non-space-filling behaviors, see, for example, <xref rid="b18" ref-type="bibr">18</xref>.</p><p>Some efforts have been made to uncover quasi-equivalence relations between optimal designs for prediction and for estimation, compared with <xref rid="b24" ref-type="bibr">24</xref> or <xref rid="b25" ref-type="bibr">25</xref>. However, it was shown in <xref rid="b26" ref-type="bibr">26</xref> that a strict equivalence between (<xref ref-type="disp-formula" rid="m11">1</xref>) and (<xref ref-type="disp-formula" rid="m35">3</xref>) does not hold, although optimal designs for one of the criteria tend to perform well under the other, as the example in the succeeding text shows.</p><p><italic>Example 1 (continued)</italic>Assume the model in Example 1 and consider 1000 i.i.d. random designs with <italic>n</italic> = 7 points. Each design is a random Latin hypercube (Lh); see, for example, <xref rid="b27" ref-type="bibr">27</xref>, where each component is independently perturbed by the addition of a normal random variable with zero mean and standard deviation 0.1 complemented by truncation to [0,1]. Figure <xref ref-type="fig" rid="fig03">3</xref>, left, shows the values of the two D-optimality criteria log|<italic>M</italic><sub><italic>β</italic></sub>(·,<italic>θ</italic>)| and log|<italic>M</italic><sub><italic>ν</italic></sub>(·,<italic>ν</italic>)| for these 1000 random designs. It is quite apparent that these two criteria are antagonistic. The star in the Figure corresponds to the values of the two optimality criteria for the maximin and minimax distance optimal Lh design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu48.jpg" mimetype="image"></inline-graphic></inline-formula>; see (<xref ref-type="disp-formula" rid="m21">2</xref>). As anticipated, the space-filling <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu49.jpg" mimetype="image"></inline-graphic></inline-formula> yields a precise estimation of <italic>β</italic> but is extremely poor for estimating <italic>ν</italic>. We also computed, for each of the random designs, the value of the EK criterion. Figure <xref ref-type="fig" rid="fig03">3</xref> (right) presents the values of −<italic>J</italic><sub><italic>α</italic></sub>(·) for <italic>α</italic> = 0.75 against those of <italic>M</italic><italic>E</italic><italic>K</italic>(·) for the same set of designs (the evaluation of MEK uses <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu50.jpg" mimetype="image"></inline-graphic></inline-formula>). The first thing that we can observe is the good correlation of the two criteria for this choice of <italic>α</italic>. Again, we note that <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu51.jpg" mimetype="image"></inline-graphic></inline-formula> is the worst design for both criteria (they should be minimized). Points in the bottom left corner correspond to designs that are nearly simultaneously optimal for both criteria, confirming the conjecture about the possibility of inferring EK-optimality from the two D-optimality criteria.</p><fig id="fig03" position="float"><label>Figure 3</label><caption><p>Values of log|<italic>M</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>)| against log|<italic>M</italic><sub><italic>β</italic></sub>(<italic>ξ</italic>,<italic>θ</italic>)| (left) and of −<italic>J</italic><sub>0.75</sub>(<italic>ξ</italic>) against <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic>) (right) for 1000 random Lh designs in Example 1 (the star corresponds to <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu52.jpg" mimetype="image"></inline-graphic></inline-formula> given by (<xref ref-type="disp-formula" rid="m21">2</xref>)).</p></caption><graphic xlink:href="asmb0031-0178-f3"></graphic></fig><p>However, the correlation between <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic>) and <italic>J</italic><sub><italic>α</italic></sub>(<italic>ξ</italic>) observed in the example earlier can be much weaker for other values of <italic>α</italic>, and the determination, without evaluating <italic>M</italic><italic>E</italic><italic>K</italic>(·), of an <italic>α</italic><sup>⋆</sup> such that the maximization of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu53.jpg" mimetype="image"></inline-graphic></inline-formula> yields a design close to optimality for <italic>M</italic><italic>E</italic><italic>K</italic>(·) is a difficult open problem. An expression with a structure analogous to criterion (<xref ref-type="disp-formula" rid="m35">3</xref>) can be obtained if we search for the design that minimizes the entropy of the posterior distribution of the predicted field. The comparative analysis of the expressions of the two criteria leads to the conclusion that reasonable values of <italic>α</italic> must be constrained to the interval [0.5,1].</p></sec><sec><title>3. Pareto-optimal designs</title><p>In Section 2, we argued that finding designs <italic>ξ</italic> that minimize the EK criterion (<xref ref-type="disp-formula" rid="m11">1</xref>) should be intimately related to finding designs that optimize a suitable combination of the D-optimality criteria for <italic>β</italic> and <italic>ν</italic>. However, our ability to define a constructive experimental design method based on <italic>J</italic><sub><italic>α</italic></sub>(·) is hampered by the lack of an efficient methodology to select <italic>α</italic>.</p><p>In this section, we present two methods that overcome this difficulty and that effectively lead to design algorithms with complexity compatible with applications to real-case scenarios, as the one considered in Section 4. The idea underlying both algorithms is to consider the individual values of the two criteria log|<italic>M</italic><sub><italic>β</italic></sub>(<italic>ξ</italic>,<italic>θ</italic>)| and log|<italic>M</italic><sub><italic>ν</italic></sub>(<italic>ξ</italic>,<italic>ν</italic>)|, and to constrain the candidate set <italic>Ξ</italic> for the minimization of (<xref ref-type="disp-formula" rid="m11">1</xref>) to the set of non-dominated designs for the corresponding multicriteria optimization problem. The algorithms differ in the manner they approximate in this non-dominated solution set. The EK criterion (<xref ref-type="disp-formula" rid="m11">1</xref>) will thus play the role of a preference function for choosing designs in the reduced candidate set <italic>Ξ</italic>.</p><p>Other authors have addressed experimental design as a multicriteria optimization problem, constraining the set of possible solutions to those indicated by the corresponding Pareto surface, for example, <xref rid="b28" ref-type="bibr">28</xref> and <xref rid="b29" ref-type="bibr">29</xref>, where the authors discuss its advantages over the use of scalar ‘desirability functions’ and propose methods to choose amongst the efficient solutions of the Pareto surface. The main new contribution of our paper is the identification of two specific criteria whose set of non-dominated solutions is a relevant (small) candidate set for optimization of the EK variance.</p><p>The set Ξ of non-dominated (or Pareto-optimal) designs for the multiple objective optimization problem defined by log|<italic>M</italic><sub><italic>β</italic></sub>(·,<italic>θ</italic>)|, and log|<italic>M</italic><sub><italic>ν</italic></sub>(·,<italic>ν</italic>)| is defined by</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m54.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>The solid line in Figure <xref ref-type="fig" rid="fig08">8</xref> is an example of a Pareto surface for simultaneous maximization of the two criteria.</p><p>For <italic>K</italic> functions <italic>φ</italic><sub><italic>i</italic></sub>(·) to be maximized with respect to some variables <italic>ξ</italic> and taking values that vary continuously in <italic>K</italic> intervals <italic>I</italic><sub><italic>i</italic></sub>, the Pareto surface, or Pareto front, is in general, a (<italic>K</italic> − 1)-dimensional bounded surface included in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu57.jpg" mimetype="image"></inline-graphic></inline-formula>. In our case, <italic>K</italic> = 2 and the Pareto surface <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu58.jpg" mimetype="image"></inline-graphic></inline-formula> reduces to a bounded curve—to a finite subset of a curve when <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu59.jpg" mimetype="image"></inline-graphic></inline-formula> is finite. Let <italic>P</italic>(<italic>ℓ</italic>) = (<italic>C</italic><sub><italic>β</italic></sub>(<italic>ℓ</italic>),<italic>C</italic><sub><italic>ν</italic></sub>(<italic>ℓ</italic>)) be a parametrization of the Pareto surface, with <italic>ℓ</italic> denoting an index for its points. We denote by {<italic>ξ</italic>}(<italic>ℓ</italic>) the set of designs that map to point <italic>P</italic>(<italic>ℓ</italic>) in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu60.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>In what follows, we consider only designs constructed over a finite subset <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu61.jpg" mimetype="image"></inline-graphic></inline-formula> of the compact design space <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu62.jpg" mimetype="image"></inline-graphic></inline-formula> having <italic>Q</italic> elements. <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu63.jpg" mimetype="image"></inline-graphic></inline-formula> can be, for instance, a regular grid, with <italic>Q</italic> growing with <italic>d</italic> like <italic>q</italic><sup><italic>d</italic></sup> for some <italic>q</italic>, or the points of a low-discrepancy sequence; see, for example, <xref rid="b30" ref-type="bibr">30</xref>. Also, the maximization over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu64.jpg" mimetype="image"></inline-graphic></inline-formula> in (<xref ref-type="disp-formula" rid="m11">1</xref>) will be replaced by maximization over a finite subset <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu65.jpg" mimetype="image"></inline-graphic></inline-formula> of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu66.jpg" mimetype="image"></inline-graphic></inline-formula> with <italic>Q</italic><sup>′</sup> elements. In general, we shall omit the index <italic>Q</italic> and simply write <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu67.jpg" mimetype="image"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu68.jpg" mimetype="image"></inline-graphic></inline-formula>. Unless otherwise stated, we shall take <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu69.jpg" mimetype="image"></inline-graphic></inline-formula>, but other choices are possible. Also, in this paper, we only consider designs without replications.</p><sec><title>3.1. Minimizing <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic>) over the set of Pareto-optimal designs</title><p>In general, {<italic>ξ</italic>}(<italic>ℓ</italic>) is not a singleton and <italic>M</italic><italic>E</italic><italic>K</italic>(·) is not constant over this set. Moreover, the minimizer of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu70.jpg" mimetype="image"></inline-graphic></inline-formula> does not generally belong to some {<italic>ξ</italic>}(<italic>ℓ</italic>). The minimization of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu71.jpg" mimetype="image"></inline-graphic></inline-formula> is therefore not equivalent to the minimization of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over the set of Pareto-optimal designs. However, if our belief that the two parametric estimation criteria log|<italic>M</italic><sub><italic>β</italic></sub>(·,<italic>θ</italic>)| and log|<italic>M</italic><sub><italic>ν</italic></sub>(·,<italic>ν</italic>)| yield good surrogates for the EK criterion is valid, then (<italic>i</italic>) the variation of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over each set {<italic>ξ</italic>}(<italic>ℓ</italic>) should be much smaller than its variation across distant points in the Pareto surface <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu72.jpg" mimetype="image"></inline-graphic></inline-formula> (this fact has been checked numerically on simple examples), and (<italic>i</italic><italic>i</italic>) the minimum of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over the Pareto-optimal designs should approach the minimum of <italic>M</italic><italic>E</italic><italic>K</italic>(·) over <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu73.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>The method proposed in this section is based on the identification of a finite set of Pareto-optimal designs <italic>Ξ</italic><sub><italic>P</italic></sub>, the final design being obtained by minimizing <italic>M</italic><italic>E</italic><italic>K</italic>(·) over this reduced set:</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m74.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>Because the Pareto surface is the set of maxima of all scalar functions monotone in each criterion, we can construct a finite set of candidate designs Ξ<sub><italic>P</italic></sub> by optimizing <italic>J</italic><sub><italic>α</italic></sub>(·) for a finite set of values of <italic>α</italic>. Because the maximization of <italic>J</italic><sub><italic>α</italic></sub>(·) can only give points that belong to the convex hull of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu75.jpg" mimetype="image"></inline-graphic></inline-formula>, we may thereby miss some regions of the Pareto front, but we found on numerical examples that the effect is marginal, see, for example, Figure <xref ref-type="fig" rid="fig04">4</xref>.</p><fig id="fig04" position="float"><label>Figure 4</label><caption><p>Sampled points (in grey) for generating the Pareto surface (black); seven points lie on the convex hull, the one in white being selected.</p></caption><graphic xlink:href="asmb0031-0178-f4"></graphic></fig><p>The optimization of <italic>J</italic><sub><italic>α</italic></sub>(·) for fixed <italic>α</italic> is carried out using a simulated annealing (SA) algorithm; see, for example, <xref rid="b31" ref-type="bibr">31</xref>–<xref rid="b33" ref-type="bibr">33</xref>. In the examples in the succeeding text, the following implementation of the SA algorithm has been used (remember, we want to maximize <italic>J</italic><sub><italic>α</italic></sub>(·)):</p><list list-type="simple"><list-item><p><bold>Step 0</bold><italic>Initialization</italic>. Set initial temperature <italic>T</italic><sub>0</sub> and cooling factor <italic>r</italic>∈(0,1).Draw initial design <italic>ξ</italic><sub>0</sub>∼<italic>p</italic><sub>0</sub>(<italic>ξ</italic>)∝1,<italic>e</italic><sub>0</sub>=<italic>J</italic><sub><italic>α</italic></sub>(<italic>ξ</italic><sub>0</sub>).Set current best solution <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu76.jpg" mimetype="image"></inline-graphic></inline-formula>.Set <italic>k</italic> = 0.</p></list-item><list-item><p><bold>Step 1</bold><italic>Generate candidate</italic><inline-formula><inline-graphic xlink:href="asmb0031-0178-mu77.jpg" mimetype="image"></inline-graphic></inline-formula> by random perturbation of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu78.jpg" mimetype="image"></inline-graphic></inline-formula>.</p></list-item><list-item><p><bold>Step 2</bold><italic>Perform a local optimization</italic> of <italic>J</italic><sub><italic>α</italic></sub> around <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu79.jpg" mimetype="image"></inline-graphic></inline-formula>:</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m80.jpg" mimetype="image" position="float"></graphic></disp-formula></p></list-item><list-item><p><bold>Step 3</bold><italic>Update best solution</italic>. Let <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu81.jpg" mimetype="image"></inline-graphic></inline-formula>. If <italic>e</italic><sub><italic>k</italic> + 1</sub>><italic>e</italic><sup>⋆</sup>, then <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu82.jpg" mimetype="image"></inline-graphic></inline-formula>.</p></list-item><list-item><p><bold>Step 4</bold><italic>Random acceptance</italic>. If <italic>e</italic><sub><italic>k</italic> + 1</sub>><italic>e</italic><sub><italic>k</italic></sub>, set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu83.jpg" mimetype="image"></inline-graphic></inline-formula>. Otherwise, set</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m84.jpg" mimetype="image" position="float"></graphic></disp-formula></p></list-item><list-item><p><bold>Step 5</bold><italic>Temperature update</italic>. If <italic>ξ</italic><sub><italic>k</italic> + 1</sub>=<italic>ξ</italic><sub><italic>k</italic></sub> (no change has been made in step 4), update the temperature according to a geometric cooling scheme: <italic>T</italic><sub><italic>k</italic> + 1</sub>=<italic>r</italic><italic>T</italic><sub><italic>k</italic></sub>.</p></list-item><list-item><p><bold>Step 6</bold><italic>Stopping condition</italic>. If <italic>k</italic> = <italic>N</italic><sub><italic>m</italic><italic>a</italic><italic>x</italic></sub> stop; otherwise <italic>k</italic>←<italic>k</italic> + 1, return to step 1.</p></list-item></list><p>Throughout the algorithm, we keep track of the best solution found, which is eventually reported as <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu85.jpg" mimetype="image"></inline-graphic></inline-formula>. It is also expedient to start the algorithm with a space-filling design <italic>ξ</italic><sub>0</sub> to quickly weed out the cases for which our method is obviously unnecessary.</p><p>Like most random-search algorithms, under assumptions that are easily satisfied, the SA algorithm earlier allows us to reach an arbitrary neighborhood (in terms of criterion value) of a global maximum of <italic>J</italic><sub><italic>α</italic></sub>(·) in a finite number of iterations almost surely; see, for example, <xref rid="b31" ref-type="bibr">31</xref>. However, convergence may be slow and the risk of stopping the algorithm well before reaching some reasonable neighborhood of an optimal solution cannot be neglected.</p><p>The random perturbation <italic>p</italic><sub><italic>s</italic><italic>a</italic></sub>(<italic>ξ</italic>|<italic>ξ</italic><sub><italic>k</italic></sub>) in step 1 consists in the replacement of two randomly chosen points (<italic>x</italic><sub><italic>i</italic></sub>,<italic>x</italic><sub><italic>j</italic></sub>) of <italic>ξ</italic><sub><italic>k</italic></sub> by two points uniformly drawn (without replacement) from <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu86.jpg" mimetype="image"></inline-graphic></inline-formula>. A more generalized version could replace up to <italic>n</italic> points.</p><p>In step 2, local optimization (<italic>J</italic><sub><italic>α</italic></sub>(·),<italic>ξ</italic>) is a procedure that performs iterative optimization of <italic>J</italic><sub><italic>α</italic></sub>(·), starting from design <italic>ξ</italic>. Our implementation assumes that <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu87.jpg" mimetype="image"></inline-graphic></inline-formula> is a regular rook-type grid (the clique <italic>V</italic><sub><italic>x</italic></sub> of point <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu88.jpg" mimetype="image"></inline-graphic></inline-formula> is defined as the set of its North, South, West, East neighbors in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu89.jpg" mimetype="image"></inline-graphic></inline-formula>).</p><p><italic>Local optimization</italic>(<italic>J</italic><sub><italic>α</italic></sub>(·),<italic>ξ</italic>)Do { Set <italic>ξ</italic><sub>0</sub>=<italic>ξ</italic> Set <italic>J</italic><sub>0</sub>=<italic>J</italic><sub><italic>α</italic></sub>(<italic>ξ</italic><sub>0</sub>),<italic>J</italic> = <italic>J</italic><sub>0</sub>. For all <italic>x</italic><sub><italic>i</italic></sub>∈<italic>ξ</italic><sub>0</sub> (scan all points in <italic>ξ</italic>) For <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu90.jpg" mimetype="image"></inline-graphic></inline-formula> (consider replacement by all points in the clique of <italic>x</italic><sub><italic>i</italic></sub>) Set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu91.jpg" mimetype="image"></inline-graphic></inline-formula> If <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu92.jpg" mimetype="image"></inline-graphic></inline-formula> set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu93.jpg" mimetype="image"></inline-graphic></inline-formula>} while <italic>J</italic> > <italic>J</italic><sub>0</sub>Return(<italic>ξ</italic><sub>0</sub>)</p><p><italic>Example 1 (continued)</italic> We illustrate now, for the process introduced in Example 1, the application of this method for finding seven-point designs for prediction over the finite design space <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu94.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>Figure <xref ref-type="fig" rid="fig04">4</xref> shows the seven distinct values on the Pareto surface obtained by maximization of <italic>J</italic><sub><italic>α</italic></sub> for 11 values of <italic>α</italic> uniformly spread in [0.5,1]. The following parameters were used for the SA algorithm: <italic>T</italic><sub>0</sub>=0.6,<italic>r</italic> = 0.93,<italic>N</italic><sub><italic>m</italic><italic>a</italic><italic>x</italic></sub>=5000. Tests over a large number of executions of the SA led to no noticeable variations of the Pareto-front in Figure <xref ref-type="fig" rid="fig04">4</xref>.</p><p><italic>M</italic><italic>E</italic><italic>K</italic>(·) was subsequently computed for the seven Pareto-designs and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu95.jpg" mimetype="image"></inline-graphic></inline-formula> selected as the best one:</p><p><disp-formula id="m96"><graphic xlink:href="asmb0031-0178-m96.jpg" mimetype="image" position="float"></graphic><label>(5)</label></disp-formula></p><p>In Figure <xref ref-type="fig" rid="fig05">5</xref>, left, we present a contour plot of the corrected Kriging variance for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu97.jpg" mimetype="image"></inline-graphic></inline-formula>. In the plot, the black dots indicate the design points, at which the variance is zero, the asterisk indicates the location of the maximum.</p><fig id="fig05" position="float"><label>Figure 5</label><caption><p>Corrected kriging variance for the Pareto-optimal design (left) and the empirical kriging optimal design (right). Black dots indicate the design points.</p></caption><graphic xlink:href="asmb0031-0178-f5"></graphic></fig><p>We also searched directly for the optimal <italic>EK</italic> design <italic>ξ</italic><sup>⋆</sup> by optimizing <italic>M</italic><italic>E</italic><italic>K</italic>(·) using the SA algorithm. The much higher computational complexity of criterion evaluation imposed in this case constraining the maximum number of iterations of the SA algorithm to <italic>N</italic><sub><italic>m</italic><italic>a</italic><italic>x</italic></sub>=2000. The optimal design obtained is shown in Figure <xref ref-type="fig" rid="fig05">5</xref> (right) along with the corresponding surface of corrected Kriging variance. The effectiveness of the method can be appreciated by computing the efficiency of the Pareto-optimal design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu98.jpg" mimetype="image"></inline-graphic></inline-formula> with respect to the optimal design <italic>ξ</italic><sup>⋆</sup>, which is in this case <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu99.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>Notice that the construction of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu100.jpg" mimetype="image"></inline-graphic></inline-formula> only required seven evaluations of the expensive criterion <italic>M</italic><italic>E</italic><italic>K</italic>(·). For completeness, we also simulated 10 000 random sets of seven designs <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu101.jpg" mimetype="image"></inline-graphic></inline-formula> and computed min<italic>i</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub><italic>i</italic></sub>) for each. The empirical distribution of these minima is given in Figure <xref ref-type="fig" rid="fig06">6</xref>. It shows that 98% of the random designs generated with the same effort as ours lead to a corrected kriging variance larger than the one obtained using <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu102.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><fig id="fig06" position="float"><label>Figure 6</label><caption><p>Empirical distribution of empirical kriging minima for 10 000 random sets of seven designs; vertical bar indicates <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu103.jpg" mimetype="image"></inline-graphic></inline-formula>.</p></caption><graphic xlink:href="asmb0031-0178-f6"></graphic></fig></sec><sec><title>3.2. A simplified exchange algorithm</title><p>The method proposed in this section is based on an idea suggested in <xref rid="b15" ref-type="bibr">15</xref>. Like the algorithm earlier, it makes use of the Pareto front, but, in contrast to it, is deterministic, stops after a finite number of iterations when <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu104.jpg" mimetype="image"></inline-graphic></inline-formula> is finite, and therefore cannot provide any guarantee of asymptotic convergence. We call <italic>exchange</italic> the substitution of one point <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu105.jpg" mimetype="image"></inline-graphic></inline-formula> for one point <italic>x</italic><sub><italic>i</italic></sub> of the current design <italic>ξ</italic>. For any given design <italic>ξ</italic> with <italic>n</italic> distinct points in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu106.jpg" mimetype="image"></inline-graphic></inline-formula>, there are thus <italic>n</italic> × (<italic>Q</italic> − <italic>n</italic>) possible exchanges with <italic>Q</italic> the number of elements in <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu107.jpg" mimetype="image"></inline-graphic></inline-formula>. The algorithm starts with an arbitrary design, for example, space-filling, and exchanges one point at a time; only exchanges corresponding to non-dominated solutions for the two criteria log|<italic>M</italic><sub><italic>β</italic></sub>(·,<italic>θ</italic>)| and log|<italic>M</italic><sub><italic>ν</italic></sub>(·,<italic>ν</italic>)| are retained for the evaluation of <italic>M</italic><italic>E</italic><italic>K</italic>(·); the best among them gives the design carried to the next iteration.</p><list list-type="simple"><list-item><p><bold>Step 0</bold><italic>Initialization</italic>. Choose a space-filling design <italic>ξ</italic><sub>0</sub> with <italic>n</italic> points (<italic>e.g</italic>., a Lh design), compute <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu108.jpg" mimetype="image"></inline-graphic></inline-formula>, set <italic>k</italic> = 0.</p></list-item><list-item><p><bold>Step 1</bold><italic>Construction of the Pareto front</italic>. Construct the <italic>N</italic><sub><italic>k</italic></sub> designs <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu109.jpg" mimetype="image"></inline-graphic></inline-formula> corresponding to all possible exchanges for <italic>ξ</italic><sub><italic>k</italic></sub> and compute the associated values of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu110.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu111.jpg" mimetype="image"></inline-graphic></inline-formula>; construct the subset Ξ<sub><italic>k</italic></sub> of designs <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu112.jpg" mimetype="image"></inline-graphic></inline-formula> that correspond to non-dominated solutions for log|<italic>M</italic><sub><italic>β</italic></sub>(·,<italic>θ</italic>)| and log|<italic>M</italic><sub><italic>ν</italic></sub>(·,<italic>ν</italic>)|.</p></list-item><list-item><p><bold>Step 2</bold><italic>Evaluation of the EK-criterion</italic>. Compute <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu113.jpg" mimetype="image"></inline-graphic></inline-formula> for all <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu114.jpg" mimetype="image"></inline-graphic></inline-formula> in Ξ<sub><italic>k</italic></sub>.</p></list-item><list-item><p><bold>Step 3</bold><italic>Design update</italic>. If <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu115.jpg" mimetype="image"></inline-graphic></inline-formula>, stop;otherwise set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu116.jpg" mimetype="image"></inline-graphic></inline-formula>, return to step 1.</p></list-item></list><p>At step 1, <italic>N</italic><sub>0</sub>=<italic>n</italic> × (<italic>Q</italic> − <italic>n</italic>) exchanges are considered at first iteration, but <italic>N</italic><sub><italic>k</italic></sub>=(<italic>n</italic> − 1) × (<italic>Q</italic> − <italic>n</italic>) for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu117.jpg" mimetype="image"></inline-graphic></inline-formula>, because we do not need to consider the exchange of the same point of <italic>ξ</italic> for two consecutive iterations (although this was not used for the example in the paper, one may also restrict the exchange of <italic>x</italic><sub><italic>i</italic></sub> with points in some neighborhood, which significantly reduces <italic>N</italic><sub><italic>k</italic></sub>). Note that not all <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu118.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu119.jpg" mimetype="image"></inline-graphic></inline-formula> have to be stored because the set of non-dominated solutions Ξ<sub><italic>k</italic></sub> can be constructed iteratively. A further simplification is obtained by restricting Ξ<sub><italic>k</italic></sub> to designs that correspond to points on the convex hull of the Pareto front (which can also be constructed iteratively). A continuation of Example 1 gives an illustration.</p><p><italic>Example 1 (continued)</italic>We, again, restrict <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu120.jpg" mimetype="image"></inline-graphic></inline-formula> to the 25 × 25 grid of points with coordinates in the set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu121.jpg" mimetype="image"></inline-graphic></inline-formula>. Note that this set contains the design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu122.jpg" mimetype="image"></inline-graphic></inline-formula> given by (<xref ref-type="disp-formula" rid="m21">2</xref>), which is chosen as initial design <italic>ξ</italic><sub>0</sub> (with <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub>0</sub>)≃1.9124). The algorithm earlier, with Ξ<sub><italic>k</italic></sub> given by all points on the Pareto front stops after three iterations and returns a design with <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub>3</sub>) = 1.2060, requiring 967 evaluations of the EK-criterion. When Ξ<sub><italic>k</italic></sub> is restricted to the points on the convex hull of the Pareto front, the algorithm stops after four iterations and returns the design</p><p><disp-formula id="m123"><graphic xlink:href="asmb0031-0178-m123.jpg" mimetype="image" position="float"></graphic><label>(6)</label></disp-formula></p><p>see Figure <xref ref-type="fig" rid="fig07">7</xref> (right) with <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sub>4</sub>)≃1.2080. Figure <xref ref-type="fig" rid="fig08">8</xref> shows the values of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu124.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu125.jpg" mimetype="image"></inline-graphic></inline-formula>, at the first iteration of the algorithm. There are 296 non-dominated points on the Pareto front (in solid line), but only 15 points (indicated by stars) on its convex hull. The restriction of Ξ<sub><italic>k</italic></sub> to those points thus reduces the computational cost significantly: the EK-criterion (<xref ref-type="disp-formula" rid="m11">1</xref>) is only evaluated 45 times in total when the algorithm stops. Note that although this is six times more often than the procedure of Section 3.1, it gives a slight improvement of the criterion and is still considerably quicker than the SA procedure.</p><fig id="fig07" position="float"><label>Figure 7</label><caption><p>Latin hypercube design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu28.jpg" mimetype="image"></inline-graphic></inline-formula> (<xref ref-type="disp-formula" rid="m21">2</xref>) (left)—the circles have radius <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu29.jpg" mimetype="image"></inline-graphic></inline-formula>—and design <italic>ξ</italic><sub>4</sub> (<xref ref-type="disp-formula" rid="m123">6</xref>) (right).</p></caption><graphic xlink:href="asmb0031-0178-f7"></graphic></fig><fig id="fig08" position="float"><label>Figure 8</label><caption><p>Values of <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu55.jpg" mimetype="image"></inline-graphic></inline-formula> against <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu56.jpg" mimetype="image"></inline-graphic></inline-formula>, at iteration 1 of the simplified exchange algorithm in Example 1: the stars correspond to points on the convex hull of the Pareto front, which is indicated by the solid line.</p></caption><graphic xlink:href="asmb0031-0178-f8"></graphic></fig></sec></sec><sec><title>4. Design of sensors for an oceanographic field</title><p>We consider in this section application of the proposed design methodology to a case of practical interest, where the goal is to design a network of fixed observation sensors for an oceanographic field using outputs of a formal (numerical) model.</p><p>The data used in this study were made available through a collaboration with the institute MUMM, a department of the Royal Belgian Institute of Natural Sciences. It consists of snapshots of the output of the biogeochemical oceanographic model MIRO&CO <xref rid="b34" ref-type="bibr">34</xref>, run to simulate the evolution of inorganic and organic carbon and nutrients, phytoplankton, bacteria, and zooplankton with realistic forcing conditions. The outputs available cover 5 years, with a periodicity of 1 week, yielding a total of 258 maps. The model covers the entire water column of the Southern Bight of the North Sea, but in the study presented here, we concentrate on a horizontal (sea surface) grid of 21 × 21 points corresponding to the bay of the Seine river. We remark that the area of interest has a realistic geometry, in particular, it is not convex as it can be seen in Figure <xref ref-type="fig" rid="fig09">9</xref>.</p><fig id="fig09" position="float"><label>Figure 9</label><caption><p>Ammonium field over the region of interest.</p></caption><graphic xlink:href="asmb0031-0178-f9"></graphic></fig><p>MIRO&CO results from the integration of four modules describing: (i) the dynamics of phytoplankton, (ii) zooplankton, (iii) bacteria and dissolved/particulate organic matter degradation and (iv) nutrient (nitrate (NO3), ammonium (NH4), phosphate (PO4), and dissolved silica (DSi)) regeneration in the water column and the sediment. We limited our study to design of a network for observation of the distribution of NH4, one realization of this field is illustrated in Figure <xref ref-type="fig" rid="fig09">9</xref>, more precisely to identify the seven-point design that would enable the best extrapolation of the NH4 field to all grid points. The data simulated by the model over the entire grid has been used as a proxy to evaluate actual prediction uncertainty.</p><p>The problem addressed in this section is thus representative of the design of networks of fixed oceanography stations with limited size based on the predictions of (complex and numerically expensive) computer models, and can be found in many other application areas, such as meteorology, agriculture, air quality, and pollution surveillance. Two possible future extensions of practical interest would be (i) the identification of the best observation network for a <italic>vector</italic> field, for instance, for the observation of several components of the output of model MIRO&CO, for instance NH4 <italic>and</italic> zoo plankton; and (ii) exploitation of the temporal correlation of the observed fields. Although the latter can be, at least formally, handled using the same approach that is used here, by considering time as an additional coordinate and model both spatial and temporal trend and correlation, the former requires a deeper modification of the framework used in this paper, calling for more complex kriging algorithms.</p><p>In order to have a realistic evaluation of the extrapolation errors that can be expected in realistic conditions, we split the available model outputs in two sets. One set (of size <italic>M</italic><sub><italic>L</italic></sub>) is used for learning the parameters of the statistical model, on the basis of which the design is defined. The other set, of size <italic>M</italic><sub><italic>T</italic></sub>, is reserved for assessing the actual extrapolation performance, by computing the errors affecting the extrapolation of the values at the design points to other grid points, as explained in 4.3 in the succeeding text. We believe that for realistic models—like those that are currently used for operational meteorology or oceanography, like MIRO&CO—the correlation structure of model predictions closely reflects the local correlation structure of the true fields, and thus that our error evaluation is representative of the deviations between the extrapolated and the real fields that can be expected in reality.</p><p>The study compares the performance of the following designs:</p><list list-type="bullet"><list-item><p>the optimal design for the corrected EK criterion <italic>ξ</italic><sup>⋆</sup>;</p></list-item><list-item><p>the SA-optimization based Pareto design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu126.jpg" mimetype="image"></inline-graphic></inline-formula> (Section 3.1);</p></list-item><list-item><p>the deterministic local Pareto optimisation design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu127.jpg" mimetype="image"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="asmb0031-0178-mu128.jpg" mimetype="image"></inline-graphic></inline-formula> in Section 3.2); and</p></list-item><list-item><p>the optimal simple Kriging-space-filling design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu129.jpg" mimetype="image"></inline-graphic></inline-formula>.</p></list-item></list><sec><title>4.1. Model fitting</title><p>Because our design criteria depend on the process model, we started by fitting a GP model to the outputs of the numerical model available. As explained earlier, only a part of the available data is used for model learning. As the available data spans several years, model learning will be based on the entire first year (1993) to cover the expected seasonal variations, the rest (1994–1997) being used for performance assessment.</p><p>Let <italic>M</italic><sub><italic>L</italic></sub>=49 denote the size of the learning set, and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu130.jpg" mimetype="image"></inline-graphic></inline-formula> the corresponding set of model outputs, each <italic>Z</italic>(<italic>t</italic><sub><italic>j</italic></sub>) gathering the values of the NH4 field at time (i.e., week) <italic>t</italic><sub><italic>j</italic></sub> over the 21 × 21 grid of analysis. We assume that the snapshots are all statistically independent realisations of the same GP. <italic>Z</italic>(<italic>x</italic>,<italic>t</italic><sub><italic>j</italic></sub>) is the NH4 value at time <italic>t</italic><sub><italic>j</italic></sub> at site <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu131.jpg" mimetype="image"></inline-graphic></inline-formula>, which is modeled by</p><p><disp-formula id="m132"><graphic xlink:href="asmb0031-0178-m132.jpg" mimetype="image" position="float"></graphic><label>(7)</label></disp-formula></p><p>where the <italic>β</italic> are unknown deterministic coefficients and the Gaussian fields <italic>ε</italic><sup>(<italic>j</italic>)</sup>(<italic>x</italic>) are statistically independent realisations of the same GP. Using Maximum Likelihood, we fitted model (<xref ref-type="disp-formula" rid="m132">7</xref>) with linear trend and Matérn correlation function</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m133.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>where <italic>K</italic><sub><italic>ν</italic></sub>(·) is the modified Bessel function of the second kind of order <italic>ν</italic>, to the 49 field snapshots from 1993:</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m134.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>obtaining</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m135.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p><inline-formula><inline-graphic xlink:href="asmb0031-0178-mu136.jpg" mimetype="image"></inline-graphic></inline-formula>, range parameter <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu137.jpg" mimetype="image"></inline-graphic></inline-formula>, and smoothness parameter <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu138.jpg" mimetype="image"></inline-graphic></inline-formula>. All computations were performed by using the function <italic>likfit</italic> of the R-package <italic>geoR</italic> (<xref rid="b35" ref-type="bibr">35</xref>, details of its usage can be most conveniently found in <xref rid="b36" ref-type="bibr">36</xref>. We have also employed the built in correction for geometric anisotropy, which amounts to rotating and stretching the coordinates by multiplication with</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m139.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>yieldingratio <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu140.jpg" mimetype="image"></inline-graphic></inline-formula> and angle <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu141.jpg" mimetype="image"></inline-graphic></inline-formula>. For simplicity, these parameters were fixed for later evaluation purposes as well as <italic>γ</italic> set to 5/2, which gives the much simplified</p><p><disp-formula id="m142"><graphic xlink:href="asmb0031-0178-m142.jpg" mimetype="image" position="float"></graphic><label>(8)</label></disp-formula></p></sec><sec><title>4.2. Finding the optimal designs</title><p>We have chosen <italic>n</italic> = 14 throughout this example as this was the largest design size for which the Pareto-optimal designs were still superior to the space-filling with respect to the EK-criterion. The Pareto-optimal design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu143.jpg" mimetype="image"></inline-graphic></inline-formula> for the aforementioned model in the region of analysis has then been found by the method presented in Section 3.1, where 18 distinct points were identified on the convex hull of the Pareto-surface. The parameters of the SA algorithm were set as in Example 1, again, the algorithm was started from a random initialization; the minimal EK variance was identified for a quite high <italic>α</italic> = 0.99. In Figure <xref ref-type="fig" rid="fig10">10</xref>, we plot the corrected kriging variance for this design (right) and by direct optimization of the EK criterion (left), overlaid with the corresponding optimal designs (indicated by the black dots). We can see that while the Pareto-optimal design distributes the sampling points mainly along the boundary of the region of analysis, the EK-optimal design contains several points in the interior of the design space, and is able to keep the corrected kriging variance at lower levels <italic>M</italic><italic>E</italic><italic>K</italic>(<italic>ξ</italic><sup>⋆</sup>) = 0.8845 versus <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu144.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><fig id="fig10" position="float"><label>Figure 10</label><caption><p>Empirical kriging variance contour maps for <italic>ξ</italic><sup>⋆</sup> (left) and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu145.jpg" mimetype="image"></inline-graphic></inline-formula> (right).</p></caption><graphic xlink:href="asmb0031-0178-f10"></graphic></fig><p>We likewise employed our sequential algorithm to yield <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu146.jpg" mimetype="image"></inline-graphic></inline-formula> albeit requiring 12 iterations with a total of 78 evaluations of the EK-criterion, yielding <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu147.jpg" mimetype="image"></inline-graphic></inline-formula>. For comparison, we have also calculated a 14-point space-filling design <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu148.jpg" mimetype="image"></inline-graphic></inline-formula> utilizing a minimax distance criterion, which is conveniently implemented as function <italic>cover.design</italic> in the R-package <italic>fields</italic><xref rid="b37" ref-type="bibr">37</xref>, giving <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu149.jpg" mimetype="image"></inline-graphic></inline-formula> (a depiction of this design is given in Figure <xref ref-type="fig" rid="fig11">11</xref> (right)).</p><fig id="fig11" position="float"><label>Figure 11</label><caption><p>Empirical kriging variance contour maps for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu150.jpg" mimetype="image"></inline-graphic></inline-formula> (left) and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu151.jpg" mimetype="image"></inline-graphic></inline-formula> (right).</p></caption><graphic xlink:href="asmb0031-0178-f11"></graphic></fig><p>A comparison of computing times between the SA-based Pareto algorithms with direct optimisation of the EK-criterion has been carried out. Using a standard computer (with Intel Core i5 520M / 2.4 GHz, Dual-Core) a ratio of approximately 1 : 200 in favor of the SA-based algorithm has been observed for the small example in <xref ref-type="fig" rid="fig11">Figure 11</xref>. We expect this ratio to improve as the complexity of the problem (the dimension of the input space and/or the size of the design) increases.</p></sec><sec><title>4.3. Performance evaluation</title><p>We finally evaluated the performance of our quasi-optimal designs by assessing the extrapolation error over the <italic>M</italic><sub><italic>T</italic></sub>=196 fields in the test set, which were not used for model learning.</p><p>For each snapshot <italic>Z</italic><sub><italic>j</italic></sub>=<italic>Z</italic>(<italic>t</italic><sub><italic>j</italic></sub>) in the test set <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu152.jpg" mimetype="image"></inline-graphic></inline-formula>, we have firstly computed the corrected kriging variance for the fitted model and its maximum</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m153.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>where <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu154.jpg" mimetype="image"></inline-graphic></inline-formula> denotes the ML estimator of <italic>θ</italic> = (<italic>σ</italic><sup>2</sup>,<italic>ρ</italic>), see (<xref ref-type="disp-formula" rid="m142">8</xref>), obtained from the data <italic>Z</italic><sub><italic>j</italic></sub>=<italic>Z</italic>(<italic>t</italic><sub><italic>j</italic></sub>) at the design points in <italic>ξ</italic>. It turns out, however, that these empirically derived EK values are very sensitive to the individual weekly observations and vary strongly, particularly for the space-filling designs, due to the small design size. Their respective medians over the test set, however, well reflect the inferior performance of the space-filling design as they yield <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu155.jpg" mimetype="image"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu156.jpg" mimetype="image"></inline-graphic></inline-formula>, and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu157.jpg" mimetype="image"></inline-graphic></inline-formula>, respectively.</p><p>Secondly, we computed the maximum squared error over the grid of analysis</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m158.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>In Figure <xref ref-type="fig" rid="fig12">12</xref>, we present the ratios of <italic>E</italic><sub><italic>j</italic></sub> for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu159.jpg" mimetype="image"></inline-graphic></inline-formula> (left), <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu160.jpg" mimetype="image"></inline-graphic></inline-formula> (middle), and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu161.jpg" mimetype="image"></inline-graphic></inline-formula> (right) versus <italic>E</italic><sub><italic>j</italic></sub>(<italic>ξ</italic><sup>⋆</sup>) on a log scale. Here, the two Pareto based designs seem to be comparable with the EK-optimal design, whereas the space-filling design performs clearly worse.</p><fig id="fig12" position="float"><label>Figure 12</label><caption><p>Boxplots of the ratios of <italic>E</italic><sub><italic>j</italic></sub> versus <italic>E</italic><sub><italic>j</italic></sub>(<italic>ξ</italic><sup>⋆</sup>) on a log scale for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu162.jpg" mimetype="image"></inline-graphic></inline-formula> (left), <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu163.jpg" mimetype="image"></inline-graphic></inline-formula> (middle) and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu164.jpg" mimetype="image"></inline-graphic></inline-formula> (right).</p></caption><graphic xlink:href="asmb0031-0178-f12"></graphic></fig><p>Finally, we compute an indicator of the consistency of the design criteria, by computing the empirical mean-square error over the <italic>M</italic><sub><italic>T</italic></sub> realisations in the test set:</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m165.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>Those empirical counterparts of our original design criterion can be compared for the four employed designs. Although the EK-optimal eventually yields <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu166.jpg" mimetype="image"></inline-graphic></inline-formula>, we get even better values for <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu167.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu168.jpg" mimetype="image"></inline-graphic></inline-formula>, whereas <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu169.jpg" mimetype="image"></inline-graphic></inline-formula> is clearly the worst. All of these maxima are realized at the same location close to the harbour of Le Havre, indicating that the model fails to predict the high values of the field in the South-East small region, whose correlation structure strongly departs from the smoother variation in the open sea region, invalidating the predictions of the kriging variance. Note that these errors are strong even for the EK optimal design where a design point is located near that region. Another factor that may be affecting performance of the predictors in this region is related to the fact that the region of analysis is not convex, and thus the use of a covariance model based on simple Euclidean distance, like the Matérn model, cannot capture the internal structure of the water mass, which is confined by the region bathymetry.</p><p>Note that it may be more adequate to address the spatio-temporal nature of the data by allowing the parameters in the model to vary. For simplicity, we refrain from introducing that here, although due to lower estimates for <italic>σ</italic>, we expect further improvements of the performance of the Pareto-based designs. Furthermore, various other refinements are certainly possible that will be treated in future work. We are confident that the present exposition achieves the intended illustrational purpose well.</p></sec></sec><sec><title>5. Conclusions</title><p>This paper proposes methods for identification of designs quasi-optimal for the corrected kriging variance in the context of prediction of spatial Gaussian fields. The criterion, also known as EK criterion, that takes into account the increased variance due to limited accuracy of the estimates of the covariance of the GP, is especially important when this uncertainty is expected to make the corresponding optimal designs depart from space-filling.</p><p>Two methods are presented, both based on using design criteria for the estimation of process parameters (related to the trend and to the covariance of the random term). These design criteria are to be simultaneously optimized, as surrogate criteria for the EK-minimization. They offer increased efficiency compared with direct optimization of the corrected kriging variance, by limiting the evaluation of the numerically expensive EK-criterion to the Pareto-front of the two criteria. The methods differ on how the Pareto-surface is determined: whereas one relies on the use of stochastic optimization (SA) to sample the Pareto-front by optimizing distinct convex combinations of the two design criteria, the second is deterministic and iteratively approaches this surface. They have characteristics that are dual in some sense: whereas in the first the number of sampled points of the Pareto surface is fixed by the user (who sets the number of convex combinations that are optimized), in the second, the number of evaluations of the EK criterion is not fixed in advance. The price paid for the controlled complexity of the former is a potentially poorer sampling of the Pareto surface, leading eventually to a larger EK-value for the chosen design.</p><p>The paper illustrates the two methods both in a simple simulated model and also on a real application with an oceanography data set. The results obtained show the validity of the approach underlying the two algorithms, which are able to identify designs that are close to optimal efficiency, and yield prediction variances that may be significantly lower than it would be possible using standard space filling designs. Of course, as we remark in the introductory sections of the paper, efforts to optimise the EK criterion should be limited to those situations where cost of observations is large and the impact of the estimation of the covariance parameters cannot be neglected. In these cases, the methods proposed here offer a cost-effective alternative to the prohibitive direct optimisation of the relevant EK-criterion. Other approaches, based on complementing a design optimal for the estimation of the trend parameters with a design optimal for the estimation of the covariance parameters, are under current investigation.</p></sec></body><back><ack><p>The authors express their gratitude to Petra Vogl for some computations and Jean-Marc Fédou, Bertrand Gauthier, Gilles Menez, Éric Thierry and Milan Stehlík for discussions and two referees for useful remarks that led to an improved version of the paper. We are particularly grateful to Dale Zimmerman for pointing out a potentially confusing inaccuracy in the notation for the kriging variance. This work has been supported by the project ANR-2011-IS01-001-01 ‘DESIRE’ and FWF I 833-N18.</p></ack><app-group><app id="app1"><title>Appendix A. Information matrix and empirical kriging (EK) variance for the Matérn covariance function</title><p>We analyze the model <italic>Y</italic>(<italic>x</italic>) = <italic>f</italic><sup><italic>T</italic></sup>(<italic>x</italic>)<italic>β</italic> + <italic>ε</italic>(<italic>x</italic>) with Gaussian <italic>ε</italic>(<italic>x</italic>) with zero mean and Matérn isotropic covariance (cf. e.g., <xref rid="b19" ref-type="bibr">19</xref>)</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m170.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>where <italic>d</italic><sub><italic>i</italic><italic>j</italic></sub>=∥<italic>x</italic><sub><italic>i</italic></sub>−<italic>x</italic><sub><italic>j</italic></sub>∥,Γ(·) is the gamma function, <italic>K</italic><sub><italic>γ</italic></sub>(·) is the modified Bessel function of the second kind with order <italic>γ</italic> and <italic>ν</italic> = (<italic>ρ</italic>,<italic>γ</italic>)<sup><italic>T</italic></sup> are the non-negative covariance parameters.</p><sec><title>A.1. Information matrix for the variance-covariance parameters</title><p>Let <italic>θ</italic> = (<italic>σ</italic><sup>2</sup>,<italic>ν</italic><sup><italic>T</italic></sup>)<sup><italic>T</italic></sup> be the variance-covariance parameters of the Matérn covariance function. Then, the information matrix for <italic>θ</italic> and a design <italic>ξ</italic> = (<italic>x</italic><sub>1</sub>,…,<italic>x</italic><sub><italic>n</italic></sub>) is given by (<xref ref-type="disp-formula" rid="m40">4</xref>). That means, we have to compute the derivatives</p><p><disp-formula id="m171"><graphic xlink:href="asmb0031-0178-m171.jpg" mimetype="image" position="float"></graphic><label>(A.1)</label></disp-formula></p><p>with <italic>C</italic><sub><italic>ν</italic></sub> simplified to <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu172.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p><italic>Derivative with respect to</italic><italic>ρ</italic><italic>:</italic></p><p>The computation of the derivative with respect to <italic>ρ</italic> is straightforward. We just have to apply the product rule using <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu173.jpg" mimetype="image"></inline-graphic></inline-formula> (see <xref rid="b38" ref-type="bibr">38</xref>) and then apply the following Bessel function identity (see <xref rid="b39" ref-type="bibr">39</xref>).</p><p><disp-formula id="m174"><graphic xlink:href="asmb0031-0178-m174.jpg" mimetype="image" position="float"></graphic><label>(A.2)</label></disp-formula></p><p>This gives</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m175.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p><italic>Derivative with respect to the order</italic><italic>γ</italic><italic>:</italic></p><p>The computation of the derivative with respect to <italic>γ</italic> is more complicated. Although we do not utilize it in the paper, we give it here for completeness thus allowing for a more refined version of the application. First, we have to apply the product rule using the polygamma function of order 0, <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu176.jpg" mimetype="image"></inline-graphic></inline-formula>. Finally, we again have to apply the identity (<xref ref-type="disp-formula" rid="m174">A.2</xref>)). We obtain</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m177.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>For the derivative of the modified Bessel function of the second kind, we have to compute</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m178.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>where <italic>I</italic><sub><italic>γ</italic></sub>(·) is the modified Bessel function of the first kind with order <italic>γ</italic> (see <xref rid="b38" ref-type="bibr">38</xref>,<xref rid="b39" ref-type="bibr">39</xref>).</p></sec><sec><title>A.2. Empirical kriging variance for the Matern covariance function</title><p>In order to find EK-optimal designs, we have to minimize the design space maximum of the corrected kriging variance (see Equation (<xref ref-type="disp-formula" rid="m11">1</xref>))</p><p><disp-formula id="m179"><graphic xlink:href="asmb0031-0178-m179.jpg" mimetype="image" position="float"></graphic><label>(A.3)</label></disp-formula></p><p>Here, <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu180.jpg" mimetype="image"></inline-graphic></inline-formula> is the kriging prediction for design <italic>ξ</italic> at point <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu181.jpg" mimetype="image"></inline-graphic></inline-formula>. Let <italic>σ</italic><sup>2</sup><italic>c</italic><sub><italic>n</italic></sub> be the vector of covariances between <italic>x</italic> and design points <italic>ξ</italic>, then we have <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu182.jpg" mimetype="image"></inline-graphic></inline-formula>, where <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu183.jpg" mimetype="image"></inline-graphic></inline-formula> is the design matrix for the given model.</p><p>The classic kriging variance is <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu184.jpg" mimetype="image"></inline-graphic></inline-formula> and the correction term in (<xref ref-type="disp-formula" rid="m179">A.3</xref>) equals to <inline-formula><inline-graphic xlink:href="asmb0031-0178-mu185.jpg" mimetype="image"></inline-graphic></inline-formula>, where <italic>V</italic><sub><italic>ν</italic></sub> and</p><p><disp-formula><graphic xlink:href="asmb0031-0178-m186.jpg" mimetype="image" position="float"></graphic></disp-formula></p><p>again depends on the derivatives (<xref ref-type="disp-formula" rid="m171">A.1</xref>) of the Matérn covariance function.</p></sec></app></app-group><ref-list><title>References</title><ref id="b1"><label>1</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Cressie</surname><given-names>N</given-names></name></person-group><source>Statistics for Spatial Data (Wiley Series in Probability and Statistics)</source><year>1993</year><publisher-loc>New York</publisher-loc><publisher-name>Wiley-Interscience</publisher-name><comment>(revised edn)</comment></element-citation></ref><ref id="b2"><label>2</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Fang</surname><given-names>KT</given-names></name><name><surname>Li</surname><given-names>R</given-names></name><name><surname>Sudjianto</surname><given-names>A</given-names></name></person-group><source>Design and Modeling for Computer Experiments (Chapman & Hall/CRC Computer Science & Data Analysis)</source><year>2005</year><publisher-loc>London</publisher-loc><publisher-name>Chapman and Hall/CRC</publisher-name></element-citation></ref><ref id="b3"><label>3</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Kleijnen</surname><given-names>JPC</given-names></name></person-group><source>Design and Analysis of Simulation Experiments</source><year>2009</year><publisher-loc>US</publisher-loc><publisher-name>Springer</publisher-name></element-citation></ref><ref id="b4"><label>4</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Rasmussen</surname><given-names>CE</given-names></name><name><surname>Williams</surname><given-names>CKI</given-names></name></person-group><source>Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning series)</source><year>2005</year><publisher-loc>Cambridge</publisher-loc><publisher-name>The MIT Press</publisher-name></element-citation></ref><ref id="b5"><label>5</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Santner</surname><given-names>TJ</given-names></name><name><surname>Williams</surname><given-names>BJ</given-names></name><name><surname>Notz</surname><given-names>W</given-names></name></person-group><source>The Design and Analysis of Computer Experiments (Springer Series in Statistics)</source><year>2003</year><publisher-loc>New York</publisher-loc><publisher-name>Springer</publisher-name></element-citation></ref><ref id="b6"><label>6</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Todini</surname><given-names>E</given-names></name><name><surname>Ferraresi</surname><given-names>M</given-names></name></person-group><article-title>Influence of parameter estimation uncertainty in kriging</article-title><source>Journal of Hydrology</source><year>1996</year><volume>175</volume><fpage>555</fpage><lpage>566</lpage></element-citation></ref><ref id="b7"><label>7</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Harville</surname><given-names>DA</given-names></name><name><surname>Jeske</surname><given-names>DR</given-names></name></person-group><article-title>Mean squared error of estimation or prediction under a general linear model</article-title><source>Journal of the American Statistical Association</source><year>1992</year><volume>87</volume><fpage>724</fpage><lpage>731</lpage></element-citation></ref><ref id="b8"><label>8</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Abt</surname><given-names>M</given-names></name></person-group><article-title>Estimating the prediction mean squared error in Gaussian stochastic processes with exponential correlation structure</article-title><source>Scandinavian Journal of Statistics</source><year>1999</year><volume>26</volume><fpage>563</fpage><lpage>578</lpage></element-citation></ref><ref id="b9"><label>9</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Zimmerman</surname><given-names>DL</given-names></name></person-group><article-title>Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction</article-title><source>Environmetrics</source><year>2006</year><volume>17</volume><fpage>635</fpage><lpage>652</lpage></element-citation></ref><ref id="b10"><label>10</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Zhu</surname><given-names>Z</given-names></name><name><surname>Stein</surname><given-names>ML</given-names></name></person-group><article-title>Spatial sampling design for prediction with estimated parameters</article-title><source>Journal of Agricultural, Biological, and Environmental Statistics</source><year>2006</year><volume>11</volume><fpage>24</fpage><lpage>44</lpage></element-citation></ref><ref id="b11"><label>11</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>den Hertog</surname><given-names>D</given-names></name><name><surname>Kleijnen</surname><given-names>JPC</given-names></name><name><surname>Siem</surname><given-names>AYD</given-names></name></person-group><article-title>The correct Kriging variance estimated by bootstrapping</article-title><source>Journal of the Operational Research Society</source><year>2006</year><volume>57</volume><fpage>400</fpage><lpage>409</lpage></element-citation></ref><ref id="b12"><label>12</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Sjöstedt-De-Luna</surname><given-names>S</given-names></name><name><surname>Young</surname><given-names>A</given-names></name></person-group><article-title>The bootstrap and kriging prediction intervals</article-title><source>Scandinavian Journal of Statistics</source><year>2003</year><volume>30</volume><fpage>175</fpage><lpage>192</lpage></element-citation></ref><ref id="b13"><label>13</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Benassi</surname><given-names>R</given-names></name><name><surname>Bect</surname><given-names>J</given-names></name><name><surname>Vazquez</surname><given-names>E</given-names></name></person-group><article-title>Robust Gaussian process-based global optimization using a fully Bayesian expected improvement criterion</article-title><source>Learning and Intelligent Optimization</source><year>2011</year><volume>6683</volume><publisher-loc>Berlin, Heidelberg</publisher-loc><publisher-name>Springer</publisher-name><fpage>176</fpage><lpage>190</lpage><comment>Lecture Notes in Computer Science</comment></element-citation></ref><ref id="b14"><label>14</label><element-citation publication-type="other"><person-group person-group-type="author"><name><surname>Gauthier</surname><given-names>B</given-names></name><name><surname>Pronzato</surname><given-names>L</given-names></name></person-group><year>2013</year><comment>Approximation of IMSE-optimal designs via quadrature rules and spectral decomposition, to appear in Communications in Statistics - Simulation and Computation</comment></element-citation></ref><ref id="b15"><label>15</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Pronzato</surname><given-names>L</given-names></name><name><surname>Müller</surname><given-names>WG</given-names></name></person-group><article-title>Design of computer experiments: space filling and beyond</article-title><source>Statistics and Computing</source><year>2012</year><volume>22</volume><fpage>681</fpage><lpage>701</lpage></element-citation></ref><ref id="b16"><label>16</label><element-citation publication-type="other"><person-group person-group-type="author"><name><surname>Vazquez</surname><given-names>E</given-names></name><name><surname>Bect</surname><given-names>E</given-names></name></person-group><article-title>Sequential search based on kriging: convergence analysis of some algorithms</article-title><year>2011</year><comment><italic>Proceedings of 58th ISI World Statistics Congress</italic> Dublin, Ireland, 7 pages</comment></element-citation></ref><ref id="b17"><label>17</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Kiefer</surname><given-names>J</given-names></name><name><surname>Wolfowitz</surname><given-names>J</given-names></name></person-group><article-title>The equivalence of two extremum problems</article-title><source>Canadian Journal of Mathematics</source><year>1960</year><volume>12</volume><fpage>363</fpage><lpage>366</lpage></element-citation></ref><ref id="b18"><label>18</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Müller</surname><given-names>WG</given-names></name><name><surname>Stehlík</surname><given-names>M</given-names></name></person-group><article-title>Compound optimal spatial designs</article-title><source>Environmetrics</source><year>2010</year><volume>21</volume><fpage>354</fpage><lpage>364</lpage></element-citation></ref><ref id="b19"><label>19</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Stein</surname><given-names>ML</given-names></name></person-group><source>Interpolation of Spatial Data. Some Theory for Kriging</source><year>1999</year><publisher-loc>Heidelberg</publisher-loc><publisher-name>Springer</publisher-name></element-citation></ref><ref id="b20"><label>20</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Zhang</surname><given-names>H</given-names></name><name><surname>Zimmerman</surname><given-names>DL</given-names></name></person-group><article-title>Towards reconciling two asymptotic frameworks in spatial statistics</article-title><source>Biometrika</source><year>2005</year><volume>92</volume><fpage>921</fpage><lpage>936</lpage></element-citation></ref><ref id="b21"><label>21</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Zhu</surname><given-names>Z</given-names></name><name><surname>Zhang</surname><given-names>H</given-names></name></person-group><article-title>Spatial sampling design under the infill asymptotic framework</article-title><source>Environmetrics</source><year>2006</year><volume>17</volume><fpage>323</fpage><lpage>337</lpage></element-citation></ref><ref id="b22"><label>22</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Mardia</surname><given-names>KV</given-names></name><name><surname>Marshall</surname><given-names>RJ</given-names></name></person-group><article-title>Maximum likelihood estimation of models for residual covariance in spatial regression</article-title><source>Biometrika</source><year>1984</year><volume>71</volume><fpage>135</fpage><lpage>146</lpage></element-citation></ref><ref id="b23"><label>23</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Smirnov</surname><given-names>OA</given-names></name></person-group><article-title>Computation of the information matrix for models with spatial interaction on a lattice</article-title><source>Journal of Computational and Graphical Statistics</source><year>2005</year><volume>14</volume><fpage>910</fpage><lpage>927</lpage></element-citation></ref><ref id="b24"><label>24</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Baldi Antognini</surname><given-names>A</given-names></name><name><surname>Zagoraiou</surname><given-names>M</given-names></name></person-group><article-title>Exact optimal designs for computer experiments via Kriging metamodelling</article-title><source>Journal of Statistical Planning and Inference</source><year>2010</year><volume>140</volume><fpage>2607</fpage><lpage>2617</lpage></element-citation></ref><ref id="b25"><label>25</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Müller</surname><given-names>WG</given-names></name><name><surname>Pronzato</surname><given-names>L</given-names></name><name><surname>Waldl</surname><given-names>H</given-names></name></person-group><article-title>Beyond space-filling: An illustrative case</article-title><source>Procedia Environmental Sciences</source><year>2011</year><volume>7</volume><fpage>14</fpage><lpage>19</lpage></element-citation></ref><ref id="b26"><label>26</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Müller</surname><given-names>WG</given-names></name><name><surname>Pronzato</surname><given-names>L</given-names></name><name><surname>Waldl</surname><given-names>H</given-names></name></person-group><article-title>Relations between designs for prediction and estimation in random fields: an illustrative case</article-title><source>Advances and Challenges in Space-time Modelling of Natural Events (Lecture Notes in Statistics) 207</source><year>2012</year><publisher-loc>Berlin, Heidelberg</publisher-loc><publisher-name>Springer</publisher-name><fpage>125</fpage><lpage>139</lpage></element-citation></ref><ref id="b27"><label>27</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>McKay</surname><given-names>MD</given-names></name><name><surname>Beckman</surname><given-names>RJ</given-names></name><name><surname>Conover</surname><given-names>WJ</given-names></name></person-group><article-title>A comparison of three methods for selecting values of input variables in the analysis of output from a computer code</article-title><source>Technometrics</source><year>1979</year><volume>21</volume><fpage>239</fpage><lpage>245</lpage></element-citation></ref><ref id="b28"><label>28</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lu</surname><given-names>L</given-names></name><name><surname>Anderson-Cook</surname><given-names>CM</given-names></name><name><surname>Robinson</surname><given-names>TJ</given-names></name></person-group><article-title>Optimization of designed experiments based on multiple criteria utilizing a Pareto frontier</article-title><source>Technometrics</source><year>2011</year><volume>53</volume><fpage>353</fpage><lpage>365</lpage></element-citation></ref><ref id="b29"><label>29</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lu</surname><given-names>L</given-names></name><name><surname>Anderson-Cook</surname><given-names>CM</given-names></name><name><surname>Robinson</surname><given-names>TJ</given-names></name></person-group><article-title>A case study to demonstrate a pareto frontier for selecting a best response surface design while simultaneously optimizing multiple criteria</article-title><source>Applied Stochastic Models in Business and Industry</source><year>2012</year><volume>28</volume><issue>3</issue><fpage>206</fpage><lpage>221</lpage></element-citation></ref><ref id="b30"><label>30</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Fang</surname><given-names>KT</given-names></name><name><surname>Wang</surname><given-names>Y</given-names></name></person-group><source>Number-Theoretic Methods in Statistics (Chapman & Hall/CRC Monographs on Statistics & Applied Probability)</source><year>1993</year><publisher-loc>London</publisher-loc><publisher-name>Chapman and Hall/CRC</publisher-name></element-citation></ref><ref id="b31"><label>31</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Auffray</surname><given-names>Y</given-names></name><name><surname>Barbillon</surname><given-names>P</given-names></name><name><surname>Marin</surname><given-names>JM</given-names></name></person-group><article-title>Maximin design on non-hypercube domain and kernel interpolation</article-title><source>Statistics and Computing</source><year>2012</year><volume>22</volume><fpage>703</fpage><lpage>712</lpage></element-citation></ref><ref id="b32"><label>32</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Bohachevsky</surname><given-names>IO</given-names></name><name><surname>Johnson</surname><given-names>ME</given-names></name><name><surname>Stein</surname><given-names>ML</given-names></name></person-group><article-title>Generalized simulated annealing for function optimization</article-title><source>Technometrics</source><year>1986</year><volume>28</volume><fpage>209</fpage><lpage>217</lpage></element-citation></ref><ref id="b33"><label>33</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Jin</surname><given-names>R</given-names></name><name><surname>Chen</surname><given-names>W</given-names></name><name><surname>Sudjianto</surname><given-names>A</given-names></name></person-group><article-title>An efficient algorithm for constructing optimal design of computer experiments</article-title><source>Journal of Statistical Planning and Inference</source><year>2005</year><volume>134</volume><fpage>268</fpage><lpage>287</lpage></element-citation></ref><ref id="b34"><label>34</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lacroix</surname><given-names>G</given-names></name><name><surname>Ruddick</surname><given-names>K</given-names></name><name><surname>Park</surname><given-names>Y</given-names></name><name><surname>Gypens</surname><given-names>N</given-names></name><name><surname>Lancelot</surname><given-names>C</given-names></name></person-group><article-title>Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll <italic>α</italic> images</article-title><source>Journal of Marine Systems</source><year>2007</year><volume>64</volume><fpage>66</fpage><lpage>88</lpage></element-citation></ref><ref id="b35"><label>35</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ribeiro</surname><given-names>PJ</given-names></name><name><surname>Diggle</surname><given-names>PJ</given-names></name></person-group><article-title>geoR: A package for geostatistical analysis</article-title><source>R-News</source><year>2001</year><volume>1</volume><issue>2</issue><fpage>14</fpage><lpage>18</lpage><comment>Available for download at: <ext-link ext-link-type="uri" xlink:href="http://cran.r-project.org/doc/Rnews">http://cran.r-project.org/doc/Rnews</ext-link> [Accessed on 12 March 2010]</comment></element-citation></ref><ref id="b36"><label>36</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Bivand</surname><given-names>RS</given-names></name><name><surname>Pebesma</surname><given-names>EJ</given-names></name><name><surname>Gmez-Rubio</surname><given-names>V</given-names></name></person-group><source>Applied Spatial Data Analysis with R (Use R!)</source><year>2008</year><publisher-loc>New York</publisher-loc><publisher-name>Springer</publisher-name></element-citation></ref><ref id="b37"><label>37</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Fields Development Team</surname></name></person-group><source>fields: Tools for Spatial Data</source><year>2006</year><publisher-loc>Boulder, CO</publisher-loc><publisher-name>National Center for Atmospheric Research</publisher-name><comment><ext-link ext-link-type="uri" xlink:href="http://www.cgd.ucar.edu/Software/Fields">http://www.cgd.ucar.edu/Software/Fields</ext-link> [Accessed on 09 February 2009]</comment></element-citation></ref><ref id="b38"><label>38</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Abramowitz</surname><given-names>M</given-names></name><name><surname>Stegun</surname><given-names>IA</given-names></name></person-group><source>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</source><year>1972</year><publisher-loc>Mineola</publisher-loc><publisher-name>Dover, 9th printing</publisher-name><fpage>374</fpage><lpage>377</lpage></element-citation></ref><ref id="b39"><label>39</label><element-citation publication-type="other"><person-group person-group-type="author"><name><surname>Weisstein</surname><given-names>EW</given-names></name></person-group><year>2013</year><comment>Modified Bessel function of the second kind, From MathWorld–A Wolfram Web Resource, <ext-link ext-link-type="uri" xlink:href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html">http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html</ext-link> ) [Accessed on 01 February 2013]</comment></element-citation></ref></ref-list></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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