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| The PLS Procedure |
The following example demonstrates issues in spectrometric calibration. The data (Umetrics 1995) consist of spectrographic readings on 33 samples containing known concentrations of two amino acids, tyrosine and tryptophan. The spectra are measured at 30 frequencies across the overall range of frequencies. For example, Figure 51.11 shows the observed spectra for three samples, one with only tryptophan, one with only tyrosine, and one with a mixture of the two, all at a total concentration of 10-6.
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Of the 33 samples, 18 are used as a training set and 15 as a test set. The data originally appear in McAvoy et al. (1989).
These data were created in a lab, with the concentrations fixed in order to provide a wide range of applicability for the model. You want to use a linear function of the logarithms of the spectra to predict the logarithms of tyrosine and tryptophan concentration, as well as the logarithm of the total concentration. Actually, because of the possibility of zeros in both the responses and the predictors, slightly different transformations are used. The following statements create SAS data sets containing the training and test data, named ftrain and ftest, respectively:
data ftrain;
input obsnam $ tot tyr f1-f30 @@;
try = tot - tyr;
if (tyr) then tyr_log = log10(tyr); else tyr_log = -8;
if (try) then try_log = log10(try); else try_log = -8;
tot_log = log10(tot);
datalines;
17mix35 0.00003 0
-6.215 -5.809 -5.114 -3.963 -2.897 -2.269 -1.675 -1.235
-0.900 -0.659 -0.497 -0.395 -0.335 -0.315 -0.333 -0.377
-0.453 -0.549 -0.658 -0.797 -0.878 -0.954 -1.060 -1.266
-1.520 -1.804 -2.044 -2.269 -2.496 -2.714
19mix35 0.00003 3E-7
-5.516 -5.294 -4.823 -3.858 -2.827 -2.249 -1.683 -1.218
-0.907 -0.658 -0.501 -0.400 -0.345 -0.323 -0.342 -0.387
-0.461 -0.554 -0.665 -0.803 -0.887 -0.960 -1.072 -1.272
-1.541 -1.814 -2.058 -2.289 -2.496 -2.712
21mix35 0.00003 7.5E-7
-5.519 -5.294 -4.501 -3.863 -2.827 -2.280 -1.716 -1.262
-0.939 -0.694 -0.536 -0.444 -0.384 -0.369 -0.377 -0.421
-0.495 -0.596 -0.706 -0.824 -0.917 -0.988 -1.103 -1.294
-1.565 -1.841 -2.084 -2.320 -2.521 -2.729
23mix35 0.00003 1.5E-6
-5.294 -4.705 -4.262 -3.605 -2.726 -2.239 -1.681 -1.250
-0.925 -0.697 -0.534 -0.437 -0.381 -0.359 -0.369 -0.426
-0.499 -0.591 -0.701 -0.843 -0.925 -0.989 -1.109 -1.310
-1.579 -1.852 -2.090 -2.316 -2.521 -2.743
25mix35 0.00003 3E-6
-4.600 -4.069 -3.764 -3.262 -2.598 -2.191 -1.680 -1.273
-0.958 -0.729 -0.573 -0.470 -0.422 -0.407 -0.422 -0.468
-0.538 -0.639 -0.753 -0.887 -0.968 -1.037 -1.147 -1.357
-1.619 -1.886 -2.141 -2.359 -2.585 -2.792
27mix35 0.00003 7.5E-6
-3.812 -3.376 -3.026 -2.726 -2.249 -1.919 -1.541 -1.198
-0.951 -0.764 -0.639 -0.570 -0.528 -0.525 -0.550 -0.606
-0.689 -0.781 -0.909 -1.031 -1.126 -1.191 -1.303 -1.503
-1.784 -2.058 -2.297 -2.507 -2.727 -2.970
29mix35 0.00003 0.000015
-3.053 -2.641 -2.382 -2.194 -1.977 -1.913 -1.728 -1.516
-1.317 -1.158 -1.029 -0.963 -0.919 -0.915 -0.933 -0.981
-1.055 -1.157 -1.271 -1.409 -1.505 -1.546 -1.675 -1.880
-2.140 -2.415 -2.655 -2.879 -3.075 -3.319
28mix35 0.00003 0.0000225
-2.626 -2.248 -2.004 -1.839 -1.742 -1.791 -1.786 -1.772
-1.728 -1.666 -1.619 -1.591 -1.575 -1.580 -1.619 -1.671
-1.754 -1.857 -1.982 -2.114 -2.210 -2.258 -2.379 -2.570
-2.858 -3.117 -3.347 -3.568 -3.764 -4.012
26mix35 0.00003 0.000027
-2.370 -1.990 -1.754 -1.624 -1.560 -1.655 -1.772 -1.899
-1.982 -2.074 -2.157 -2.211 -2.267 -2.317 -2.369 -2.460
-2.545 -2.668 -2.807 -2.951 -3.030 -3.075 -3.214 -3.376
-3.685 -3.907 -4.129 -4.335 -4.501 -4.599
24mix35 0.00003 0.0000285
-2.326 -1.952 -1.702 -1.583 -1.507 -1.629 -1.771 -1.945
-2.115 -2.297 -2.448 -2.585 -2.696 -2.808 -2.913 -3.030
-3.163 -3.265 -3.376 -3.534 -3.642 -3.721 -3.858 -4.012
-4.262 -4.501 -4.704 -4.822 -4.956 -5.292
22mix35 0.00003 0.00002925
-2.277 -1.912 -1.677 -1.556 -1.487 -1.630 -1.791 -1.969
-2.203 -2.437 -2.655 -2.844 -3.032 -3.214 -3.378 -3.503
-3.646 -3.812 -3.958 -4.129 -4.193 -4.262 -4.415 -4.501
-4.823 -5.111 -5.113 -5.294 -5.290 -5.294
20mix35 0.00003 0.0000297
-2.266 -1.912 -1.688 -1.546 -1.500 -1.640 -1.801 -2.011
-2.277 -2.545 -2.823 -3.094 -3.376 -3.572 -3.812 -4.012
-4.262 -4.415 -4.501 -4.705 -4.823 -4.823 -4.956 -5.111
-5.111 -5.516 -5.524 -5.806 -5.806 -5.806
18mix35 0.00003 0.00003
-2.258 -1.900 -1.666 -1.524 -1.479 -1.621 -1.803 -2.043
-2.308 -2.626 -2.895 -3.214 -3.568 -3.907 -4.193 -4.423
-4.825 -5.111 -5.111 -5.516 -5.516 -5.516 -5.516 -5.806
-5.806 -5.806 -5.806 -5.806 -6.210 -6.215
trp2 0.0001 0
-5.922 -5.435 -4.366 -3.149 -2.124 -1.392 -0.780 -0.336
-0.002 0.233 0.391 0.490 0.540 0.563 0.541 0.488
0.414 0.313 0.203 0.063 -0.028 -0.097 -0.215 -0.411
-0.678 -0.953 -1.208 -1.418 -1.651 -1.855
mix5 0.0001 0.00001
-3.932 -3.411 -2.964 -2.462 -1.836 -1.308 -0.796 -0.390
-0.076 0.147 0.294 0.394 0.446 0.460 0.443 0.389
0.314 0.220 0.099 -0.033 -0.128 -0.197 -0.308 -0.506
-0.785 -1.050 -1.313 -1.529 -1.745 -1.970
mix4 0.0001 0.000025
-2.996 -2.479 -2.099 -1.803 -1.459 -1.126 -0.761 -0.424
-0.144 0.060 0.195 0.288 0.337 0.354 0.330 0.274
0.206 0.105 -0.009 -0.148 -0.242 -0.306 -0.424 -0.626
-0.892 -1.172 -1.425 -1.633 -1.877 -2.071
mix3 0.0001 0.00005
-2.128 -1.661 -1.344 -1.160 -0.996 -0.877 -0.696 -0.495
-0.313 -0.165 -0.042 0.032 0.069 0.079 0.050 -0.006
-0.082 -0.179 -0.295 -0.436 -0.523 -0.584 -0.706 -0.898
-1.178 -1.446 -1.696 -1.922 -2.128 -2.350
mix6 0.0001 0.00009
-1.140 -0.757 -0.497 -0.362 -0.329 -0.412 -0.513 -0.647
-0.772 -0.877 -0.958 -1.040 -1.104 -1.162 -1.233 -1.317
-1.425 -1.543 -1.661 -1.804 -1.877 -1.959 -2.034 -2.249
-2.502 -2.732 -2.964 -3.142 -3.313 -3.576
;
data ftest;
input obsnam $ tot tyr f1-f30 @@;
try = tot - tyr;
if (tyr) then tyr_log = log10(tyr); else tyr_log = -8;
if (try) then try_log = log10(try); else try_log = -8;
tot_log = log10(tot);
datalines;
43trp6 1E-6 0
-5.915 -5.918 -6.908 -5.428 -4.117 -5.103 -4.660 -4.351
-4.023 -3.849 -3.634 -3.634 -3.572 -3.513 -3.634 -3.572
-3.772 -3.772 -3.844 -3.932 -4.017 -4.023 -4.117 -4.227
-4.492 -4.660 -4.855 -5.428 -5.103 -5.428
59mix6 1E-6 1E-7
-5.903 -5.903 -5.903 -5.082 -4.213 -5.083 -4.838 -4.639
-4.474 -4.213 -4.001 -4.098 -4.001 -4.001 -3.907 -4.001
-4.098 -4.098 -4.206 -4.098 -4.213 -4.213 -4.335 -4.474
-4.639 -4.838 -4.837 -5.085 -5.410 -5.410
51mix6 1E-6 2.5E-7
-5.907 -5.907 -5.415 -4.843 -4.213 -4.843 -4.843 -4.483
-4.343 -4.006 -4.006 -3.912 -3.830 -3.830 -3.755 -3.912
-4.006 -4.001 -4.213 -4.213 -4.335 -4.483 -4.483 -4.642
-4.841 -5.088 -5.088 -5.415 -5.415 -5.415
49mix6 1E-6 5E-7
-5.419 -5.091 -5.091 -4.648 -4.006 -4.846 -4.648 -4.483
-4.343 -4.220 -4.220 -4.220 -4.110 -4.110 -4.110 -4.220
-4.220 -4.343 -4.483 -4.483 -4.650 -4.650 -4.846 -4.846
-5.093 -5.091 -5.419 -5.417 -5.417 -5.907
53mix6 1E-6 7.5E-7
-5.083 -4.837 -4.837 -4.474 -3.826 -4.474 -4.639 -4.838
-4.837 -4.639 -4.639 -4.641 -4.641 -4.639 -4.639 -4.837
-4.838 -4.838 -5.083 -5.082 -5.083 -5.410 -5.410 -5.408
-5.408 -5.900 -5.410 -5.903 -5.900 -6.908
57mix6 1E-6 9E-7
-5.082 -4.836 -4.639 -4.474 -3.826 -4.636 -4.638 -4.638
-4.837 -5.082 -5.082 -5.408 -5.082 -5.080 -5.408 -5.408
-5.408 -5.408 -5.408 -5.408 -5.408 -5.900 -5.900 -5.900
-5.900 -5.900 -5.900 -5.900 -6.908 -6.908
41tyro6 1E-6 1E-6
-5.104 -4.662 -4.662 -4.358 -3.705 -4.501 -4.662 -4.859
-5.104 -5.431 -5.433 -5.918 -5.918 -5.918 -5.431 -5.918
-5.918 -5.918 -5.918 -5.918 -5.918 -5.918 -5.918 -6.908
-5.918 -5.918 -6.908 -6.908 -5.918 -5.918
28trp5 0.00001 0
-5.937 -5.937 -5.937 -4.526 -3.544 -3.170 -2.573 -2.115
-1.792 -1.564 -1.400 -1.304 -1.244 -1.213 -1.240 -1.292
-1.373 -1.453 -1.571 -1.697 -1.801 -1.873 -2.008 -2.198
-2.469 -2.706 -2.990 -3.209 -3.384 -3.601
37mix5 0.00001 1E-6
-5.109 -4.865 -4.501 -4.029 -3.319 -3.070 -2.569 -2.207
-1.895 -1.684 -1.516 -1.423 -1.367 -1.348 -1.374 -1.415
-1.503 -1.596 -1.718 -1.839 -1.927 -1.997 -2.118 -2.333
-2.567 -2.874 -3.106 -3.313 -3.579 -3.781
33mix5 0.00001 2.5E-6
-4.366 -4.129 -3.781 -3.467 -3.037 -2.939 -2.593 -2.268
-1.988 -1.791 -1.649 -1.565 -1.520 -1.509 -1.524 -1.580
-1.665 -1.758 -1.882 -2.037 -2.090 -2.162 -2.284 -2.465
-2.761 -3.037 -3.270 -3.520 -3.709 -3.937
31mix5 0.00001 5E-6
-3.790 -3.373 -3.119 -2.915 -2.671 -2.718 -2.555 -2.398
-2.229 -2.085 -1.971 -1.902 -1.860 -1.837 -1.881 -1.949
-2.009 -2.127 -2.230 -2.381 -2.455 -2.513 -2.624 -2.827
-3.117 -3.373 -3.586 -3.785 -4.040 -4.366
35mix5 0.00001 7.5E-6
-3.321 -2.970 -2.765 -2.594 -2.446 -2.548 -2.616 -2.617
-2.572 -2.550 -2.508 -2.487 -2.488 -2.487 -2.529 -2.593
-2.688 -2.792 -2.908 -3.037 -3.149 -3.189 -3.273 -3.467
-3.781 -4.029 -4.241 -4.501 -4.669 -4.865
39mix5 0.00001 9E-6
-3.142 -2.812 -2.564 -2.404 -2.281 -2.502 -2.589 -2.706
-2.842 -2.964 -3.068 -3.103 -3.182 -3.268 -3.361 -3.411
-3.517 -3.576 -3.705 -3.849 -3.932 -3.932 -4.029 -4.234
-4.501 -4.664 -4.860 -5.104 -5.431 -5.433
26tyro5 0.00001 0.00001
-3.037 -2.696 -2.464 -2.321 -2.239 -2.444 -2.602 -2.823
-3.144 -3.396 -3.742 -4.063 -4.398 -4.699 -4.893 -5.138
-5.140 -5.461 -5.463 -5.945 -5.461 -5.138 -5.140 -5.138
-5.138 -5.463 -5.461 -5.461 -5.461 -5.461
tyro2 0.0001 0.0001
-1.081 -0.710 -0.470 -0.337 -0.327 -0.433 -0.602 -0.841
-1.119 -1.423 -1.750 -2.121 -2.449 -2.818 -3.110 -3.467
-3.781 -4.029 -4.241 -4.366 -4.501 -4.366 -4.501 -4.501
-4.668 -4.668 -4.865 -4.865 -5.109 -5.111
;
The following statements fit a PLS model with 10 factors.
proc pls data=ftrain nfac=10;
model tot_log tyr_log try_log = f1-f30;
run;
The table shown in Output 51.3.1 indicates that only three or four factors are required to explain almost all of the variation in both the predictors and the responses.
Output 51.3.1: Amount of Training Set Variation Explained
proc pls data=ftrain nfac=10 cv=testset(ftest)
cvtest(stat=press seed=12345);
model tot_log tyr_log try_log = f1-f30;
run;
The results of the test set validation are shown in Output 51.3.2. They indicate that, although five PLS factors give the minimum predicted residual sum of squares, the residuals for four factors are insignificantly different from those for five. Thus, the smaller model is preferred.
Output 51.3.2: Test Set Validation for the Number of PLS Factors|
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ods listing close;
ods output XLoadings=xloadings;
proc pls data=ftrain nfac=4 details method=pls;
model tot_log tyr_log try_log = f1-f30;
run;
ods listing;
proc transpose data=xloadings(drop=NumberOfFactors)
out =xloadings;
data xloadings; set xloadings;
n = _n_;
rename col1=Factor1 col2=Factor2
col3=Factor3 col4=Factor4;
run;
goptions border;
axis1 label=("Loading" ) major=(number=5) minor=none;
axis2 label=("Frequency") minor=none;
symbol1 v=none i=join c=red l=1;
symbol2 v=none i=join c=green l=1 /*l= 3*/;
symbol3 v=none i=join c=blue l=1 /*l=34*/;
symbol4 v=none i=join c=yellow l=1 /*l=46*/;
legend1 label=none cborder=black;
proc gplot data=xloadings;
plot (Factor1 Factor2 Factor3 Factor4)*n
/ overlay legend=legend1 vaxis=axis1
haxis=axis2 vref=0 lvref=2 frame cframe=ligr;
run; quit;
The resulting plot is shown in Output 51.3.3.
Output 51.3.3: Predictor Loadings Across Frequencies
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