PLS scores values

We now have enough information to find our PLS Scores matrix and PLS Loadings matrix. First of all the PLS Loadings matrix is simply the right singular values matrix or the V matrix this matrix is referred to as the P matrix in principal components analysis and partial least squares terminology. The PLS Scores matrix is calculated as... 

For the blend uniformity application, the primary concern is overall heterogeneity within the sample. This is directly indicated by the width of the distribution and is quantitatively measured as the percent standard deviation relative to the mean (%SD). The %SD value obtained from the PLS scores distribution from each of the six imaged tablets is noted on the histograms in Figure 8.15(A-F). The value of %STD consistently decreases as the blending quality of the sample improves. 

Figure 7.11 (a-f) Histograms of the PLS score images showing the statistical distribution of the API class. Heterogeneity in the blend is indicated by deviations from a normal distribution and can be expressed as the percent standard deviation (%STD) metric calculated by dividing the standard deviation by the mean value of the histogram distributions. 

Table 2.3 Individual normalized standard deviation values of histogram distributions from PLS score images a measure of sample heterogeneity for individual group A and B samples...

In the preceding description of the Mahalanobis distance, the number of coordinates in the distance metric is equal to the number of spectral frequencies. As discussed earlier in the section on principal component analysis, the intensities at many frequencies are dependent, and by using the full spectrum, we fit the noise in addition to the real information. In recent years, Mahalanobis distance has been defined with PCA or PLS scores instead of the spectral frequencies because these techniques eliminate or at least reduce most of the overfitting problem. The overall application of the Mahalanobis distance metric is the same except that the rt intensity values are replaced by the scores from PCA or PLS. An example of a Mahalanobis distance calculation on a set of Raman spectra for 25 carbohydrates is shown in Fig. 5-11. The 25 spectra were first subjected to PCA, and it was found that the first three principal components could account for most of the variance in the spectra. It was first assumed that all 25 spectra belonged to the same class because they were all carbohydrates. However, as shown in the three-dimensional plot in Fig. 5-11, the spectra can be clearly divided into three separate classes, with two of the spectra almost equal distance from each of the three classes. Most of the components in the upper left class in the two-dimensional plot were sugars however, some sugars were found in the other two classes. For unknowns, scores have to be calculated from the principal components and processed in the same way as the spectral intensities. 

To characterize the reaction system for the PLS analysis, the principal property score values were used as descriptors. For the ketones, two additional descriptors were used to describe the steric environment of the carbonyl group. The v parameter given by Charton[14] was used to describe the size of the ketone side chain. This parameter is a measure of the van der Waals radius of the substituent, and can be regarded as a measure of how large the side chain appears to be when... 

Since the value of the response can be predicted from the PLS scores as ... 

Let us now consider a new set of values measured for the various X-variables, collected in a supplementary row vector x. From this we want to derive a row vector y of expected T-values using the predictive PLS model. To do this, the same sequence of operations is followed transforming x into a set of factor scores r, t 2, t A pertaining to this new observation. From these t -scores y can be... 

An examination of the sample distributions observed in principal components projections using isomer concentration data expressed as fractional composition, as well as the clustering of samples by similar values of their second principal component score term, revealed consistent differences existed in sample profiles. The next step in this data evaluation is to statistically analyze correlations of the PLS components from analyses with the external variables such as percent sand, clay and silt, and total organic matter in samples. These correlations may play an important role in identifying factors resulting in changes in PCB composition and enable one to more clearly understand the forces determining the distribution and fate of PCB in a complex ecosystem. 

As a result, it is critical to evaluate process samples in real-time for their appropriateness of use with the empirical model. For models built using PC A, PLS, PCR and other factor based methods, the mechanism for such a model health monitor is already built into the model. Equations 12.21 and 12.22 can be used to calculate Hotelling and Q residual statistics for each process sample using the sample s on-line analytical profile (xp) and the loadings (P) and scores (T) of the model. An abnormally high value would indicate... 

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