Figure 4.13:

Predicted versus measured outputs for the validation dataset. Top: PCR using \textit {four} principal components. Bottom: PLSR using \textit {two} latent variables. Left: first output. Right: second output.

Code for Figure 4.13

Text of the GNU GPL.

main.m

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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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# 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
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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

pca_pls_data.dat

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#     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