Chapter Contents Previous Next
 The PLS Procedure

## Example 51.3: Choosing a PLS Model by Test Set Validation

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.

Figure 51.11: Spectra for Three Samples of Tyrosine and Tryptophan

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

 The PLS Procedure

 Percent Variation Accounted for by PartialLeast Squares Factors Number ofExtractedFactors Model Effects Dependent Variables Current Total Current Total 1 81.1654 81.1654 48.3385 48.3385 2 16.8113 97.9768 32.5465 80.8851 3 1.7639 99.7407 11.4438 92.3289 4 0.1951 99.9357 3.8363 96.1652 5 0.0276 99.9634 1.6880 97.8532 6 0.0132 99.9765 0.7247 98.5779 7 0.0052 99.9817 0.2926 98.8705 8 0.0053 99.9870 0.1252 98.9956 9 0.0049 99.9918 0.1067 99.1023 10 0.0034 99.9952 0.1684 99.2707

In order to choose the optimal number of PLS factors, you can explore how well models based on the training data with different numbers of factors fit the test data. To do so, use the CV=TESTSET option, with an argument pointing to the test data set ftest, as in the following statements:

```   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

 The PLS Procedure

 Test Set Validation forthe Number of ExtractedFactors Number ofExtractedFactors Root Mean PRESS Prob > PRESS 0 3.056797 <.0001 1 2.630561 <.0001 2 1.00706 0.0070 3 0.664603 0.0020 4 0.521578 0.3800 5 0.500034 1.0000 6 0.513561 0.5100 7 0.501431 0.6870 8 1.055791 0.1530 9 1.435085 0.1010 10 1.720389 0.0320

 Minimum root mean PRESS 0.5 Minimizing number of factors 5 Smallest number of factors with p > 0.1 4

 The PLS Procedure

 Percent Variation Accounted for by PartialLeast Squares Factors Number ofExtractedFactors Model Effects Dependent Variables Current Total Current Total 1 81.1654 81.1654 48.3385 48.3385 2 16.8113 97.9768 32.5465 80.8851 3 1.7639 99.7407 11.4438 92.3289 4 0.1951 99.9357 3.8363 96.1652

The factor loadings show how the PLS factors are constructed from the centered and scaled predictors. For spectral calibration, it is useful to plot the loadings against the frequency. In many cases, the physical meanings that can be attached to factor loadings help to validate the scientific interpretation of the PLS model. You can use the following statements to plot the loadings for the four PLS factors against frequency.
```   ods listing close;
proc pls data=ftrain nfac=4 details method=pls;
model tot_log tyr_log try_log = f1-f30;
run;
ods listing;

n = _n_;
rename col1=Factor1 col2=Factor2
col3=Factor3 col4=Factor4;
run;
goptions border;
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;