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Figure 1. Usually, data can be organized as a table, a matrix, here denoted X, with N rows=observations and K columns=variables. Provided the variation between the observations is moderate, X can be approximated by a bilinear model (latent variable model) with just a few components (A in number). The smaller the variation between the observations, the fewer components are needed to well approximate X. A residual matrix, E (not shown) comprises the difference between the data matrix, X, and the model, that is, E=X-T P’. The interpretation of the model is helped by the layers, tap, , appearing in the order of importance, with the first layer (a= 1) explaining more of the variation in X than the second layer (a=2), and so on. And the expansion is terminated when the components, layers, no longer explain significant parts of X as determined, for instance, by cross-validation (CV). The number

Figure 1 Usually, data can be organized as a table, a matrix, here denoted X, with N rows=observations and K columns=variables. Provided the variation between the observations is moderate, X can be approximated by a bilinear model (latent variable model) with just a few components (A in number). The smaller the variation between the observations, the fewer components are needed to well approximate X. A residual matrix, E (not shown) comprises the difference between the data matrix, X, and the model, that is, E=X-T P’. The interpretation of the model is helped by the layers, tap, , appearing in the order of importance, with the first layer (a= 1) explaining more of the variation in X than the second layer (a=2), and so on. And the expansion is terminated when the components, layers, no longer explain significant parts of X as determined, for instance, by cross-validation (CV). The number

Related Figures (12)

Figure 3. A time series plot of the total analysis index (TAI). Nine sam- ples are marked for reference purposes (see text).
Figure 3. A time series plot of the total analysis index (TAI). Nine sam- ples are marked for reference purposes (see text).
Figure 6. Combined control chart with T2 and DModxX versus time order. Tolerance levels corresponding to 95% and 99% are given for T2, and the critical distance for DModX is plotted. Seven samples are marked for comparison with Figure 3. It is interesting to compare Figures 3 and 6, with the same seven samples (167, 195, 228, 338, 399, 577, and 611) marked. These are, according to the TAI scale, among the most impure copper samples (Figure 3). The combined control charts in Figure 6 also identifies these but differentiates them into two categories. Five samples are abnormal according to T? and two according to DModx. Thus, the former five samples (167, 195, 228, 399, and 577) conform to the dominating correlation struc- ture. This means that their impurity profiles have normal shape but are unacceptably large. The latter (338 and 611), however, break the normal correlation structure and hence indicates an abnormal type of impurity. This information is not available in overall aggregates, such as TAI. Further interpretation of the samples—in the model plane—from the center in the score plot (Figure 4). This can be used to construct elliptic tolerance regions for the scores, as displayed in Figure 4 and as a control chart in Figure 6.
Figure 6. Combined control chart with T2 and DModxX versus time order. Tolerance levels corresponding to 95% and 99% are given for T2, and the critical distance for DModX is plotted. Seven samples are marked for comparison with Figure 3. It is interesting to compare Figures 3 and 6, with the same seven samples (167, 195, 228, 338, 399, 577, and 611) marked. These are, according to the TAI scale, among the most impure copper samples (Figure 3). The combined control charts in Figure 6 also identifies these but differentiates them into two categories. Five samples are abnormal according to T? and two according to DModx. Thus, the former five samples (167, 195, 228, 399, and 577) conform to the dominating correlation struc- ture. This means that their impurity profiles have normal shape but are unacceptably large. The latter (338 and 611), however, break the normal correlation structure and hence indicates an abnormal type of impurity. This information is not available in overall aggregates, such as TAI. Further interpretation of the samples—in the model plane—from the center in the score plot (Figure 4). This can be used to construct elliptic tolerance regions for the scores, as displayed in Figure 4 and as a control chart in Figure 6.
Figure 5. Principal components analysis loading plot corresponding to Figure 4. Each single quality variable gives rise to one plotted point. The interpretation suggests that the first principal component reflects “average” impurity, because all loadings are positive. The second princi- pal component models deviations from this “average” impurity level.
Figure 5. Principal components analysis loading plot corresponding to Figure 4. Each single quality variable gives rise to one plotted point. The interpretation suggests that the first principal component reflects “average” impurity, because all loadings are positive. The second princi- pal component models deviations from this “average” impurity level.
Figure 4. Principal components analysis score plot with t2 versus t1 accounting for 67% of the total variation. Each copper sample is shown as a unique plot mark. Samples 111 and 302 are highlighted for reference purposes. The “acceptable” Hotelling’s T region is indicated by the ellipse (95% tolerance region).
Figure 4. Principal components analysis score plot with t2 versus t1 accounting for 67% of the total variation. Each copper sample is shown as a unique plot mark. Samples 111 and 302 are highlighted for reference purposes. The “acceptable” Hotelling’s T region is indicated by the ellipse (95% tolerance region).
Figure 7. Relationship between measured and estimated lambda proportions for the 102 samples of the training set (left). Partial least squares (PLS) predictions for the 26 samples of the test set (right). The similarity of root mean square error of estimation, root mean square error of cross-validation, and root mean square error of prediction indicates that the PLS model has sound explanatory and predictive powers of the lambda content in the sam- ples. Similar results are obtainable using PLS to predict the remaining four carrageenan constituents (no plots shown). Partial least squares trees (PLS-Tress) are constructed similarly to regression trees [44] with the difference that one searches the score space of X instead of individual X-variables. This allows tree-type models to be applied to complex data matrices with many and collinear, even incomplete, variables [45]. Other To minimize the effect of the baseline variation, the spectral data were preprocessed using standard normal variate transform filtering [41]. Thereafter, the spectral data were Pareto scaled, and the Y-block (comprising the proportions of the five carra- geenan species) was UV-scaled.
Figure 7. Relationship between measured and estimated lambda proportions for the 102 samples of the training set (left). Partial least squares (PLS) predictions for the 26 samples of the test set (right). The similarity of root mean square error of estimation, root mean square error of cross-validation, and root mean square error of prediction indicates that the PLS model has sound explanatory and predictive powers of the lambda content in the sam- ples. Similar results are obtainable using PLS to predict the remaining four carrageenan constituents (no plots shown). Partial least squares trees (PLS-Tress) are constructed similarly to regression trees [44] with the difference that one searches the score space of X instead of individual X-variables. This allows tree-type models to be applied to complex data matrices with many and collinear, even incomplete, variables [45]. Other To minimize the effect of the baseline variation, the spectral data were preprocessed using standard normal variate transform filtering [41]. Thereafter, the spectral data were Pareto scaled, and the Y-block (comprising the proportions of the five carra- geenan species) was UV-scaled.
Figure 8. Hierarchical model structure used when separating Y-predictive and Y-orthogonal spectral variation. J means a predictive component expressing joint information. Ux means an orthogonal-in-X component expressing structure that is unique to X. Uy means an orthogonal-in-Y component expressing structure that is unique to Y. Figure 8 shows that the first base level OPLS model between the NIR block and the Y-block contains four predictive compo- nents expressing the joint information. Additionally, the model contains 2 orthogonal components accounting for systematic structure in the NIR data that is unrelated to the Y-block (the five concentration variables). The corresponding results for the
Figure 8. Hierarchical model structure used when separating Y-predictive and Y-orthogonal spectral variation. J means a predictive component expressing joint information. Ux means an orthogonal-in-X component expressing structure that is unique to X. Uy means an orthogonal-in-Y component expressing structure that is unique to Y. Figure 8 shows that the first base level OPLS model between the NIR block and the Y-block contains four predictive compo- nents expressing the joint information. Additionally, the model contains 2 orthogonal components accounting for systematic structure in the NIR data that is unrelated to the Y-block (the five concentration variables). The corresponding results for the
Figure 9. Scores (left) and loadings (right) of the first two components of the top level OPLS. The design structure of the underlying design is easy to see. In the loading plot, $M15 refers to the near infrared base level model, $M16 to the infrared model, and $M17 to the Raman model.
Figure 9. Scores (left) and loadings (right) of the first two components of the top level OPLS. The design structure of the underlying design is easy to see. In the loading plot, $M15 refers to the near infrared base level model, $M16 to the infrared model, and $M17 to the Raman model.
Figure 10. Score plot of the two orthogonal scores of the lower level near infrared model (upper left). Score plot of the two first orthogonal scores of the lower level infrared model (upper right). Score plot of the two first orthogonal scores of the lower level Raman model (lower left). Score plot of the two scores of the top level PCA model (lower right). The same observations (samples) are marked throughout all four plots.
Figure 10. Score plot of the two orthogonal scores of the lower level near infrared model (upper left). Score plot of the two first orthogonal scores of the lower level infrared model (upper right). Score plot of the two first orthogonal scores of the lower level Raman model (lower left). Score plot of the two scores of the top level PCA model (lower right). The same observations (samples) are marked throughout all four plots.
Figure 11. Score line plot of t1 of the hierarchical principal components analysis model. Systematic shifts between the different sampling days are obvious.
Figure 11. Score line plot of t1 of the hierarchical principal components analysis model. Systematic shifts between the different sampling days are obvious.
Figure 12. Group contribution plot between the day 3 and day 4 samples. This plot confirms that the “jump” seen in t1 arises because of anomalies in the near infrared and infrared spectra.
Figure 12. Group contribution plot between the day 3 and day 4 samples. This plot confirms that the “jump” seen in t1 arises because of anomalies in the near infrared and infrared spectra.
Figure 13. Line plot of po1—the loading of the first orthogonal-in-X component—of the lower level near infrared model. This plot suggests the region around 1970 nm to contain a large fraction of the noncorrelating variation.
Figure 13. Line plot of po1—the loading of the first orthogonal-in-X component—of the lower level near infrared model. This plot suggests the region around 1970 nm to contain a large fraction of the noncorrelating variation.
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