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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62446c485d5dd7338891fd96/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IMM</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws0">INFORMA<span class="_ _0"></span>TICS<span class="_"> </span>AND<span class="_"> </span>MA<span class="_ _0"></span>THEMA<span class="_ _0"></span>TICAL<span class="_"> </span>MODELLING</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">T<span class="_ _0"></span>echnical<span class="_"> </span>University<span class="_"> </span>of<span class="_"> </span>Denmark</div><div class="t m0 x4 h3 y4 ff1 fs1 fc0 sc0 ls0 ws0">DK-2800<span class="_ _1"> </span>Kgs.<span class="_"> </span>L<span class="_ _2"></span>yngby<span class="_ _1"> </span>–<span class="_ _1"> </span>Denmark</div><div class="t m0 x5 h4 y5 ff2 fs2 fc0 sc0 ls1 ws0">DAC<span class="_ _3"></span>E</div><div class="t m0 x6 h5 y6 ff2 fs3 fc0 sc0 ls2 ws0">AM<span class="_ _4"></span><span class="fs4 ls3">A<span class="_ _5"></span>TLAB<span class="_ _6"> </span><span class="fs3 ls0">Kriging<span class="_"> </span>T<span class="_ _5"></span>oolbo<span class="_ _7"></span>x</span></span></div><div class="t m0 x7 h6 y7 ff2 fs5 fc0 sc0 ls0 ws0">V<span class="_ _7"></span>ersion<span class="_"> </span>2.0,<span class="_ _8"> </span>A<span class="_ _2"></span>ugust<span class="_"> </span>1,<span class="_"> </span>2002</div><div class="t m0 x8 h6 y8 ff2 fs5 fc0 sc0 ls0 ws0">Søren<span class="_"> </span>N.<span class="_"> </span>Lophaven</div><div class="t m0 x9 h6 y9 ff2 fs5 fc0 sc0 ls0 ws0">Hans<span class="_"> </span>Bruun<span class="_"> </span>Nielsen</div><div class="t m0 xa h6 ya ff2 fs5 fc0 sc0 ls0 ws0">Jacob<span class="_"> </span>Søndergaar<span class="_ _7"></span>d</div><div class="c xb yb w2 h7"><div class="t m0 xc h8 yc ff3 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 xd h8 yd ff3 fs6 fc0 sc0 ls0 ws0">20</div><div class="t m0 xe h8 ye ff3 fs6 fc0 sc0 ls0 ws0">40</div><div class="t m0 xf h8 yf ff3 fs6 fc0 sc0 ls0 ws0">60</div><div class="t m0 x10 h8 y10 ff3 fs6 fc0 sc0 ls0 ws0">80</div><div class="t m0 x11 h8 y11 ff3 fs6 fc0 sc0 ls0 ws0">100</div><div class="t m0 x12 h8 y12 ff3 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x13 h8 y13 ff3 fs6 fc0 sc0 ls0 ws0">20</div><div class="t m0 x14 h8 y14 ff3 fs6 fc0 sc0 ls0 ws0">40</div><div class="t m0 x15 h8 y15 ff3 fs6 fc0 sc0 ls0 ws0">60</div><div class="t m0 x16 h8 y16 ff3 fs6 fc0 sc0 ls0 ws0">80</div><div class="t m0 x0 h8 y17 ff3 fs6 fc0 sc0 ls0 ws0">100</div><div class="t m0 x17 h8 y18 ff3 fs6 fc0 sc0 ls0 ws0">34</div><div class="t m0 x17 h8 y19 ff3 fs6 fc0 sc0 ls0 ws0">36</div><div class="t m0 x17 h8 y1a ff3 fs6 fc0 sc0 ls0 ws0">38</div><div class="t m0 x17 h8 y1b ff3 fs6 fc0 sc0 ls0 ws0">40</div><div class="t m0 x17 h8 y1c ff3 fs6 fc0 sc0 ls0 ws0">42</div><div class="t m0 x17 h8 y1d ff3 fs6 fc0 sc0 ls0 ws0">44</div><div class="t m0 x17 h8 y1e ff3 fs6 fc0 sc0 ls0 ws0">46</div></div><div class="t m0 x18 h9 y1f ff2 fs7 fc0 sc0 ls0 ws0">T<span class="_ _7"></span>echnical<span class="_"> </span>Report<span class="_"> </span>IMM-TR-2002-12</div><div class="t m0 x19 ha y20 ff4 fs7 fc0 sc0 ls0 ws0">Please<span class="_ _9"> </span>direct<span class="_ _9"> </span>comm<span class="_ _2"></span>unication<span class="_ _9"> </span>to<span class="_ _9"> </span>Hans<span class="_ _9"> </span>Bruun<span class="_ _9"> </span>Nielsen<span class="_ _9"> </span>(<span class="ff5">hbn@imm.dtu.dk</span>)</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62446c485d5dd7338891fd96/bg2.jpg"><div class="t m0 x1a hb y21 ff6 fs8 fc0 sc0 ls0 ws0">Con<span class="_ _2"></span>ten<span class="_ _7"></span>ts</div><div class="t m0 x1a hc y22 ff6 fs7 fc0 sc0 ls0 ws0">1.<span class="_ _a"> </span>In<span class="_ _2"></span>tro<span class="_ _b"></span>duction<span class="_ _c"> </span>1</div><div class="t m0 x1a hc y23 ff6 fs7 fc0 sc0 ls0 ws0">2.<span class="_ _a"> </span>Mo<span class="_ _b"></span>delling<span class="_ _d"> </span>and<span class="_ _d"> </span>Prediction<span class="_ _e"> </span>1</div><div class="t m0 x1b ha y24 ff4 fs7 fc0 sc0 ls0 ws0">2.1.<span class="_ _8"> </span>The<span class="_ _9"> </span>Kriging<span class="_ _9"> </span>Predictor<span class="_ _f"> </span><span class="ls4">..........................<span class="_ _10"> </span>2</span></div><div class="t m0 x1b ha y25 ff4 fs7 fc0 sc0 ls0 ws0">2.2.<span class="_ _8"> </span>Regression<span class="_ _9"> </span>Mo<span class="_ _b"></span>dels<span class="_ _11"> </span><span class="ls4">............................<span class="_ _10"> </span>5</span></div><div class="t m0 x1b ha y26 ff4 fs7 fc0 sc0 ls0 ws0">2.3.<span class="_ _8"> </span>Correlation<span class="_ _9"> </span>Mo<span class="_ _b"></span>dels<span class="_ _12"> </span><span class="ls4">............................<span class="_ _10"> </span>6</span></div><div class="t m0 x1a hc y27 ff6 fs7 fc0 sc0 ls0 ws0">3.<span class="_ _a"> </span>Generalized<span class="_ _d"> </span>Least<span class="_ _d"> </span>Squares<span class="_ _d"> </span>Fit<span class="_ _13"> </span>9</div><div class="t m0 x1b ha y28 ff4 fs7 fc0 sc0 ls0 ws0">3.1.<span class="_ _8"> </span>Computational<span class="_ _9"> </span>Asp<span class="_ _b"></span>ects<span class="_ _9"> </span><span class="ls4">..........................<span class="_ _d"> </span>1<span class="_ _14"></span>0</span></div><div class="t m0 x1a hc y29 ff6 fs7 fc0 sc0 ls0 ws0">4.<span class="_ _a"> </span>Exp<span class="_ _b"></span>erimen<span class="_ _2"></span>tal<span class="_ _d"> </span>Design<span class="_ _15"> </span>12</div><div class="t m0 x1b ha y2a ff4 fs7 fc0 sc0 ls0 ws0">4.1.<span class="_ _8"> </span>Rectangular<span class="_ _9"> </span>Grid<span class="_ _12"> </span><span class="ls4">.............................<span class="_ _d"> </span>1<span class="_ _16"></span>2</span></div><div class="t m0 x1b ha y2b ff4 fs7 fc0 sc0 ls0 ws0">4.2.<span class="_ _8"> </span>Latin<span class="_ _9"> </span>Hyp<span class="_ _b"></span>ercub<span class="_ _b"></span>e<span class="_ _9"> </span>Sampling<span class="_ _17"> </span><span class="ls4">........................<span class="_ _d"> </span>1<span class="_ _16"></span>2</span></div><div class="t m0 x1a hc y2c ff6 fs7 fc0 sc0 ls0 ws0">5.<span class="_ _a"> </span>Reference<span class="_ _d"> </span>Man<span class="_ _2"></span>ual<span class="_ _18"> </span>13</div><div class="t m0 x1b ha y2d ff4 fs7 fc0 sc0 ls0 ws0">5.1.<span class="_ _8"> </span>Mo<span class="_ _b"></span>del<span class="_ _9"> </span>Construction<span class="_ _9"> </span><span class="ls4">............................<span class="_ _d"> </span>1<span class="_ _16"></span>4</span></div><div class="t m0 x1b ha y2e ff4 fs7 fc0 sc0 ls0 ws0">5.2.<span class="_ _8"> </span>Ev<span class="_ _7"></span>aluate<span class="_ _9"> </span>the<span class="_ _9"> </span>Mo<span class="_ _b"></span>del<span class="_ _1"> </span><span class="ls4">............................<span class="_ _d"> </span>1<span class="_ _14"></span>5</span></div><div class="t m0 x1b ha y2f ff4 fs7 fc0 sc0 ls0 ws0">5.3.<span class="_ _8"> </span>Regression<span class="_ _9"> </span>Mo<span class="_ _b"></span>dels<span class="_ _11"> </span><span class="ls4">............................<span class="_ _d"> </span>1<span class="_ _16"></span>6</span></div><div class="t m0 x1b ha y30 ff4 fs7 fc0 sc0 ls0 ws0">5.4.<span class="_ _8"> </span>Correlation<span class="_ _9"> </span>Mo<span class="_ _b"></span>dels<span class="_ _12"> </span><span class="ls4">............................<span class="_ _d"> </span>1<span class="_ _16"></span>7</span></div><div class="t m0 x1b ha y31 ff4 fs7 fc0 sc0 ls0 ws0">5.5.<span class="_ _8"> </span>Exp<span class="_ _b"></span>erimen<span class="_ _2"></span>tal<span class="_ _9"> </span>Design<span class="_ _8"> </span><span class="ls4">...........................<span class="_ _d"> </span>1<span class="_ _16"></span>8</span></div><div class="t m0 x1b ha y32 ff4 fs7 fc0 sc0 ls0 ws0">5.6.<span class="_ _8"> </span>Auxiliary<span class="_ _9"> </span>F<span class="_ _7"></span>unctions<span class="_ _17"> </span><span class="ls4">............................<span class="_ _d"> </span>1<span class="_ _16"></span>9</span></div><div class="t m0 x1b ha y33 ff4 fs7 fc0 sc0 ls0 ws0">5.7.<span class="_ _8"> </span>Data<span class="_ _9"> </span>Files<span class="_ _a"> </span><span class="ls4">.................................<span class="_ _d"> </span>2<span class="_ _16"></span>0</span></div><div class="t m0 x1a hc y34 ff6 fs7 fc0 sc0 ls0 ws0">6.<span class="_ _a"> </span>Examples<span class="_ _d"> </span>of<span class="_ _d"> </span>Usage<span class="_ _19"> </span>21</div><div class="t m0 x1b ha y35 ff4 fs7 fc0 sc0 ls0 ws0">6.1.<span class="_ _8"> </span>W<span class="_ _7"></span>ork-through<span class="_ _9"> </span>Example<span class="_ _9"> </span><span class="ls4">..........................<span class="_ _d"> </span>2<span class="_ _14"></span>1</span></div><div class="t m0 x1b ha y36 ff4 fs7 fc0 sc0 ls0 ws0">6.2.<span class="_ _8"> </span>Adding<span class="_ _9"> </span>a<span class="_ _9"> </span>Regression<span class="_ _9"> </span>F<span class="_ _7"></span>unction<span class="_ _a"> </span><span class="ls4">......................<span class="_ _d"> </span>2<span class="_ _16"></span>4</span></div><div class="t m0 x1a hc y37 ff6 fs7 fc0 sc0 ls0 ws0">7.<span class="_ _a"> </span>Notation<span class="_ _1a"> </span>24</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62446c485d5dd7338891fd96/bg3.jpg"><div class="t m0 x19 hb y21 ff6 fs8 fc0 sc0 ls0 ws0">1.<span class="_ _1b"> </span>In<span class="_ _2"></span>tro<span class="_ _b"></span>duction</div><div class="t m0 x19 ha y38 ff4 fs7 fc0 sc0 ls0 ws0">This<span class="_ _17"> </span>rep<span class="_ _b"></span>ort<span class="_ _17"> </span>describ<span class="_ _b"></span>es<span class="_ _17"> </span>the<span class="_ _17"> </span>bac<span class="_ _2"></span>kground<span class="_ _17"> </span>for<span class="_ _6"> </span>and<span class="_ _17"> </span>use<span class="_ _17"> </span>of<span class="_ _17"> </span>the<span class="_ _17"> </span>softw<span class="_ _2"></span>are<span class="_ _17"> </span>pack<span class="_ _7"></span>age<span class="_ _17"> </span>D<span class="_ _2"></span>ACE</div><div class="t m0 x19 ha y39 ff4 fs7 fc0 sc0 ls0 ws0">(Design<span class="_ _6"> </span>and<span class="_ _6"> </span>Analysis<span class="_ _1"> </span>of<span class="_ _6"> </span>Computer<span class="_ _6"> </span>Exp<span class="_ _b"></span>erimen<span class="_ _2"></span>ts),<span class="_ _1c"> </span>whic<span class="_ _2"></span>h<span class="_ _6"> </span>is<span class="_ _1"> </span>a<span class="_ _6"> </span>Matlab<span class="_ _6"> </span>to<span class="_ _b"></span>olb<span class="_ _b"></span>o<span class="_ _2"></span>x<span class="_ _6"> </span>for</div><div class="t m0 x19 ha y3a ff4 fs7 fc0 sc0 ls0 ws0">w<span class="_ _2"></span>orking<span class="_ _9"> </span>with<span class="_ _9"> </span>kriging<span class="_ _9"> </span>approximations<span class="_ _9"> </span>to<span class="_ _9"> </span>computer<span class="_ _9"> </span>models.</div><div class="t m0 x19 ha y3b ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _1d"> </span>t<span class="_ _2"></span>ypical<span class="_ _1d"> </span>use<span class="_ _1d"> </span>of<span class="_ _1d"> </span>this<span class="_ _1d"> </span>softw<span class="_ _2"></span>are<span class="_ _1d"> </span>is<span class="_ _1d"> </span>to<span class="_ _1d"> </span>construct<span class="_ _1d"> </span>a<span class="_ _1d"> </span>kriging<span class="_ _1d"> </span>appro<span class="_ _2"></span>ximation<span class="_ _1d"> </span>mo<span class="_ _b"></span>del<span class="_ _1d"> </span>based</div><div class="t m0 x19 ha y3c ff4 fs7 fc0 sc0 ls0 ws0">on<span class="_ _17"> </span>data<span class="_ _6"> </span>from<span class="_ _17"> </span>a<span class="_ _6"> </span>computer<span class="_ _17"> </span>exp<span class="_ _b"></span>eriment,<span class="_ _6"> </span>and<span class="_ _17"> </span>to<span class="_ _6"> </span>use<span class="_ _17"> </span>this<span class="_ _6"> </span>appro<span class="_ _2"></span>ximation<span class="_ _6"> </span>mo<span class="_ _b"></span>del<span class="_ _17"> </span>as<span class="_ _17"> </span>a</div><div class="t m0 x19 ha y3d ff4 fs7 fc0 sc0 ls0 ws0">surrogate<span class="_ _d"> </span>for<span class="_ _d"> </span>the<span class="_ _d"> </span>computer<span class="_ _d"> </span>mo<span class="_ _b"></span>del.<span class="_ _a"> </span>Here,<span class="_ _d"> </span>a<span class="_ _d"> </span>computer<span class="_ _d"> </span>exp<span class="_ _b"></span>erimen<span class="_ _2"></span>t<span class="_ _d"> </span>is<span class="_ _d"> </span>a<span class="_ _d"> </span>collection<span class="_ _d"> </span>of</div><div class="t m0 x19 ha y3e ff4 fs7 fc0 sc0 ls0 ws0">pairs<span class="_ _d"> </span>of<span class="_ _d"> </span>input<span class="_ _17"> </span>and<span class="_ _d"> </span>responses<span class="_ _17"> </span>from<span class="_ _d"> </span>runs<span class="_ _d"> </span>of<span class="_ _d"> </span>a<span class="_ _17"> </span>computer<span class="_ _d"> </span>model.<span class="_ _a"> </span>Both<span class="_ _d"> </span>the<span class="_ _17"> </span>input<span class="_ _d"> </span>and</div><div class="t m0 x19 ha y3f ff4 fs7 fc0 sc0 ls0 ws0">the<span class="_ _9"> </span>resp<span class="_ _b"></span>onse<span class="_ _9"> </span>from<span class="_ _9"> </span>the<span class="_ _9"> </span>computer<span class="_ _9"> </span>model<span class="_ _9"> </span>are<span class="_ _9"> </span>likely<span class="_ _9"> </span>to<span class="_ _9"> </span>be<span class="_ _9"> </span>high<span class="_ _9"> </span>dimensional.</div><div class="t m0 x19 ha y40 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _6"> </span>computer<span class="_ _1"> </span>mo<span class="_ _b"></span>dels<span class="_ _6"> </span>we<span class="_ _6"> </span>address<span class="_ _6"> </span>are<span class="_ _1"> </span>deterministic,<span class="_ _1"> </span>and<span class="_ _1"> </span>thus<span class="_ _6"> </span>a<span class="_ _6"> </span>resp<span class="_ _b"></span>onse<span class="_ _6"> </span>from<span class="_ _1"> </span>a</div><div class="t m0 x19 ha y41 ff4 fs7 fc0 sc0 ls0 ws0">mo<span class="_ _b"></span>del<span class="_ _d"> </span>lac<span class="_ _2"></span>ks<span class="_ _d"> </span>random<span class="_ _d"> </span>error,<span class="_ _17"> </span>i.e.,<span class="_ _d"> </span>rep<span class="_ _b"></span>eated<span class="_ _d"> </span>runs<span class="_ _d"> </span>for<span class="_ _d"> </span>the<span class="_ _d"> </span>same<span class="_ _d"> </span>input<span class="_ _d"> </span>parameters<span class="_ _17"> </span>giv<span class="_ _2"></span>es</div><div class="t m0 x19 ha y42 ff4 fs7 fc0 sc0 ls0 ws0">the<span class="_ _9"> </span>same<span class="_ _9"> </span>resp<span class="_ _b"></span>onse<span class="_ _9"> </span>from<span class="_ _9"> </span>the<span class="_ _9"> </span>model.</div><div class="t m0 x19 ha y43 ff4 fs7 fc0 sc0 ls0 ws0">Often<span class="_ _1e"> </span>the<span class="_ _1e"> </span>appro<span class="_ _2"></span>ximation<span class="_ _1e"> </span>mo<span class="_ _b"></span>dels<span class="_ _1e"> </span>are<span class="_ _1e"> </span>needed<span class="_ _1e"> </span>as<span class="_ _1e"> </span>a<span class="_ _1e"> </span>part<span class="_ _1e"> </span>of<span class="_ _1e"> </span>a<span class="_ _1e"> </span>design<span class="_ _1e"> </span>problem,<span class="_ _1e"> </span>in<span class="_ _1e"> </span>whic<span class="_ _2"></span>h</div><div class="t m0 x19 ha y44 ff4 fs7 fc0 sc0 ls0 ws0">the<span class="_ _6"> </span>b<span class="_ _b"></span>est<span class="_ _6"> </span>set<span class="_ _6"> </span>of<span class="_ _6"> </span>parameters<span class="_ _6"> </span>for<span class="_ _6"> </span>running<span class="_ _6"> </span>the<span class="_ _6"> </span>computer<span class="_ _6"> </span>mo<span class="_ _b"></span>del<span class="_ _17"> </span>is<span class="_ _6"> </span>determined.<span class="_ _1f"> </span>This</div><div class="t m0 x19 ha y45 ff4 fs7 fc0 sc0 ls0 ws0">is<span class="_ _17"> </span>for<span class="_ _d"> </span>example<span class="_ _17"> </span>problems<span class="_ _17"> </span>where<span class="_ _17"> </span>a<span class="_ _d"> </span>computer<span class="_ _17"> </span>mo<span class="_ _b"></span>del<span class="_ _17"> </span>is<span class="_ _d"> </span>fitted<span class="_ _17"> </span>to<span class="_ _17"> </span>ph<span class="_ _2"></span>ysical<span class="_ _17"> </span>data.<span class="_ _12"> </span>This</div><div class="t m0 x19 ha y46 ff4 fs7 fc0 sc0 ls0 ws0">design<span class="_ _9"> </span>problem<span class="_ _d"> </span>is<span class="_ _9"> </span>related<span class="_ _d"> </span>to<span class="_ _9"> </span>the<span class="_ _d"> </span>more<span class="_ _9"> </span>general<span class="_ _d"> </span>problem<span class="_ _9"> </span>of<span class="_ _9"> </span>predicting<span class="_ _d"> </span>output<span class="_ _9"> </span>from<span class="_ _d"> </span>a</div><div class="t m0 x19 ha y47 ff4 fs7 fc0 sc0 ls0 ws0">computer<span class="_ _9"> </span>mo<span class="_ _b"></span>del<span class="_ _1e"> </span>at<span class="_ _9"> </span>untried<span class="_ _1e"> </span>inputs.</div><div class="t m0 x19 ha y48 ff4 fs7 fc0 sc0 ls0 ws0">In<span class="_ _9"> </span>Section<span class="_ _d"> </span>2<span class="_ _9"> </span>we<span class="_ _9"> </span>consider<span class="_ _9"> </span>mo<span class="_ _b"></span>dels<span class="_ _9"> </span>for<span class="_ _d"> </span>computer<span class="_ _9"> </span>exp<span class="_ _b"></span>erimen<span class="_ _2"></span>ts<span class="_ _d"> </span>and<span class="_ _9"> </span>efficien<span class="_ _2"></span>t<span class="_ _d"> </span>predictors,</div><div class="t m0 x19 ha y49 ff4 fs7 fc0 sc0 ls0 ws0">Section<span class="_ _6"> </span>3<span class="_ _6"> </span>discusses<span class="_ _17"> </span>generalized<span class="_ _6"> </span>least<span class="_ _6"> </span>squares<span class="_ _6"> </span>and<span class="_ _6"> </span>implemen<span class="_ _2"></span>tation<span class="_ _6"> </span>asp<span class="_ _b"></span>ects,<span class="_ _6"> </span>and<span class="_ _6"> </span>in</div><div class="t m0 x19 ha y4a ff4 fs7 fc0 sc0 ls0 ws0">Section<span class="_ _1d"> </span>4<span class="_ _20"> </span>w<span class="_ _2"></span>e<span class="_ _20"> </span>consider<span class="_ _1d"> </span>exp<span class="_ _b"></span>erimen<span class="_ _2"></span>tal<span class="_ _1d"> </span>design<span class="_ _20"> </span>for<span class="_ _1d"> </span>the<span class="_ _20"> </span>predictors.<span class="_ _17"> </span>Section<span class="_ _1d"> </span>5<span class="_ _20"> </span>is<span class="_ _1d"> </span>a<span class="_ _20"> </span>reference</div><div class="t m0 x19 ha y4b ff4 fs7 fc0 sc0 ls0 ws0">man<span class="_ _2"></span>ual<span class="_ _9"> </span>for<span class="_ _1e"> </span>the<span class="_ _1e"> </span>to<span class="_ _b"></span>olb<span class="_ _b"></span>o<span class="_ _2"></span>x,<span class="_ _9"> </span>and<span class="_ _1e"> </span>finally<span class="_ _9"> </span>examples<span class="_ _1e"> </span>of<span class="_ _9"> </span>usage<span class="_ _1e"> </span>and<span class="_ _1e"> </span>list<span class="_ _9"> </span>of<span class="_ _1e"> </span>notation<span class="_ _9"> </span>are<span class="_ _1e"> </span>given</div><div class="t m0 x19 ha y4c ff4 fs7 fc0 sc0 ls0 ws0">in<span class="_ _9"> </span>Sections<span class="_ _9"> </span>6<span class="_ _9"> </span>and<span class="_ _9"> </span>7.</div><div class="t m0 x19 hb y4d ff6 fs8 fc0 sc0 ls0 ws0">2.<span class="_ _1b"> </span>Mo<span class="_ _b"></span>delling<span class="_ _21"> </span>and<span class="_ _21"> </span>Prediction</div><div class="t m0 x19 ha y4e ff4 fs7 fc0 sc0 ls0 ws0">Giv<span class="_ _2"></span>en<span class="_ _21"> </span>a<span class="_ _a"> </span>set<span class="_ _21"> </span>of<span class="_ _21"> </span><span class="ff7">m<span class="_ _21"> </span></span>design<span class="_ _22"> </span>sites<span class="_ _21"> </span><span class="ff7">S<span class="_ _8"> </span></span><span class="ls5">=[<span class="_ _23"></span><span class="ff7">s</span></span></div><div class="t m0 x1c hd y4f ff8 fs9 fc0 sc0 ls6 ws0">1</div><div class="t m0 x1d ha y4e ff9 fs7 fc0 sc0 ls7 ws0">···<span class="_ _21"> </span><span class="ff7">s</span></div><div class="t m0 x1e hd y4f ffa fs9 fc0 sc0 ls8 ws0">m</div><div class="t m0 x1f ha y4e ff4 fs7 fc0 sc0 ls7 ws0">]</div><div class="t m0 x20 he y50 ffb fs9 fc0 sc0 ls8 ws0">></div><div class="t m0 x21 ha y4e ff4 fs7 fc0 sc0 ls0 ws0">with<span class="_ _21"> </span><span class="ff7">s</span></div><div class="t m0 x22 hd y4f ffa fs9 fc0 sc0 ls0 ws0">i</div><div class="t m0 x23 ha y4e ff9 fs7 fc0 sc0 ls0 ws0">∈<span class="_ _24"></span><span class="ff4 ls9">IR</span></div><div class="t m0 x24 hd y51 ffa fs9 fc0 sc0 lsa ws0">n</div><div class="t m0 x25 ha y4e ff4 fs7 fc0 sc0 ls0 ws0">and<span class="_ _21"> </span>resp<span class="_ _b"></span>onses</div><div class="t m0 x19 ha y52 ff7 fs7 fc0 sc0 ls0 ws0">Y<span class="_ _1c"> </span><span class="ff4 lsb">=[<span class="_ _25"></span><span class="ff7">y</span></span></div><div class="t m0 x26 hd y53 ff8 fs9 fc0 sc0 lsc ws0">1</div><div class="t m0 x27 ha y54 ff9 fs7 fc0 sc0 ls7 ws0">···<span class="_ _9"> </span><span class="ff7">y</span></div><div class="t m0 x28 hd y53 ffa fs9 fc0 sc0 ls8 ws0">m</div><div class="t m0 x29 ha y54 ff4 fs7 fc0 sc0 ls7 ws0">]</div><div class="t m0 x2a he y55 ffb fs9 fc0 sc0 ls8 ws0">></div><div class="t m0 xf ha y54 ff4 fs7 fc0 sc0 ls0 ws0">with<span class="_ _17"> </span><span class="ff7">y</span></div><div class="t m0 x2b hd y53 ffa fs9 fc0 sc0 ls0 ws0">i</div><div class="t m0 x2c ha y54 ff9 fs7 fc0 sc0 ls0 ws0">∈<span class="_ _24"></span><span class="ff4 ls9">IR</span></div><div class="t m0 x2d hd y56 ffa fs9 fc0 sc0 lsa ws0">q</div><div class="t m0 x2e ha y54 ff4 fs7 fc0 sc0 ls0 ws0">.<span class="_ _12"> </span>The<span class="_ _17"> </span>data<span class="_ _17"> </span>is<span class="_ _17"> </span>assumed<span class="_ _17"> </span>to<span class="_ _17"> </span>satisfy<span class="_ _d"> </span>the<span class="_ _17"> </span>normalization</div><div class="t m0 x19 ha y57 ff4 fs7 fc0 sc0 ls0 ws0">conditions</div><div class="t m0 x2f hd y58 ff8 fs9 fc0 sc0 ls0 ws0">1</div><div class="t m0 x30 ha y59 ff7 fs7 fc0 sc0 ls0 ws0">µ<span class="ff4">[</span>S</div><div class="t m0 x31 hd y5a ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,j</span></div><div class="t m0 x10 ha y59 ff4 fs7 fc0 sc0 lsb ws0">]=0<span class="_ _25"></span><span class="ff7">,<span class="_ _26"> </span><span class="ff4">V</span></span></div><div class="t m0 x32 hf y5b ffd fs7 fc0 sc0 lsb ws0">£</div><div class="t m0 x33 h10 y59 ff7 fs7 fc0 sc0 lsb ws0">S</div><div class="t m0 x34 hd y5a ffc fs9 fc0 sc0 lsc ws0">:<span class="_ _27"></span><span class="ffa ls0">,j</span></div><div class="t m0 x35 h10 y59 ff7 fs7 fc0 sc0 ls7 ws0">,S</div><div class="t m0 x36 hd y5a ffc fs9 fc0 sc0 ls8 ws0">:<span class="_ _28"></span><span class="ffa ls0">,j</span></div><div class="t m0 x37 hf y5c ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x38 ha y5d ff4 fs7 fc0 sc0 lsb ws0">=1<span class="_ _25"></span><span class="ff7 lsd">,j<span class="_ _29"></span><span class="ff4 lsb">=1<span class="_ _25"></span><span class="ff7 ls7">,...,n<span class="_"> </span>,</span></span></span></div><div class="t m0 x30 ha y5e ff7 fs7 fc0 sc0 ls7 ws0">µ<span class="_ _2a"></span><span class="ff4">[<span class="_ _2a"></span><span class="ff7">Y</span></span></div><div class="t m0 x31 hd y5f ffc fs9 fc0 sc0 ls8 ws0">:<span class="_ _28"></span><span class="ffa ls0">,j</span></div><div class="t m0 x10 ha y60 ff4 fs7 fc0 sc0 lsb ws0">]=0<span class="_ _25"></span><span class="ff7">,<span class="_ _26"> </span><span class="ff4">V</span></span></div><div class="t m0 x32 hf y61 ffd fs7 fc0 sc0 lsb ws0">£</div><div class="t m0 x33 h10 y60 ff7 fs7 fc0 sc0 lsb ws0">Y</div><div class="t m0 x34 hd y5f ffc fs9 fc0 sc0 lsc ws0">:<span class="_ _27"></span><span class="ffa ls0">,j</span></div><div class="t m0 x35 h10 y60 ff7 fs7 fc0 sc0 ls7 ws0">,Y</div><div class="t m0 x1c hd y5f ffc fs9 fc0 sc0 ls8 ws0">:<span class="_ _28"></span><span class="ffa ls0">,j</span></div><div class="t m0 x39 hf y62 ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x3a ha y63 ff4 fs7 fc0 sc0 lse ws0">=1<span class="_ _25"></span><span class="ff7 lsd">,j<span class="_ _29"></span><span class="ff4 lse">=1<span class="_ _25"></span><span class="ff7 ls7">,...,q<span class="_ _2b"> </span>,</span></span></span></div><div class="t m0 x3b ha y64 ff4 fs7 fc0 sc0 ls0 ws0">(2.1)</div><div class="t m0 x19 ha y65 ff4 fs7 fc0 sc0 ls0 ws0">where<span class="_ _9"> </span><span class="ff7">X</span></div><div class="t m0 x3c hd y66 ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,j</span></div><div class="t m0 x3d ha y67 ff4 fs7 fc0 sc0 ls0 ws0">is<span class="_ _9"> </span>the<span class="_ _d"> </span>v<span class="_ _7"></span>ector<span class="_ _d"> </span>giv<span class="_ _2"></span>en<span class="_ _9"> </span>by<span class="_ _1e"> </span>the<span class="_ _d"> </span><span class="ff7">j<span class="_ _b"></span></span>th<span class="_ _9"> </span>column<span class="_ _d"> </span>in<span class="_ _9"> </span>matrix<span class="_ _9"> </span><span class="ff7">X<span class="_ _3"></span></span>,<span class="_ _9"> </span>and<span class="_ _9"> </span><span class="ff7">µ</span>[<span class="_ _24"> </span><span class="ff9">·<span class="_ _24"></span></span>]<span class="_ _d"> </span>and<span class="_ _1e"> </span>V</div><div class="t m0 x3e hf y68 ffd fs7 fc0 sc0 ls0 ws0">£</div><div class="t m0 x3f ha y67 ff9 fs7 fc0 sc0 ls0 ws0">·<span class="ff7">,<span class="_ _24"> </span></span>·</div><div class="t m0 x40 hf y68 ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x19 ha y69 ff4 fs7 fc0 sc0 ls0 ws0">denote<span class="_ _9"> </span>resp<span class="_ _b"></span>ectiv<span class="_ _2"></span>ely<span class="_ _9"> </span>the<span class="_ _9"> </span>mean<span class="_ _9"> </span>and<span class="_ _9"> </span>the<span class="_ _9"> </span>co<span class="_ _2"></span>v<span class="_ _7"></span>ariance.</div><div class="t m0 x19 ha y6a ff4 fs7 fc0 sc0 ls0 ws0">F<span class="_ _0"></span>ollowing<span class="_ _2c"> </span>[9]<span class="_ _2c"> </span>we<span class="_ _2b"> </span>adopt<span class="_ _2c"> </span>a<span class="_ _20"> </span>model<span class="_ _1e"> </span>ˆ<span class="_ _2d"></span><span class="ff7">y<span class="_ _20"> </span><span class="ff4">that<span class="_ _2c"> </span>expresses<span class="_ _20"> </span>the<span class="_ _2b"> </span>deterministic<span class="_ _20"> </span>response<span class="_ _20"> </span></span>y<span class="_ _b"></span><span class="ff4">(</span>x<span class="ff4">)<span class="_ _24"> </span><span class="ff9">∈<span class="_ _24"></span></span><span class="ls9">IR</span></span></span></div><div class="t m0 x41 hd y6b ffa fs9 fc0 sc0 lsa ws0">q</div><div class="t m0 x42 ha y6c ff4 fs7 fc0 sc0 ls9 ws0">,</div><div class="t m0 x19 ha y6d ff4 fs7 fc0 sc0 ls0 ws0">for<span class="_ _1d"> </span>an<span class="_ _1e"> </span><span class="ff7">n<span class="_ _1d"> </span></span>dimensional<span class="_ _1e"> </span>input<span class="_ _1d"> </span><span class="ff7">x<span class="_ _2b"> </span><span class="ff9 lsf">∈D⊆</span></span><span class="ls9">IR</span></div><div class="t m0 x43 hd y6e ffa fs9 fc0 sc0 lsa ws0">n</div><div class="t m0 x44 ha y6f ff4 fs7 fc0 sc0 ls0 ws0">,<span class="_ _1e"> </span>as<span class="_ _1d"> </span>a<span class="_ _1e"> </span>realization<span class="_ _1d"> </span>of<span class="_ _1e"> </span>a<span class="_ _1d"> </span>regression<span class="_ _1e"> </span>mo<span class="_ _b"></span>del<span class="_ _1d"> </span><span class="ff9">F<span class="_ _17"> </span></span>and</div><div class="t m0 x45 h11 y70 ffe fsa fc0 sc0 ls0 ws0">1</div><div class="t m0 x46 h12 y71 fff fs1 fc0 sc0 ls0 ws0">The<span class="_ _2c"> </span>user<span class="_ _2c"> </span>does<span class="_ _2c"> </span>not<span class="_ _2c"> </span>hav<span class="_ _2"></span>e<span class="_ _2c"> </span>to<span class="_ _2c"> </span>think<span class="_ _2c"> </span>of<span class="_ _2c"> </span>this:<span class="_ _9"> </span>The<span class="_ _2c"> </span>first<span class="_ _2c"> </span>step<span class="_ _2c"> </span>in<span class="_ _2c"> </span>the<span class="_ _2c"> </span>mo<span class="_ _b"></span>del<span class="_ _2b"> </span>construction<span class="_ _2c"> </span>is<span class="_ _2c"> </span>to<span class="_ _2c"> </span>normalize</div><div class="t m0 x19 h12 y72 fff fs1 fc0 sc0 ls0 ws0">the<span class="_ _1d"> </span>giv<span class="_ _2"></span>en<span class="_ _1d"> </span><span class="ff10 ls10">S,<span class="_ _24"> </span>Y<span class="_ _6"> </span></span>so<span class="_ _1d"> </span>that<span class="_ _1d"> </span>(2.1)<span class="_ _1d"> </span>is<span class="_ _1d"> </span>satisfied,<span class="_ _1d"> </span>see<span class="_ _1d"> </span>(5.1)<span class="_ _1d"> </span>b<span class="_ _b"></span>elow.</div><div class="t m0 x47 ha y73 ff4 fs7 fc0 sc0 ls0 ws0">1</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62446c485d5dd7338891fd96/bg4.jpg"><div class="t m0 x1a ha y21 ff4 fs7 fc0 sc0 ls0 ws0">a<span class="_ _9"> </span>random<span class="_ _9"> </span>function<span class="_ _9"> </span>(sto<span class="_ _b"></span>c<span class="_ _2"></span>hastic<span class="_ _9"> </span>pro<span class="_ _b"></span>cess),</div><div class="t m0 x48 ha y74 ff4 fs7 fc0 sc0 ls0 ws0">ˆ<span class="_ _2d"></span><span class="ff7">y</span></div><div class="t m0 x6 hd y75 ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 xb ha y76 ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">x</span><span class="ls11">)=<span class="ff9">F<span class="_ _16"></span><span class="ff4">(<span class="_ _2e"></span><span class="ff7">β</span></span></span></span></div><div class="t m0 x49 hd y75 ffc fs9 fc0 sc0 ls12 ws0">:<span class="_ _16"></span><span class="ffa ls0">,`</span></div><div class="t m0 x4a ha y76 ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4 ls13">)+<span class="ff7">z</span></span></div><div class="t m0 x43 hd y75 ffa fs9 fc0 sc0 ls14 ws0">`</div><div class="t m0 x4b ha y76 ff4 fs7 fc0 sc0 ls13 ws0">(<span class="_ _28"></span><span class="ff7">x<span class="_ _28"></span><span class="ff4">)<span class="_ _28"></span><span class="ff7 lsd">,`<span class="_ _2f"></span><span class="ff4 lse">=1<span class="_ _25"></span><span class="ff7 lsf">,...,q<span class="_ _2b"> </span>.<span class="_ _30"> </span><span class="ff4 ls0">(2.2)</span></span></span></span></span></span></div><div class="t m0 x1a ha y77 ff4 fs7 fc0 sc0 ls0 ws0">W<span class="_ _0"></span>e<span class="_ _1c"> </span>use<span class="_ _1c"> </span>a<span class="_ _1c"> </span>regression<span class="_ _1c"> </span>mo<span class="_ _b"></span>del<span class="_ _1"> </span>which<span class="_ _1"> </span>is<span class="_ _1c"> </span>a<span class="_ _1c"> </span>linear<span class="_ _1c"> </span>com<span class="_ _2"></span>bination<span class="_ _1c"> </span>of<span class="_ _1c"> </span><span class="ff7">p<span class="_ _1c"> </span></span>c<span class="_ _2"></span>hosen<span class="_ _1c"> </span>functions</div><div class="t m0 x1a h10 y78 ff7 fs7 fc0 sc0 ls0 ws0">f</div><div class="t m0 x4c hd y79 ffa fs9 fc0 sc0 ls0 ws0">j</div><div class="t m0 x4d ha y7a ff4 fs7 fc0 sc0 ls11 ws0">:I<span class="_ _29"></span>R</div><div class="t m0 x4e hd y7b ffa fs9 fc0 sc0 ls12 ws0">n</div><div class="t m0 x4f ha y7a ff9 fs7 fc0 sc0 ls0 ws0">7→<span class="_ _1d"> </span><span class="ff4 ls9">IR<span class="_ _2c"> </span>,</span></div><div class="t m0 x50 ha y7c ff9 fs7 fc0 sc0 ls9 ws0">F<span class="_ _1e"> </span><span class="ff4">(<span class="_ _2c"> </span><span class="ff7">β</span></span></div><div class="t m0 x51 hd y7d ffc fs9 fc0 sc0 lsa ws0">:<span class="_ _b"></span><span class="ffa ls0">,`</span></div><div class="t m0 x52 ha y7e ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4 ls15">)=<span class="ff7">β</span></span></div><div class="t m0 x53 hd y7d ff8 fs9 fc0 sc0 ls16 ws0">1<span class="_ _4"></span><span class="ffa ls0">,`</span></div><div class="t m0 x32 h10 y7e ff7 fs7 fc0 sc0 ls0 ws0">f</div><div class="t m0 x33 hd y7d ff8 fs9 fc0 sc0 ls0 ws0">1</div><div class="t m0 x54 ha y7e ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">x</span><span class="ls17">)+<span class="ff9 ls7">···<span class="ff7">β</span></span></span></div><div class="t m0 x55 hd y7d ffa fs9 fc0 sc0 ls0 ws0">p,`</div><div class="t m0 x56 h10 y7e ff7 fs7 fc0 sc0 ls0 ws0">f</div><div class="t m0 x57 hd y7d ffa fs9 fc0 sc0 ls0 ws0">p</div><div class="t m0 x58 ha y7e ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">x</span>)</div><div class="t m0 x11 ha y7f ff4 fs7 fc0 sc0 ls18 ws0">=[<span class="_ _23"></span><span class="ff7">f</span></div><div class="t m0 x59 hd y80 ff8 fs9 fc0 sc0 ls19 ws0">1</div><div class="t m0 x5a ha y81 ff4 fs7 fc0 sc0 ls18 ws0">(<span class="_ _23"></span><span class="ff7">x<span class="_ _23"></span><span class="ff4">)<span class="_ _2a"></span><span class="ff9 ls7">···<span class="_ _1e"> </span><span class="ff7">f</span></span></span></span></div><div class="t m0 x39 hd y80 ffa fs9 fc0 sc0 ls8 ws0">p</div><div class="t m0 x5b ha y81 ff4 fs7 fc0 sc0 ls7 ws0">(<span class="_ _2a"></span><span class="ff7">x<span class="_ _2a"></span><span class="ff4 ls0">)]<span class="_ _2c"> </span><span class="ff7">β</span></span></span></div><div class="t m0 x5c hd y80 ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,`</span></div><div class="t m0 x11 ha y82 ff9 fs7 fc0 sc0 ls0 ws0">≡<span class="_ _12"> </span><span class="ff7">f<span class="_ _3"></span><span class="ff4">(</span>x<span class="ff4">)</span></span></div><div class="t m0 x33 he y83 ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x34 h10 y82 ff7 fs7 fc0 sc0 ls0 ws0">β</div><div class="t m0 x5d hd y84 ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,`</span></div><div class="t m0 x5e ha y82 ff7 fs7 fc0 sc0 ls0 ws0">.<span class="_ _31"> </span><span class="ff4">(2.3)</span></div><div class="t m0 x1a ha y85 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _9"> </span>co<span class="_ _b"></span>efficien<span class="_ _2"></span>ts<span class="_ _9"> </span><span class="ff9">{<span class="ff7">β</span></span></div><div class="t m0 xe hd y86 ffa fs9 fc0 sc0 ls1a ws0">k,`</div><div class="t m0 x29 ha y87 ff9 fs7 fc0 sc0 ls1b ws0">}<span class="_ _1e"> </span><span class="ff4 ls0">are<span class="_ _9"> </span>regression<span class="_ _9"> </span>parameters.</span></div><div class="t m0 x1a ha y88 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _9"> </span>random<span class="_ _9"> </span>pro<span class="_ _b"></span>cess<span class="_ _1e"> </span><span class="ff7">z<span class="_ _d"> </span></span>is<span class="_ _9"> </span>assumed<span class="_ _9"> </span>to<span class="_ _9"> </span>hav<span class="_ _2"></span>e<span class="_ _9"> </span>mean<span class="_ _9"> </span>zero<span class="_ _1e"> </span>and<span class="_ _9"> </span>cov<span class="_ _7"></span>ariance</div><div class="t m0 xe ha y89 ff4 fs7 fc0 sc0 ls0 ws0">E</div><div class="t m0 x5f hf y8a ffd fs7 fc0 sc0 ls0 ws0">£</div><div class="t m0 x60 h10 y89 ff7 fs7 fc0 sc0 ls0 ws0">z</div><div class="t m0 xb hd y8b ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 x50 ha y8c ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">w<span class="_ _b"></span></span>)<span class="ff7">z</span></div><div class="t m0 x61 hd y8b ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 x62 ha y8c ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">x</span>)</div><div class="t m0 x2c hf y8d ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x11 ha y8c ff4 fs7 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span><span class="ff7">σ</span></div><div class="t m0 x53 hd y8e ff8 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 x53 hd y8f ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 x5a ha y8c ff9 fs7 fc0 sc0 ls0 ws0">R<span class="ff4">(<span class="ff7 ls1b">θ,<span class="_ _32"> </span>w,<span class="_ _32"> </span>x<span class="_ _2"></span><span class="ff4">)<span class="ff7 lsd">,`<span class="_ _33"></span><span class="ff4 lse">=1<span class="_ _25"></span><span class="ff7 ls7">,...,q<span class="_ _34"> </span><span class="ff4 ls0">(2.4)</span></span></span></span></span></span></span></div><div class="t m0 x1a ha y90 ff4 fs7 fc0 sc0 ls0 ws0">b<span class="_ _b"></span>et<span class="_ _2"></span>w<span class="_ _2"></span>een<span class="_ _6"> </span><span class="ff7">z<span class="_ _35"></span></span>(<span class="ff7">w<span class="_ _b"></span></span>)<span class="_ _6"> </span>and<span class="_ _6"> </span><span class="ff7">z<span class="_ _b"></span></span>(<span class="ff7">x</span>),<span class="_ _1"> </span>where<span class="_ _1"> </span><span class="ff7">σ</span></div><div class="t m0 x11 hd y91 ff8 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11 hd y92 ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 x4a ha y93 ff4 fs7 fc0 sc0 ls0 ws0">is<span class="_ _6"> </span>the<span class="_ _6"> </span>pro<span class="_ _b"></span>cess<span class="_ _6"> </span>v<span class="_ _7"></span>ariance<span class="_ _1"> </span>for<span class="_ _6"> </span>the<span class="_ _6"> </span><span class="ff7">`</span>th<span class="_ _6"> </span>comp<span class="_ _b"></span>onen<span class="_ _2"></span>t</div><div class="t m0 x1a ha y94 ff4 fs7 fc0 sc0 ls0 ws0">of<span class="_ _1"> </span>the<span class="_ _1c"> </span>resp<span class="_ _b"></span>onse<span class="_ _1"> </span>and<span class="_ _1"> </span><span class="ff9">R</span>(<span class="ff7 ls1b">θ,<span class="_ _32"> </span>w,<span class="_ _32"> </span>x</span>)<span class="_ _6"> </span>is<span class="_ _1c"> </span>the<span class="_ _1"> </span>correlation<span class="_ _1c"> </span>mo<span class="_ _b"></span>del<span class="_ _1"> </span>with<span class="_ _1"> </span>parameters<span class="_ _1c"> </span><span class="ff7">θ<span class="_ _b"></span></span><span class="ls1c">.A<span class="_ _33"></span>n</span></div><div class="t m0 x1a ha y95 ff4 fs7 fc0 sc0 ls0 ws0">in<span class="_ _2"></span>terpretation<span class="_ _2c"> </span>of<span class="_ _20"> </span>the<span class="_ _2b"> </span>mo<span class="_ _b"></span>del<span class="_ _2c"> </span>(2.2)<span class="_ _20"> </span>is<span class="_ _2c"> </span>that<span class="_ _2c"> </span>deviations<span class="_ _2c"> </span>from<span class="_ _2c"> </span>the<span class="_ _20"> </span>regression<span class="_ _2b"> </span>mo<span class="_ _b"></span>del,<span class="_ _20"> </span>though</div><div class="t m0 x1a ha y96 ff4 fs7 fc0 sc0 ls0 ws0">the<span class="_ _17"> </span>resp<span class="_ _b"></span>onse<span class="_ _17"> </span>is<span class="_ _17"> </span>deterministic,<span class="_ _6"> </span>ma<span class="_ _2"></span>y<span class="_ _17"> </span>resemble<span class="_ _17"> </span>a<span class="_ _17"> </span>sample<span class="_ _17"> </span>path<span class="_ _17"> </span>of<span class="_ _17"> </span>a<span class="_ _17"> </span>(suitably<span class="_ _17"> </span>chosen)</div><div class="t m0 x1a ha y97 ff4 fs7 fc0 sc0 ls0 ws0">sto<span class="_ _b"></span>c<span class="_ _2"></span>hastic<span class="_ _9"> </span>pro<span class="_ _b"></span>cess<span class="_ _1e"> </span><span class="ff7">z<span class="_ _35"></span></span>.<span class="_ _6"> </span>In<span class="_ _9"> </span>the<span class="_ _1e"> </span>following<span class="_ _1e"> </span>we<span class="_ _1e"> </span>will<span class="_ _9"> </span>fo<span class="_ _b"></span>cus<span class="_ _9"> </span>on<span class="_ _9"> </span>the<span class="_ _9"> </span>kriging<span class="_ _9"> </span>predictor<span class="_ _9"> </span>for<span class="_ _9"> </span><span class="ff7">y<span class="_ _b"></span></span>.</div><div class="t m0 x1a ha y98 ff4 fs7 fc0 sc0 ls0 ws0">First,<span class="_ _9"> </span>ho<span class="_ _2"></span>wev<span class="_ _2"></span>er,<span class="_ _1e"> </span>we<span class="_ _1e"> </span>must<span class="_ _1e"> </span>b<span class="_ _b"></span>ear<span class="_ _9"> </span>in<span class="_ _9"> </span>mind<span class="_ _9"> </span>that<span class="_ _9"> </span>the<span class="_ _9"> </span>true<span class="_ _9"> </span>v<span class="_ _7"></span>alue<span class="_ _9"> </span>can<span class="_ _9"> </span>b<span class="_ _b"></span>e<span class="_ _9"> </span>written<span class="_ _9"> </span>as</div><div class="t m0 x63 h10 y99 ff7 fs7 fc0 sc0 ls0 ws0">y</div><div class="t m0 x31 hd y9a ffa fs9 fc0 sc0 ls0 ws0">`</div><div class="t m0 x64 ha y9b ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">x</span><span class="ls11">)=<span class="ff9">F<span class="_ _16"></span><span class="ff4">(<span class="_ _2e"></span><span class="ff7">β</span></span></span></span></div><div class="t m0 x65 hd y9a ffc fs9 fc0 sc0 ls12 ws0">:<span class="_ _16"></span><span class="ffa ls0">,`</span></div><div class="t m0 x66 ha y9b ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4 ls13">)+<span class="ff7">α<span class="_ _28"></span><span class="ff4">(<span class="_ _28"></span><span class="ff7">β</span></span></span></span></div><div class="t m0 x67 hd y9a ffc fs9 fc0 sc0 ls14 ws0">:<span class="_ _25"></span><span class="ffa ls0">,`</span></div><div class="t m0 x55 ha y9b ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4">)<span class="_"> </span><span class="ff7">,<span class="_ _36"> </span></span><span class="ls0">(2.5)</span></span></div><div class="t m0 x1a ha y9c ff4 fs7 fc0 sc0 ls0 ws0">where<span class="_ _d"> </span><span class="ff7">α<span class="_ _9"> </span></span>is<span class="_ _d"> </span>the<span class="_ _9"> </span>approximation<span class="_ _9"> </span>error.<span class="_ _37"> </span>The<span class="_ _d"> </span>assumption<span class="_ _9"> </span>is<span class="_ _d"> </span>that<span class="_ _9"> </span>by<span class="_ _9"> </span>prop<span class="_ _b"></span>er<span class="_ _9"> </span>choice<span class="_ _9"> </span>of<span class="_ _d"> </span><span class="ff7">β</span></div><div class="t m0 x1a ha y9d ff4 fs7 fc0 sc0 ls0 ws0">this<span class="_ _9"> </span>error<span class="_ _9"> </span>b<span class="_ _b"></span>eha<span class="_ _2"></span>v<span class="_ _2"></span>es<span class="_ _9"> </span>like<span class="_ _1e"> </span>“white<span class="_ _9"> </span>noise”<span class="_ _9"> </span>in<span class="_ _9"> </span>the<span class="_ _9"> </span>region<span class="_ _9"> </span>of<span class="_ _9"> </span>in<span class="_ _2"></span>terest,<span class="_ _9"> </span>i.e.,<span class="_ _9"> </span>for<span class="_ _9"> </span><span class="ff7">x<span class="_ _24"> </span><span class="ff9 lsf">∈D<span class="_ _5"></span><span class="ff4">.</span></span></span></div><div class="t m0 x1a h13 y9e ff6 fsb fc0 sc0 ls0 ws0">2.1.<span class="_ _38"> </span>The<span class="_ _6"> </span>Kriging<span class="_ _1"> </span>Predictor</div><div class="t m0 x1a ha y9f ff4 fs7 fc0 sc0 ls0 ws0">F<span class="_ _0"></span>or<span class="_ _1"> </span>the<span class="_ _1"> </span>set<span class="_ _6"> </span><span class="ff7">S<span class="_ _37"> </span></span>of<span class="_ _6"> </span>design<span class="_ _1"> </span>sites<span class="_ _1"> </span>w<span class="_ _2"></span>e<span class="_ _1"> </span>ha<span class="_ _2"></span>ve<span class="_ _6"> </span>the<span class="_ _6"> </span>expanded<span class="_ _1"> </span><span class="ff7">m<span class="ff9">×</span>p<span class="_ _6"> </span></span>design<span class="_ _1"> </span>matrix<span class="_ _6"> </span><span class="ff7">F<span class="_ _a"> </span></span>with</div><div class="t m0 x1a h10 ya0 ff7 fs7 fc0 sc0 ls0 ws0">F</div><div class="t m0 x68 hd ya1 ffa fs9 fc0 sc0 ls0 ws0">ij</div><div class="t m0 x1b ha ya2 ff4 fs7 fc0 sc0 ls0 ws0">=<span class="_ _1d"> </span><span class="ff7">f</span></div><div class="t m0 x69 hd ya1 ffa fs9 fc0 sc0 ls0 ws0">j</div><div class="t m0 x6a ha ya2 ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">s</span></div><div class="t m0 x6b hd ya1 ffa fs9 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6c ha ya2 ff4 fs7 fc0 sc0 ls0 ws0">),</div><div class="t m0 x62 ha ya3 ff7 fs7 fc0 sc0 ls0 ws0">F<span class="_ _8"> </span><span class="ff4 ls1d">=[<span class="_ _2e"></span><span class="ff7">f<span class="_ _14"></span><span class="ff4">(<span class="_ _39"></span><span class="ff7">s</span></span></span></span></div><div class="t m0 x59 hd ya4 ff8 fs9 fc0 sc0 ls1e ws0">1</div><div class="t m0 x6d ha ya5 ff4 fs7 fc0 sc0 ls1d ws0">)<span class="_ _5"></span><span class="ff9 ls7">···<span class="_ _9"> </span><span class="ff7">f<span class="_ _7"></span><span class="ff4">(<span class="_ _2a"></span><span class="ff7">s</span></span></span></span></div><div class="t m0 x39 hd ya4 ffa fs9 fc0 sc0 ls8 ws0">m</div><div class="t m0 x3a ha ya5 ff4 fs7 fc0 sc0 ls0 ws0">)]</div><div class="t m0 x6e he ya6 ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x6f ha ya5 ff7 fs7 fc0 sc0 ls0 ws0">,<span class="_ _3a"> </span><span class="ff4">(2.6)</span></div><div class="t m0 x1a ha ya7 ff4 fs7 fc0 sc0 ls0 ws0">with<span class="_ _9"> </span><span class="ff7">f<span class="_ _32"> </span></span>(<span class="ff7">x</span>)<span class="_ _9"> </span>defined<span class="_ _9"> </span>in<span class="_ _d"> </span>(2.3).<span class="_ _1"> </span>F<span class="_ _7"></span>urther,<span class="_ _9"> </span>define<span class="_ _d"> </span><span class="ff7">R<span class="_ _9"> </span></span>as<span class="_ _d"> </span>the<span class="_ _9"> </span>matrix<span class="_ _9"> </span><span class="ff7">R<span class="_ _d"> </span></span>of<span class="_ _9"> </span>sto<span class="_ _b"></span>chastic-process</div><div class="t m0 x1a ha ya8 ff4 fs7 fc0 sc0 ls0 ws0">correlations<span class="_ _9"> </span>b<span class="_ _b"></span>et<span class="_ _2"></span>w<span class="_ _2"></span>een<span class="_ _9"> </span><span class="ff7">z<span class="_ _b"></span></span>’s<span class="_ _9"> </span>at<span class="_ _9"> </span>design<span class="_ _9"> </span>sites,</div><div class="t m0 x70 h10 ya9 ff7 fs7 fc0 sc0 ls0 ws0">R</div><div class="t m0 x71 hd yaa ffa fs9 fc0 sc0 ls0 ws0">ij</div><div class="t m0 x61 ha yab ff4 fs7 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span><span class="ff9">R</span>(<span class="ff7 ls1b">θ,<span class="_ _32"> </span>s</span></div><div class="t m0 x72 hd yaa ffa fs9 fc0 sc0 ls1a ws0">i</div><div class="t m0 x73 h10 yab ff7 fs7 fc0 sc0 ls7 ws0">,s</div><div class="t m0 x65 hd yaa ffa fs9 fc0 sc0 ls8 ws0">j</div><div class="t m0 x74 ha yab ff4 fs7 fc0 sc0 ls7 ws0">)<span class="_ _2a"></span><span class="ff7 lsd">,i<span class="_ _3b"></span>,<span class="_ _3c"></span>j<span class="_ _29"></span><span class="ff4 lse">=1<span class="_ _25"></span><span class="ff7 lsf">,...,m<span class="_"> </span>.<span class="_ _3d"> </span><span class="ff4 ls0">(2.7)</span></span></span></span></div><div class="t m0 x1a ha yac ff4 fs7 fc0 sc0 ls0 ws0">A<span class="_ _2"></span>t<span class="_ _9"> </span>an<span class="_ _9"> </span>untried<span class="_ _1e"> </span>p<span class="_ _b"></span>oin<span class="_ _2"></span>t<span class="_ _9"> </span><span class="ff7">x<span class="_ _9"> </span></span>let</div><div class="t m0 x75 ha yad ff7 fs7 fc0 sc0 ls0 ws0">r<span class="_ _b"></span><span class="ff4">(</span>x<span class="ff4 ls1d">)=[<span class="_ _39"></span><span class="ff9">R<span class="_ _39"></span><span class="ff4">(<span class="_ _39"></span><span class="ff7 ls1b">θ,<span class="_ _32"> </span>s</span></span></span></span></div><div class="t m0 x53 hd yae ff8 fs9 fc0 sc0 ls1a ws0">1</div><div class="t m0 x76 ha yaf ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4">)<span class="_ _9"> </span><span class="ff9">···<span class="_ _9"> </span>R<span class="_ _2a"></span><span class="ff4">(<span class="_ _2a"></span><span class="ff7 ls1b">θ,<span class="_ _32"> </span>s</span></span></span></span></div><div class="t m0 x77 hd yae ffa fs9 fc0 sc0 ls1a ws0">m</div><div class="t m0 x1e ha yaf ff7 fs7 fc0 sc0 ls7 ws0">,x<span class="_ _2a"></span><span class="ff4 ls0">)]</span></div><div class="t m0 x78 he yb0 ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x79 ha yaf ff4 fs7 fc0 sc0 ls0 ws0">(2.8)</div><div class="t m0 x1a ha yb1 ff4 fs7 fc0 sc0 ls0 ws0">b<span class="_ _b"></span>e<span class="_ _1e"> </span>the<span class="_ _9"> </span>vector<span class="_ _1e"> </span>of<span class="_ _9"> </span>correlations<span class="_ _9"> </span>b<span class="_ _b"></span>et<span class="_ _2"></span>ween<span class="_ _1e"> </span><span class="ff7">z<span class="_ _b"></span></span>’s<span class="_ _9"> </span>at<span class="_ _9"> </span>design<span class="_ _9"> </span>sites<span class="_ _9"> </span>and<span class="_ _9"> </span><span class="ff7">x</span>.</div><div class="t m0 x1a ha yb2 ff4 fs7 fc0 sc0 ls0 ws0">No<span class="_ _2"></span>w,<span class="_ _6"> </span>for<span class="_ _6"> </span>the<span class="_ _17"> </span>sak<span class="_ _2"></span>e<span class="_ _17"> </span>of<span class="_ _6"> </span>con<span class="_ _2"></span>venience,<span class="_ _17"> </span>assume<span class="_ _17"> </span>that<span class="_ _6"> </span><span class="ff7">q<span class="_ _2b"> </span></span>=<span class="_ _24"> </span>1,<span class="_ _6"> </span>implying<span class="_ _17"> </span>that<span class="_ _17"> </span><span class="ff7">β<span class="_ _1c"> </span></span>=<span class="_ _6"> </span><span class="ff7">β</span></div><div class="t m0 x7a hd yb3 ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,<span class="ff8">1</span></span></div><div class="t m0 x7b ha yb4 ff4 fs7 fc0 sc0 ls0 ws0">and</div><div class="t m0 x1a ha yb5 ff7 fs7 fc0 sc0 ls0 ws0">Y<span class="_ _1c"> </span><span class="ff4">=<span class="_ _1d"> </span></span>Y</div><div class="t m0 x12 hd yb6 ffc fs9 fc0 sc0 ls0 ws0">:<span class="_ _5"></span><span class="ffa">,<span class="ff8">1</span></span></div><div class="t m0 x45 ha yb7 ff4 fs7 fc0 sc0 ls0 ws0">,<span class="_ _9"> </span>and<span class="_ _9"> </span>consider<span class="_ _9"> </span>the<span class="_ _9"> </span>linear<span class="_ _9"> </span>predictor</div><div class="t m0 x7c ha yb8 ff4 fs7 fc0 sc0 ls0 ws0">ˆ<span class="_ _2d"></span><span class="ff7">y<span class="_ _b"></span><span class="ff4">(</span>x<span class="ff4 ls11">)=<span class="ff7">c</span></span></span></div><div class="t m0 x5d he yb9 ffb fs9 fc0 sc0 ls12 ws0">></div><div class="t m0 x7d ha yba ff7 fs7 fc0 sc0 ls1f ws0">Y,<span class="_ _3e"> </span><span class="ff4 ls0">(2.9)</span></div><div class="t m0 x7e ha ybb ff4 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62446c485d5dd7338891fd96/bg5.jpg"><div class="t m0 x19 ha y21 ff4 fs7 fc0 sc0 ls0 ws0">with<span class="_ _9"> </span><span class="ff7">c<span class="_ _1d"> </span></span>=<span class="_ _1d"> </span><span class="ff7">c</span>(<span class="ff7">x</span>)<span class="_ _24"> </span><span class="ff9">∈<span class="_ _24"></span></span><span class="ls9">IR</span></div><div class="t m0 x50 hd ybc ffa fs9 fc0 sc0 lsa ws0">m</div><div class="t m0 x7f ha y21 ff4 fs7 fc0 sc0 ls0 ws0">.<span class="_ _6"> </span>The<span class="_ _9"> </span>error<span class="_ _9"> </span>is</div><div class="t m0 x71 ha ybd ff4 fs7 fc0 sc0 ls0 ws0">ˆ<span class="_ _2d"></span><span class="ff7">y<span class="_ _b"></span><span class="ff4">(</span>x<span class="ff4">)<span class="_ _2b"> </span><span class="ff9">−<span class="_ _20"> </span></span></span>y<span class="_ _b"></span><span class="ff4">(</span>x<span class="ff4 ls20">)=<span class="ff7">c</span></span></span></div><div class="t m0 x80 he ybe ffb fs9 fc0 sc0 ls21 ws0">></div><div class="t m0 x34 ha ybf ff7 fs7 fc0 sc0 ls20 ws0">Y<span class="_ _28"></span><span class="ff9">−<span class="_ _3f"></span><span class="ff7">y<span class="_ _39"></span><span class="ff4">(<span class="_ _23"></span><span class="ff7">x<span class="_ _23"></span><span class="ff4">)</span></span></span></span></span></div><div class="t m0 x73 ha yc0 ff4 fs7 fc0 sc0 ls20 ws0">=<span class="ff7">c</span></div><div class="t m0 x80 he yc1 ffb fs9 fc0 sc0 ls21 ws0">></div><div class="t m0 x34 ha yc2 ff4 fs7 fc0 sc0 ls20 ws0">(<span class="_ _23"></span><span class="ff7 ls22">Fβ<span class="_"> </span><span class="ff4">+<span class="_ _3"></span></span>Z<span class="_ _7"></span><span class="ff4">)<span class="_ _35"></span><span class="ff9">−<span class="_ _3"></span></span>(<span class="_ _5"></span><span class="ff7">f<span class="_ _2"></span><span class="ff4">(<span class="_ _5"></span><span class="ff7">x<span class="_ _2a"></span><span class="ff4">)</span></span></span></span></span></span></div><div class="t m0 x81 he yc1 ffb fs9 fc0 sc0 ls23 ws0">></div><div class="t m0 x82 ha yc2 ff7 fs7 fc0 sc0 ls22 ws0">β<span class="_"> </span><span class="ff4">+<span class="_ _3"></span></span>z<span class="_ _0"></span><span class="ff4">)</span></div><div class="t m0 x73 ha yc3 ff4 fs7 fc0 sc0 ls22 ws0">=<span class="_ _37"> </span><span class="ff7">c</span></div><div class="t m0 x80 he yc4 ffb fs9 fc0 sc0 ls23 ws0">></div><div class="t m0 x34 ha yc5 ff7 fs7 fc0 sc0 ls22 ws0">Z<span class="_ _32"> </span><span class="ff9">−<span class="_ _3"></span></span>z<span class="_"> </span><span class="ff4">+</span></div><div class="t m0 x83 hf yc6 ffd fs7 fc0 sc0 ls22 ws0">¡</div><div class="t m0 x84 h10 yc5 ff7 fs7 fc0 sc0 ls22 ws0">F</div><div class="t m0 x85 he yc4 ffb fs9 fc0 sc0 ls23 ws0">></div><div class="t m0 x86 ha yc5 ff7 fs7 fc0 sc0 ls22 ws0">c<span class="_ _3"></span><span class="ff9">−<span class="_ _35"></span></span>f<span class="ff4">(<span class="_ _2a"></span><span class="ff7">x<span class="_ _5"></span><span class="ff4">)</span></span></span></div><div class="t m0 x87 hf yc6 ffd fs7 fc0 sc0 ls22 ws0">¢</div><div class="t m0 x88 he yc7 ffb fs9 fc0 sc0 ls23 ws0">></div><div class="t m0 x22 h10 yc5 ff7 fs7 fc0 sc0 ls24 ws0">β,</div><div class="t m0 x19 ha yc8 ff4 fs7 fc0 sc0 ls0 ws0">where<span class="_ _17"> </span><span class="ff7">Z<span class="_ _6"> </span></span><span class="ls25">=[<span class="_ _40"></span><span class="ff7">z</span></span></div><div class="t m0 x28 hd yc9 ff8 fs9 fc0 sc0 ls26 ws0">1</div><div class="t m0 x89 h10 yca ff7 fs7 fc0 sc0 ls7 ws0">...<span class="_ _17"> </span>z</div><div class="t m0 x7 hd yc9 ffa fs9 fc0 sc0 ls8 ws0">m</div><div class="t m0 x62 ha yca ff4 fs7 fc0 sc0 ls7 ws0">]</div><div class="t m0 x8a he ycb ffb fs9 fc0 sc0 ls8 ws0">></div><div class="t m0 x8b ha yca ff4 fs7 fc0 sc0 ls0 ws0">are<span class="_ _17"> </span>the<span class="_ _d"> </span>errors<span class="_ _17"> </span>at<span class="_ _17"> </span>the<span class="_ _d"> </span>design<span class="_ _17"> </span>sites.<span class="_ _12"> </span>T<span class="_ _0"></span>o<span class="_ _17"> </span>k<span class="_ _2"></span>eep<span class="_ _17"> </span>the<span class="_ _d"> </span>predictor</div><div class="t m0 x19 ha ycc ff4 fs7 fc0 sc0 ls0 ws0">un<span class="_ _2"></span>biased<span class="_ _9"> </span>we<span class="_ _1e"> </span>demand<span class="_ _9"> </span>that<span class="_ _9"> </span><span class="ff7">F</span></div><div class="t m0 x11 he ycd ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x8c ha yce ff7 fs7 fc0 sc0 ls0 ws0">c<span class="_ _2c"> </span><span class="ff9">−<span class="_ _2c"> </span></span>f<span class="_ _3"></span><span class="ff4">(</span>x<span class="ff4">)<span class="_ _1d"> </span>=<span class="_ _1d"> </span>0,<span class="_ _9"> </span>or</span></div><div class="t m0 x8d h10 ycf ff7 fs7 fc0 sc0 ls0 ws0">F</div><div class="t m0 x8e he yd0 ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x33 ha yd1 ff7 fs7 fc0 sc0 ls0 ws0">c<span class="ff4">(</span>x<span class="ff4 ls11">)=<span class="ff7">f<span class="_ _16"></span><span class="ff4">(<span class="_ _2e"></span><span class="ff7">x<span class="_ _39"></span><span class="ff4">)<span class="_ _25"></span><span class="ff7">.<span class="_ _41"> </span><span class="ff4 ls0">(2.10)</span></span></span></span></span></span></span></div><div class="t m0 x19 ha yd2 ff4 fs7 fc0 sc0 ls0 ws0">Under<span class="_ _9"> </span>this<span class="_ _9"> </span>condition<span class="_ _9"> </span>the<span class="_ _9"> </span>mean<span class="_ _9"> </span>squared<span class="_ _9"> </span>error<span class="_ _1e"> </span>(MSE)<span class="_ _9"> </span>of<span class="_ _9"> </span>the<span class="_ _9"> </span>predictor<span class="_ _9"> </span>(2.9)<span class="_ _9"> </span>is</div><div class="t m0 x8f ha yd3 ff7 fs7 fc0 sc0 ls0 ws0">ϕ<span class="ff4">(</span>x<span class="ff4 ls20">)=E</span></div><div class="t m0 x74 hf yd4 ffd fs7 fc0 sc0 ls20 ws0">£</div><div class="t m0 x33 ha yd3 ff4 fs7 fc0 sc0 ls27 ws0">(ˆ<span class="_ _39"></span><span class="ff7">y<span class="_ _2"></span><span class="ff4">(<span class="_ _7"></span><span class="ff7">x<span class="_ _7"></span><span class="ff4">)<span class="_ _32"> </span><span class="ff9">−<span class="_ _24"></span><span class="ff7">y<span class="_ _2"></span><span class="ff4">(<span class="_ _7"></span><span class="ff7">x<span class="_ _7"></span><span class="ff4 ls0">))</span></span></span></span></span></span></span></span></span></div><div class="t m0 x90 hd yd5 ff8 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5c hf yd6 ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x91 ha yd7 ff4 fs7 fc0 sc0 ls20 ws0">=E</div><div class="t m0 x74 hf yd8 ffd fs7 fc0 sc0 ls0 ws0">£¡</div><div class="t m0 x54 h10 yd7 ff7 fs7 fc0 sc0 ls0 ws0">c</div><div class="t m0 x92 he yd9 ffb fs9 fc0 sc0 ls0 ws0">></div><div class="t m0 x35 ha yda ff7 fs7 fc0 sc0 ls0 ws0">Z<span class="_ _1d"> </span><span class="ff9">−<span class="_ _2c"> </span></span>z</div><div class="t m0 x3a hf ydb ffd fs7 fc0 sc0 ls0 ws0">¢</div><div class="t m0 x83 hd ydc ff8 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 x84 hf ydd ffd fs7 fc0 sc0 ls0 ws0">¤</div><div class="t m0 x91 ha yde ff4 fs7 fc0 sc0 ls20 ws0">=E</div><div class="t m0 x74 hf ydf ffd fs7 fc0 sc0 ls20 ws0">£</div><div class="t m0 x33 h10 yde ff7 fs7 fc0 sc0 ls20 ws0">z</div><div class="t m0 x54 hd ye0 ff8 fs9 fc0 sc0 ls21 ws0">2</div><div class="t m0 x93 ha ye1 ff4 fs7 fc0 sc0 ls20 ws0">+<span class="_ _4"></span><span class="ff7">c</span></div><div class="t m0 x36 he ye0 ffb fs9 fc0 sc0 ls21 ws0">></div><div class="t m0 x94 h10 ye1 ff7 fs7 fc0 sc0 ls28 ws0">ZZ</div><div class="t m0 x95 he ye0 ffb fs9 fc0 sc0 ls29 ws0">></div><div class="t m0 x77 ha ye1 ff7 fs7 fc0 sc0 ls28 ws0">c<span class="_ _32"> </span><span class="ff9">−<span class="_ _24"></span><span class="ff4">2<span class="_ _7"></span><span class="ff7">c</span></span></span></div><div class="t m0 x96 he ye0 ffb fs9 fc0 sc0 ls29 ws0">></div><div class="t m0 x81 h10 ye1 ff7 fs7 fc0 sc0 ls28 ws0">Zz</div><div class="t m0 x97 hf ye2 ffd fs7 fc0 sc0 ls28 ws0">¤</div><div class="t m0 x91 ha ye3 ff4 fs7 fc0 sc0 ls28 ws0">=<span class="_ _22"> </span><span class="ff7">σ</span></div><div class="t m0 x98 hd ye4 ff8 fs9 fc0 sc0 ls29 ws0">2</div><div class="t m0 x33 hf ye5 ffd fs7 fc0 sc0 ls28 ws0">¡</div><div class="t m0 x43 ha ye6 ff4 fs7 fc0 sc0 ls17 ws0">1+<span class="ff7">c</span></div><div class="t m0 x99 he ye4 ffb fs9 fc0 sc0 ls2a ws0">></div><div class="t m0 x39 ha ye7 ff7 fs7 fc0 sc0 ls2b ws0">Rc<span class="_ _2b"> </span><span class="ff9">−<span class="_ _2c"> </span><span class="ff4">2</span></span>c</div><div class="t m0 x1f he ye4 ffb fs9 fc0 sc0 ls2c ws0">></div><div class="t m0 x9a h10 ye7 ff7 fs7 fc0 sc0 ls2b ws0">r</div><div class="t m0 x9b hf ye8 ffd fs7 fc0 sc0 ls2b ws0">¢</div><div class="t m0 x9c ha ye7 ff7 fs7 fc0 sc0 ls2b ws0">.<span class="_ _42"> </span><span class="ff4 ls0">(2.11)</span></div><div class="t m0 x19 ha ye9 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _6"> </span>Lagrangian<span class="_ _17"> </span>function<span class="_ _6"> </span>for<span class="_ _6"> </span>the<span class="_ _17"> </span>problem<span class="_ _6"> </span>of<span class="_ _6"> </span>minimizing<span class="_ _17"> </span><span class="ff7">ϕ<span class="_ _6"> </span></span>with<span class="_ _6"> </span>resp<span class="_ _b"></span>ect<span class="_ _17"> </span>to<span class="_ _6"> </span><span class="ff7">c<span class="_ _6"> </span></span>and</div><div class="t m0 x19 ha yea ff4 fs7 fc0 sc0 ls0 ws0">sub<span class="_ _b"></span>ject<span class="_ _9"> </span>to<span class="_ _9"> </span>the<span class="_ _9"> </span>constraint<span class="_ _1e"> </span>(2.10)<span class="_ _9"> </span>is</div><div class="t m0 xb ha yeb ff7 fs7 fc0 sc0 ls0 ws0">L<span class="ff4">(</span>c,<span class="_ _24"> </span>λ<span class="ff4 ls11">)=<span class="ff7">σ</span></span></div><div class="t m0 x9d hd yec ff8 fs9 fc0 sc0 ls12 ws0">2</div><div class="t m0 x2e hf yed ffd fs7 fc0 sc0 ls11 ws0">¡</div><div class="t m0 x9e ha yee ff4 fs7 fc0 sc0 ls13 ws0">1+<span class="ff7">c</span></div><div class="t m0 x9f he yec ffb fs9 fc0 sc0 ls14 ws0">></div><div class="t m0 x4b ha yef ff7 fs7 fc0 sc0 ls2b ws0">Rc<span class="_ _2b"> </span><span class="ff9">−<span class="_ _2c"> </span><span class="ff4">2</span></span>c</div><div class="t m0 xa0 he yec ffb fs9 fc0 sc0 ls2c ws0">></div><div class="t m0 x84 h10 yef ff7 fs7 fc0 sc0 ls2b ws0">r</div><div class="t m0 xa1 hf yf0 ffd fs7 fc0 sc0 ls2b ws0">¢</div><div class="t m0 x5c ha yef ff9 fs7 fc0 sc0 ls2b ws0">−<span class="_ _2b"> </span><span class="ff7">λ</span></div><div class="t m0 x78 he yec ffb fs9 fc0 sc0 ls2c ws0">></div><div class="t m0 xa2 ha yef ff4 fs7 fc0 sc0 ls2b ws0">(<span class="ff7">F</span></div><div class="t m0 xa3 he yec ffb fs9 fc0 sc0 ls2c ws0">></div><div class="t m0 xa4 ha yef ff7 fs7 fc0 sc0 ls2b ws0">c<span class="_ _2b"> </span><span class="ff9">−<span class="_ _2c"> </span></span>f<span class="_ _3"></span><span class="ff4">)<span class="_ _9"> </span></span>.<span class="_ _43"> </span><span class="ff4 ls0">(2.12)</span></div><div class="t m0 x19 ha yf1 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _9"> </span>gradien<span class="_ _2"></span>t<span class="_ _9"> </span>of<span class="_ _17"> </span>(2.12)<span class="_ _9"> </span>with<span class="_ _9"> </span>resp<span class="_ _b"></span>ect<span class="_ _1e"> </span>to<span class="_ _9"> </span><span class="ff7">c<span class="_ _9"> </span></span>is</div><div class="t m0 x2c h10 yf2 ff7 fs7 fc0 sc0 ls0 ws0">L</div><div class="t m0 x8 he yf3 ffb fs9 fc0 sc0 ls0 ws0">0</div><div class="t m0 x8 hd yf4 ffa fs9 fc0 sc0 ls0 ws0">c</div><div class="t m0 x11 ha yf5 ff4 fs7 fc0 sc0 ls0 ws0">(<span class="ff7">c,<span class="_ _24"> </span>λ</span><span class="ls1d">)=2<span class="_ _39"></span><span class="ff7">σ</span></span></div><div class="t m0 xa5 hd yf3 ff8 fs9 fc0 sc0 ls1e ws0">2</div><div class="t m0 x1c ha yf5 ff4 fs7 fc0 sc0 ls1d ws0">(<span class="_ _39"></span><span class="ff7 ls2b">Rc<span class="_ _2c"> </span><span class="ff9">−<span class="_ _2c"> </span></span>r<span class="ff4">)<span class="_ _2c"> </span><span class="ff9">−<span class="_ _2c"> </span></span></span><span class="ls22">Fλ<span class="_ _2b"> </span>,</span></span></div><div class="t m0 x19 ha yf6 ff4 fs7 fc0 sc0 ls0 ws0">and<span class="_ _2c"> </span>from<span class="_ _20"> </span>the<span class="_ _20"> </span>first<span class="_ _2c"> </span>order<span class="_ _20"> </span>necessary<span class="_ _2c"> </span>conditions<span class="_ _20"> </span>for<span class="_ _2c"> </span>optimality<span class="_ _2c"> </span>(see<span class="_ _20"> </span>e.g.<span class="_ _2c"> </span>[7,<span class="_ _1d"> </span>Section<span class="_ _20"> </span>12.2])</div><div class="t m0 x19 ha yf7 ff4 fs7 fc0 sc0 ls0 ws0">w<span class="_ _2"></span>e<span class="_ _9"> </span>get<span class="_ _9"> </span>the<span class="_ _9"> </span>following<span class="_ _1e"> </span>system<span class="_ _9"> </span>of<span class="_ _9"> </span>equations</div><div class="t m0 x9 hf yf8 ffd fs7 fc0 sc0 ls0 ws0">·</div><div class="t m0 x8c h10 yf9 ff7 fs7 fc0 sc0 ls2d ws0">RF</div><div class="t m0 x7c h10 yfa ff7 fs7 fc0 sc0 ls2d ws0">F</div><div class="t m0 x2e he yfb ffb fs9 fc0 sc0 ls2e ws0">></div><div class="t m0 x8e ha yfc ff4 fs7 fc0 sc0 ls2d ws0">0</div><div class="t m0 x54 hf yfd ffd fs7 fc0 sc0 ls7 ws0">¸·</div><div class="t m0 xa6 h10 yfe ff7 fs7 fc0 sc0 ls7 ws0">c</div><div class="t m0 xa6 ha yff ff4 fs7 fc0 sc0 ls7 ws0">˜</div><div class="t m0 xa7 h10 y100 ff7 fs7 fc0 sc0 ls7 ws0">λ</div><div class="t m0 xa8 hf yfd ffd fs7 fc0 sc0 ls7 ws0">¸</div><div class="t m0 xa9 ha y101 ff4 fs7 fc0 sc0 ls7 ws0">=</div><div class="t m0 x85 hf yfd ffd fs7 fc0 sc0 ls7 ws0">·</div><div class="t m0 x1f h10 y102 ff7 fs7 fc0 sc0 ls7 ws0">r</div><div class="t m0 xaa h10 yfc ff7 fs7 fc0 sc0 ls7 ws0">f</div><div class="t m0 x78 hf yfd ffd fs7 fc0 sc0 ls7 ws0">¸</div><div class="t m0 x9c ha y101 ff7 fs7 fc0 sc0 ls7 ws0">,<span class="_ _44"> </span><span class="ff4 ls0">(2.13)</span></div><div class="t m0 x19 ha y103 ff4 fs7 fc0 sc0 ls0 ws0">where<span class="_ _9"> </span>w<span class="_ _2"></span>e<span class="_ _9"> </span>hav<span class="_ _2"></span>e<span class="_ _1e"> </span>defined</div><div class="t m0 x74 ha y104 ff4 fs7 fc0 sc0 ls0 ws0">˜</div><div class="t m0 x98 ha y105 ff7 fs7 fc0 sc0 ls0 ws0">λ<span class="_ _a"> </span><span class="ff4">=<span class="_ _a"> </span><span class="ff9">−</span></span></div><div class="t m0 x5b h10 y106 ff7 fs7 fc0 sc0 ls0 ws0">λ</div><div class="t m0 x39 ha y107 ff4 fs7 fc0 sc0 ls0 ws0">2<span class="ff7">σ</span></div><div class="t m0 xa0 hd y108 ff8 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 xab h10 y109 ff7 fs7 fc0 sc0 ls0 ws0">.</div><div class="t m0 x19 ha y10a ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _9"> </span>solution<span class="_ _9"> </span>to<span class="_ _9"> </span>(2.13)<span class="_ _9"> </span>is</div><div class="t m0 x8f ha y10b ff4 fs7 fc0 sc0 ls0 ws0">˜</div><div class="t m0 xac ha y10c ff7 fs7 fc0 sc0 ls0 ws0">λ<span class="_ _12"> </span><span class="ff4 ls20">=(<span class="_ _23"></span><span class="ff7">F</span></span></div><div class="t m0 xad he y10d ffb fs9 fc0 sc0 ls21 ws0">></div><div class="t m0 x65 h10 y10e ff7 fs7 fc0 sc0 ls20 ws0">R</div><div class="t m0 x66 hd y10d ffb fs9 fc0 sc0 ls21 ws0">−<span class="_ _4"></span><span class="ff8">1</span></div><div class="t m0 x92 ha y10e ff7 fs7 fc0 sc0 ls20 ws0">F<span class="_ _16"></span><span class="ff4">)</span></div><div class="t m0 xa5 hd y10d ffb fs9 fc0 sc0 ls21 ws0">−<span class="_ _4"></span><span class="ff8">1</span></div><div class="t m0 xae ha y10e ff4 fs7 fc0 sc0 ls20 ws0">(<span class="_ _23"></span><span class="ff7">F</span></div><div class="t m0 x67 he y10d ffb fs9 fc0 sc0 ls21 ws0">></div><div class="t m0 xaf h10 y10e ff7 fs7 fc0 sc0 ls20 ws0">R</div><div class="t m0 x6f hd y10d ffb fs9 fc0 sc0 ls21 ws0">−<span class="_ _4"></span><span class="ff8">1</span></div><div class="t m0 xb0 ha y10e ff7 fs7 fc0 sc0 ls20 ws0">r<span class="_ _3f"></span><span class="ff9">−<span class="_ _3f"></span><span class="ff7">f<span class="_ _2d"></span><span class="ff4">)<span class="_ _45"></span><span class="ff7">,</span></span></span></span></div><div class="t m0 xb1 ha y10f ff7 fs7 fc0 sc0 ls20 ws0">c<span class="ff4">=</span>R</div><div class="t m0 x9e hd y110 ffb fs9 fc0 sc0 ls21 ws0">−<span class="_ _4"></span><span class="ff8">1</span></div><div class="t m0 x65 ha y111 ff4 fs7 fc0 sc0 ls20 ws0">(<span class="_ _23"></span><span class="ff7">r<span class="_ _3f"></span><span class="ff9">−<span class="_ _3f"></span><span class="ff7">F</span></span></span></div><div class="t m0 xa5 ha y112 ff4 fs7 fc0 sc0 ls20 ws0">˜</div><div class="t m0 xa5 ha y111 ff7 fs7 fc0 sc0 ls20 ws0">λ<span class="_ _23"></span><span class="ff4">)<span class="_ _45"></span><span class="ff7">.</span></span></div><div class="t m0 xb2 ha y113 ff4 fs7 fc0 sc0 ls0 ws0">(2.14)</div><div class="t m0 x19 ha y114 ff4 fs7 fc0 sc0 ls0 ws0">The<span class="_ _9"> </span>matrix<span class="_ _9"> </span><span class="ff7">R<span class="_ _9"> </span></span>and<span class="_ _9"> </span>therefore<span class="_ _9"> </span><span class="ff7">R</span></div><div class="t m0 xb3 hd y115 ffb fs9 fc0 sc0 ls0 ws0">−<span class="ff8">1</span></div><div class="t m0 x5 ha yba ff4 fs7 fc0 sc0 ls0 ws0">is<span class="_ _9"> </span>symmetric,<span class="_ _9"> </span>and<span class="_ _9"> </span>b<span class="_ _2"></span>y<span class="_ _9"> </span>means<span class="_ _9"> </span>of<span class="_ _17"> </span>(2.9)<span class="_ _9"> </span>w<span class="_ _2"></span>e<span class="_ _9"> </span>find</div><div class="t m0 x47 ha ybb ff4 fs7 fc0 sc0 ls0 ws0">3</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>