Figure 4.12:
The sum of squares validation error for PCR and PLSR versus the number of principal components/latent variables \ell ; note that only two latent variables are required versus four principal components.
Code for Figure 4.12
Text of the GNU GPL.
main.m
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115 | n = 100;
p = 2;
q = 5;
% reproducible data!
rand("seed",-1);
% create the data set
% true parameters
B = [zeros(3,2); [-1 1]; [0 0]];
X = rand(n, q-1);
% add a nearly dependent final column
xq = X*ones(q-1,1)/sqrt(q-1) + 0.01*rand(n,1);
% fitting data
X = [X, xq];
Y = X*B + 0.1*rand(n,p);
% cross validation data
X2 = rand(n, q);
Y2 = X2*B + 0.1*rand(n,p);
Xdat = [X;X2]; Ydat = [Y;Y2];
% write out low precision data for posting on website
myfile = fopen('pca_pls_data.dat', 'w');
fprintf(myfile, '% x1 x2 x3 x4 x5 y1 y2')
fprintf(myfile, '\n')
for i = 1: 2*n
fprintf(myfile, '%8.3f', Xdat(i,:), Ydat(i,:))
fprintf(myfile, '\n')
end
fclose(myfile);
% reload low precision data so we estimate with the same data
Data = load ('pca_pls_data.dat');
X = Data(1:n,1:q); Y = Data(1:n,q+1:q+p);
X2 = Data(n+1:end,1:q); Y2 = Data(n+1:end,q+1:q+p);
% full least squares
Bls = inv(X'*X)*X'*Y;
% PCR or SVD
[U,S,V] = svd(X);
% check all numbers of principal components (singular values)
Bsvd(1:q) = {zeros(q,p)};
Ysvd(1:q) = {zeros(n,p)};
for l = 1:q
Ul=U(:,1:l); Sl=S(1:l,1:l); Vl=V(:,1:l);
Bsvd(l) = {Vl*inv(Sl)*Ul'*Y};
% compare prediction on cross-validation data
Ysvd(l) = {X*cell2mat(Bsvd(l))};
errsvd(l) = norm(Y-cell2mat(Ysvd(l)));
end
%PLS algorithm
U = []; T=[]; P=[]; Q=[];
E = X; F = Y;
Bpls(1:q) = {zeros(q,p)};
Ypls(1:q) = {zeros(n,p)};
for i = 1:q
[UU,SS,VV] = svd(E'*F);
u = UU(:,1); v=VV(:,1);
t = E*u; w = F*v;
t = t/sqrt(t'*t);
pp = E'*t; qq = F'*t;
E = E - t*pp'; F = F - t*qq';
U = [U u]; T = [T t]; P = [P pp]; Q=[Q qq];
R = U / (P'*U); % i.e., R = inv(P'U)*U
%Bpls = R*inv(T'*T)*T'*Y
Bpls(i) = {R*Q'};
Ypls(i) = {X*cell2mat(Bpls(i))};
errpls(i) = norm(Y-cell2mat(Ypls(i)));
end
figure()
plot(1:q, errsvd, '-o', 1:q, errpls, '-+')
%cross validation
for i = 1:q
Y2svd(i) = {X2*cell2mat(Bsvd(i))};
err2svd(i) = norm(Y2-cell2mat(Y2svd(i)));
Y2pls(i) = {X2*cell2mat(Bpls(i))};
err2pls(i) = norm(Y2-cell2mat(Y2pls(i)));
end
figure()
plot(1:q, err2svd, '-o', 1:q, err2pls, '-+')
for i = 1:q
figure()
tmp = cell2mat(Y2pls(i));
plot(Y2(:,1), tmp(:,1), 'o')
end
for i = 1:q
figure()
tmp = cell2mat(Y2svd(i))
plot(Y2(:,1), tmp(:,1), 'o')
end
% save data for book figure()
tab1 = [(1:q)', errsvd'];
tab2 = [(1:q)', errpls'];
tab3 = [Y, cell2mat(Ysvd)];
tab4 = [Y, cell2mat(Ypls)];
tab5 = [(1:q)', err2svd'];
tab6 = [(1:q)', err2pls'];
tab7 = [Y2, cell2mat(Y2svd)];
tab8 = [Y2, cell2mat(Y2pls)];
save pca_pls.dat tab1 tab2 tab3 tab4 tab5 tab6 tab7 tab8
|
pca_pls.dat
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467 | # Created by Octave 8.3.0, Mon Jan 22 21:53:01 2024 PST <jbraw@telemark>
# name: tab1
# type: matrix
# rows: 5
# columns: 2
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# name: tab2
# type: matrix
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# name: tab3
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-0.34100000000000003 0.46200000000000002 -0.36905245784747831 0.43668834082973118 -0.36406469366727673 0.4317941629665934 -0.35428175539828294 0.40864059333365604 -0.35672434346033444 0.40803421833039388 -0.79832688662711004 0.2513218049749612
0.028000000000000001 0.13600000000000001 -0.46730170975153762 0.55294363703338745 0.010413367520610785 0.084192013740320176 0.016613576895350188 0.069517794650968204 0.020091162984616551 0.070381108980954787 -0.18635332106360364 -0.0028802746313573843
-0.159 0.32500000000000001 -0.22816859950735116 0.26998483556905917 -0.084469332559209173 0.12898182383118265 -0.087774298456225877 0.13680378446581004 -0.088179219360916883 0.13670326243398748 -1.0173221024987438 -0.19302360581360589
-0.872 0.94299999999999995 -0.47717969554818862 0.56463195162540647 -0.73531835429962833 0.81792710826016057 -0.73331243117537614 0.81317963061503518 -0.73855510860973306 0.8118781305398689 -1.9031041741214323 0.39861218053325959
-0.51200000000000001 0.64800000000000002 -0.30443477577832162 0.36022823936983267 -0.35170966786001556 0.4066161041423616 -0.34993127443091027 0.40240712776293847 -0.3544848281459585 0.40127670336116789 -2.2306581309231692 -0.26452482043171033
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pca_pls_data.dat
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201 | # x1 x2 x3 x4 x5 y1 y2
0.129 0.020 0.149 0.461 0.386 -0.430 0.513
0.585 0.650 0.000 0.495 0.870 -0.419 0.518
0.271 0.898 0.536 0.736 1.229 -0.646 0.829
0.177 0.579 0.435 0.529 0.862 -0.484 0.549
0.236 0.571 0.041 0.347 0.602 -0.322 0.371
0.795 0.152 0.381 0.737 1.042 -0.686 0.773
0.861 0.718 0.181 0.576 1.169 -0.516 0.638
0.583 0.664 0.700 0.281 1.115 -0.222 0.340
0.887 0.440 0.080 0.064 0.745 -0.050 0.088
0.450 0.898 0.295 0.672 1.158 -0.576 0.732
0.254 0.591 0.300 0.590 0.874 -0.569 0.602
0.563 0.584 0.995 0.459 1.304 -0.399 0.550
0.460 0.816 0.400 0.251 0.971 -0.227 0.254
0.524 0.654 0.993 0.208 1.199 -0.117 0.261
0.898 0.128 0.228 0.917 1.085 -0.906 0.967
0.503 0.918 0.427 0.363 1.115 -0.348 0.374
0.988 0.435 0.341 0.819 1.300 -0.788 0.857
0.311 0.909 0.204 0.642 1.041 -0.578 0.715
0.614 0.385 0.300 0.881 1.098 -0.793 0.882
0.441 0.555 0.693 0.075 0.886 0.011 0.158
0.746 0.470 0.755 0.361 1.167 -0.291 0.400
0.704 0.313 0.542 0.537 1.053 -0.531 0.537
0.058 0.529 0.843 0.728 1.085 -0.690 0.730
0.261 0.908 0.869 0.008 1.032 0.055 0.018
0.672 0.668 0.744 0.910 1.502 -0.839 0.948
0.172 0.871 0.145 0.883 1.036 -0.870 0.968
0.878 0.520 0.149 0.626 1.093 -0.588 0.690
0.830 0.465 0.361 0.718 1.190 -0.714 0.810
0.170 0.052 0.618 0.399 0.625 -0.343 0.470
0.004 0.849 0.559 0.349 0.886 -0.291 0.393
0.864 0.990 0.507 0.881 1.630 -0.823 0.942
0.781 0.876 0.127 0.256 1.023 -0.183 0.275
0.001 0.226 0.776 0.871 0.947 -0.831 0.931
0.480 0.225 0.006 0.600 0.658 -0.543 0.695
0.995 0.074 0.708 0.351 1.069 -0.259 0.432
0.922 0.224 0.708 0.835 1.350 -0.757 0.835
0.630 0.393 0.287 0.402 0.859 -0.311 0.466
0.144 0.276 0.593 0.599 0.811 -0.502 0.696
0.531 0.357 0.195 0.692 0.892 -0.609 0.727
0.210 0.541 0.182 0.792 0.873 -0.765 0.806
0.614 0.645 0.649 0.531 1.222 -0.473 0.542
0.685 0.134 0.220 0.887 0.972 -0.848 0.978
0.889 0.320 0.983 0.921 1.560 -0.899 0.978
0.306 0.398 0.402 0.467 0.796 -0.445 0.550
0.765 0.590 0.733 0.784 1.445 -0.752 0.820
0.930 0.661 0.786 0.013 1.201 0.043 0.079
0.396 0.431 0.894 0.193 0.961 -0.178 0.208
0.572 0.594 0.869 0.615 1.330 -0.556 0.672
0.900 0.610 0.957 0.575 1.528 -0.533 0.656
0.749 0.413 0.654 0.380 1.108 -0.370 0.475
0.390 0.582 0.964 0.211 1.080 -0.199 0.224
0.562 0.681 0.109 0.866 1.110 -0.847 0.914
0.771 0.820 0.631 0.054 1.139 -0.034 0.140
0.322 0.064 0.707 0.117 0.613 -0.018 0.179
0.778 0.213 0.424 0.071 0.748 -0.036 0.114
0.547 0.180 0.990 0.578 1.150 -0.535 0.674
0.726 0.819 0.037 0.484 1.038 -0.419 0.511
0.974 0.870 0.010 0.356 1.106 -0.347 0.443
0.913 0.306 0.603 0.723 1.277 -0.680 0.726
0.641 0.911 0.810 0.611 1.487 -0.521 0.627
0.932 0.902 0.114 0.949 1.455 -0.902 0.961
0.535 0.393 0.868 0.497 1.150 -0.454 0.533
0.518 0.406 0.833 0.407 1.088 -0.327 0.437
0.891 0.096 0.704 0.576 1.135 -0.556 0.631
0.507 0.334 0.921 0.435 1.107 -0.382 0.472
0.852 0.572 0.619 0.039 1.047 0.029 0.121
0.979 0.525 0.332 0.873 1.357 -0.820 0.879
0.175 0.217 0.639 0.081 0.561 -0.001 0.099
0.981 0.764 0.665 0.657 1.541 -0.580 0.680
0.301 0.657 0.976 0.683 1.313 -0.622 0.756
0.220 0.555 0.897 0.667 1.176 -0.663 0.696
0.354 0.725 0.741 0.823 1.327 -0.779 0.850
0.636 0.728 0.478 0.099 0.975 -0.038 0.136
0.391 0.451 0.672 0.675 1.104 -0.621 0.715
0.018 0.756 0.258 0.242 0.647 -0.214 0.300
0.066 0.278 0.822 0.591 0.886 -0.498 0.680
0.968 0.744 0.707 0.573 1.502 -0.569 0.630
0.584 0.394 0.261 0.624 0.936 -0.529 0.637
0.065 0.626 0.625 0.599 0.966 -0.550 0.675
0.048 0.041 0.826 0.435 0.679 -0.418 0.524
0.358 0.357 0.054 0.068 0.427 0.011 0.088
0.068 0.029 0.681 0.699 0.747 -0.640 0.798
0.953 0.506 0.338 0.963 1.381 -0.931 1.055
0.528 0.238 0.015 0.188 0.490 -0.168 0.234
0.478 0.582 0.392 0.032 0.751 0.030 0.124
0.782 0.852 0.044 0.440 1.066 -0.355 0.453
0.605 0.090 0.451 0.599 0.877 -0.581 0.630
0.366 0.218 0.119 0.666 0.691 -0.624 0.674
0.750 0.526 0.124 0.200 0.804 -0.145 0.223
0.004 0.634 0.480 0.891 1.010 -0.824 0.933
0.932 0.113 0.050 0.131 0.617 -0.131 0.132
0.391 0.327 0.841 0.223 0.900 -0.183 0.247
0.180 0.023 0.821 0.098 0.560 -0.052 0.107
0.101 0.791 0.923 0.582 1.202 -0.526 0.599
0.651 0.371 0.464 0.891 1.192 -0.821 0.927
0.882 0.321 0.573 0.803 1.299 -0.771 0.804
0.920 0.115 0.266 0.025 0.666 0.005 0.057
0.232 0.672 0.888 0.887 1.342 -0.860 0.975
0.931 0.020 0.577 0.884 1.211 -0.866 0.907
0.392 0.173 0.901 0.589 1.035 -0.567 0.615
0.291 0.410 0.238 0.275 0.319 -0.233 0.327
0.924 0.091 0.551 0.320 0.419 -0.319 0.374
0.740 0.151 0.585 0.821 0.091 -0.725 0.856
0.939 0.164 0.118 0.680 0.658 -0.652 0.752
0.206 0.329 0.163 0.999 0.114 -0.996 1.006
0.504 0.056 0.328 0.928 0.540 -0.894 0.981
0.449 0.441 0.273 0.714 0.154 -0.633 0.769
0.776 0.737 0.610 0.335 0.188 -0.293 0.414
0.844 0.048 0.368 0.607 0.653 -0.511 0.634
0.903 0.964 0.673 0.096 0.721 -0.057 0.144
0.763 0.958 0.535 0.169 0.456 -0.093 0.230
0.960 0.565 0.518 0.090 0.798 -0.044 0.141
0.510 0.975 0.983 0.429 0.615 -0.413 0.435
0.664 0.429 0.137 0.341 0.411 -0.308 0.441
0.227 0.424 0.536 0.297 0.422 -0.220 0.357
0.540 0.670 0.065 0.113 0.680 -0.101 0.201
0.201 0.434 0.134 0.795 0.473 -0.778 0.869
0.881 0.929 0.040 0.007 0.939 0.057 0.085
0.471 0.126 0.965 0.851 0.444 -0.794 0.893
0.498 0.182 0.530 0.852 0.051 -0.812 0.880
0.109 0.306 0.320 0.579 0.008 -0.516 0.609
0.725 0.315 0.179 0.328 0.560 -0.269 0.356
0.034 0.117 0.268 0.151 0.836 -0.054 0.208
0.648 0.332 0.011 0.470 0.693 -0.458 0.492
0.353 0.925 0.075 0.020 0.065 0.003 0.055
0.576 0.170 0.998 0.145 0.572 -0.087 0.226
0.590 0.160 0.255 0.695 0.342 -0.631 0.787
0.390 0.843 0.696 0.007 0.444 0.025 0.099
0.402 0.493 0.246 0.995 0.630 -0.981 1.026
0.666 0.868 0.574 0.620 0.916 -0.543 0.652
0.053 0.112 0.489 0.337 0.044 -0.270 0.401
0.981 0.772 0.243 0.789 0.525 -0.729 0.870
0.423 0.273 0.685 0.156 0.248 -0.066 0.167
0.741 0.247 0.969 0.193 0.043 -0.096 0.209
0.329 0.706 0.465 0.072 0.061 -0.031 0.120
0.463 0.564 0.490 0.226 0.446 -0.151 0.243
0.222 0.673 0.177 0.715 0.574 -0.656 0.780
0.622 0.938 0.474 0.843 0.067 -0.825 0.868
0.549 0.959 0.265 0.299 0.143 -0.201 0.364
0.020 0.288 0.153 0.327 0.130 -0.247 0.377
0.232 0.110 0.747 0.400 0.385 -0.347 0.466
0.189 0.943 0.470 0.229 0.252 -0.172 0.302
0.379 0.173 0.945 0.741 0.934 -0.701 0.816
0.557 0.470 0.308 0.305 0.620 -0.229 0.394
0.468 0.346 0.835 0.024 0.705 0.016 0.102
0.958 0.922 0.479 0.303 0.083 -0.228 0.356
0.708 0.834 0.708 0.815 0.674 -0.754 0.908
0.752 0.396 0.746 0.423 0.044 -0.335 0.501
0.394 0.905 0.478 0.349 0.069 -0.348 0.408
0.461 0.946 0.923 0.188 0.484 -0.160 0.214
0.687 0.270 0.470 0.228 0.103 -0.156 0.306
0.386 0.750 0.774 0.073 0.382 -0.025 0.090
0.152 0.234 0.526 0.273 0.036 -0.179 0.278
0.568 0.043 0.039 0.320 0.620 -0.281 0.344
0.255 0.834 0.354 0.090 0.920 0.003 0.178
0.848 0.812 0.244 0.653 0.217 -0.630 0.687
0.310 0.303 0.862 0.979 0.545 -0.956 1.042
0.905 0.640 0.602 0.817 0.802 -0.807 0.874
0.798 0.994 0.828 0.861 0.547 -0.846 0.933
0.583 0.024 0.161 0.218 0.734 -0.201 0.252
0.896 0.152 0.640 0.151 0.126 -0.140 0.201
0.836 0.019 0.936 0.974 0.089 -0.932 1.071
0.084 0.535 0.135 0.131 0.114 -0.086 0.180
0.595 0.737 0.314 0.519 0.035 -0.510 0.560
0.349 0.673 0.217 0.188 0.670 -0.148 0.228
0.424 0.504 0.202 0.806 0.520 -0.799 0.899
0.692 0.760 0.309 0.906 0.097 -0.814 0.973
0.877 0.571 0.450 0.273 0.851 -0.269 0.354
0.861 0.909 0.988 0.373 0.601 -0.361 0.452
0.654 0.844 0.436 0.245 0.008 -0.224 0.262
0.110 0.045 0.556 0.169 0.616 -0.103 0.204
0.403 0.564 0.687 0.208 0.407 -0.192 0.219
0.888 0.561 0.866 0.138 0.916 -0.112 0.145
0.810 0.473 0.770 0.073 0.678 -0.020 0.141
0.998 0.555 0.309 0.703 0.387 -0.646 0.748
0.413 0.566 0.634 0.297 0.214 -0.249 0.320
0.645 0.330 0.265 0.140 0.176 -0.080 0.164
0.802 0.146 0.878 0.497 0.396 -0.413 0.538
0.918 0.905 0.945 0.510 0.338 -0.453 0.563
0.539 0.891 0.802 0.997 0.063 -0.965 1.026
0.925 0.725 0.118 0.836 0.719 -0.814 0.927
0.718 0.045 0.700 0.755 0.825 -0.694 0.772
0.928 0.918 0.062 0.916 0.655 -0.884 0.918
0.522 0.883 0.132 0.898 0.789 -0.824 0.950
0.611 0.331 0.241 0.503 0.969 -0.408 0.566
0.242 0.875 0.627 0.908 0.074 -0.853 0.931
0.855 0.432 0.015 0.154 0.119 -0.114 0.186
0.549 0.387 0.838 0.290 0.379 -0.197 0.345
0.595 0.347 0.255 0.095 0.295 -0.079 0.166
0.086 0.046 0.467 0.938 0.914 -0.907 1.027
0.687 0.010 0.472 0.061 0.660 -0.025 0.158
0.859 0.300 0.698 0.764 0.102 -0.722 0.769
0.091 0.284 0.922 0.076 0.030 0.002 0.136
0.352 0.573 0.247 0.342 0.216 -0.248 0.426
0.773 0.983 0.860 0.303 0.448 -0.280 0.400
0.944 0.392 0.022 0.438 0.685 -0.341 0.462
0.790 0.953 0.387 0.058 0.998 0.028 0.136
0.315 0.334 0.585 0.227 0.279 -0.159 0.325
0.904 0.237 0.524 0.915 0.725 -0.872 0.943
0.823 0.265 0.519 0.606 0.192 -0.512 0.648
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