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Example of EGO with linear regression

Example of EGO with linear regression. EGO is usually applied to a kriging surrogate. However it is applicable to any surrogate that provides normally distributed uncertainty about predictions.

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Example of EGO with linear regression

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  1. Example of EGO with linear regression • EGO is usually applied to a kriging surrogate. However it is applicable to any surrogate that provides normally distributed uncertainty about predictions. • With kriging we often assume zero noise, so we will never sample a point twice. With noisy data that can happen. • We will use as an example • For comparison with quadratic fit • Data is noisy with noise distributed as

  2. DOE and data x=lhsdesign(5,1); x‘=0.9474 0.2692 0.4620 0.0504 0.7099 x=sort(x); x‘= 0.0504 0.2692 0.4622 0.7099 0.9474 obj=inline('2-5*(x-0.45).^2'); f=obj(x); f‘= 1.2015 1.8365 1.9993 1.6623 0.7629 noise=randn(5,1); noise’= -2.1384 -0.8396 1.3546 -1.0722 0.9610 fnoisy=f+0.3*noise; fnoisy’= 0.5600 1.5847 2.4056 1.3406 1.0512 xg=linspace(0,1,21);fg=obj(xg); plot(xg,fg,x,fnoisy,'go','MarkerSize',20, 'MarkerFaceColor','g','LineWidth',2)

  3. Fit xx=[ones(5,1),x,x.^2]; [b,bint,r,rint,stats] = regress(fnoisy,xx); b = 0.3015 6.6325 -6.3559 bint = -1.9351 2.5381 -4.0883 17.3532 -16.6833 3.9716 stats =0.7819 3.5857 0.2181 0.2045 fit=b(1)+b(2)*xg'+b(3)*(xg.^2)‘ hold on; plot(xg,fit,'r','LineWidth',2) xlabel('x');legend('f','data','fit','Location','NorthEast') .

  4. Prediction variance • Recall equation xprimex=xx'*xx; xpxinv=inv(xprimex); xmg=[ones(21,1),xg',(xg.^2)']; variance=diag(xmg*xpxinv*xmg') *stats(4); sterr=sqrt(variance); plot(xg,fit,'r','LineWidth',2);hold on; plot(xg,2*sterr,'m','Linewidth',2); legend('fit','2*sterr','location','NorthEast') .

  5. Expected improvement • Recall • pbs=min(fnoisy); delpbs=(pbs-fit)./sterr; • phi=normpdf(delpbs); • PHI=normcdf(delpbs); • ei=(pbs-fit).*PHI+sterr.*phi; .

  6. One more iteration – new fit xnew= 0 >> x=[xnew; x];newnoise=randn(1,1) newnoise=0.1240 noise=[newnoise;noise]; fnoisy=obj(x)+0.3*noise; fnoisy’=1.0247 0.5601 1.5847 2.4056 1.3406 1.0513 %computer crash and data reinstatedfromMatlabprintout [b,bint,r,rint,stats] = regress(fnoisy,xx) b ‘= 0.7132 4.9955 -5.0270 bint = -0.3909 1.8172 -1.2927 11.2836 -11.6259 1.5719 stats = 0.6806 3.1960 0.1805 0.2114 .

  7. Prediction variance and expected improvement . .

  8. Expected feasibility example • Constraint boundary example by considering a constraint • That should lead to sampling in the middle • Recall u=(1.6-fit)./sterr; up=u+2;um=u-2; PHI=normcdf(u); PHIP=normcdf(up); PHIM=normcdf(um); phi=normpdf(u);phip=normpdf(up); phim=normpdf(um); first=(fit-1.6).*(2*PHI-PHIP-PHIM) second=-sterr.*(2*phi-phip-phim) third=2*sterr.*(PHIP-PHIM) efeas=first+second +third

  9. Plots . .

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