Variance, principal component models

Musumarra et al. [44] also identified miconazole and other drugs by principal components analysis of standardized thin-layer chromatographic data in four eluent systems and of retention indexes on SE 30. The principal component analysis of standardized R values in four eluents systems ethylacetate-methanol-30% ammonia (85 10 15), cyclohexane-toluene-diethylamine (65 25 10), ethylacetate-chloroform (50 50), and acetone with plates dipped in potassium hydroxide solution, and of gas chromatographic retention indexes in SE 30 for 277 compounds provided a two principal components model that explains 82% of the total variance. The scores plot allowed identification of unknowns or restriction of the range of inquiry to very few candidates. Comparison of these candidates with those selected from another principal components model derived from thin-layer chromatographic data only allowed identification of the drug in all the examined cases. 

The principal components model of the Aroclor seunples (Table i) preserves greater than 95% of the sample variance of the entire data set. From the 3-D seunple score plot (Figure 3) one can make these observations PCB mixtures of two Aroclors form a straight line three Aroclor mixtures form a plane and that possible mixtures of the four Aroclors are bounded by the intersection of the four planes. Samples not bounded by or inside the volume formed by the intersection of the four planes may... 

Loadings Plot (Model and Variable Diagnostic) The loading plot in Figure 4.64 reveals that the first and se< ond loadings have nonrandom features, while the third is random in nature. This suggests a two-principal component model consistent with the percent variance explained, residuals plots, and mSECV PCA results... 

PLS is related to principal components analysis (PCA) (20), This is a method used to project the matrix of the X-block, with the aim of obtaining a general survey of the distribution of the objects in the molecular space. PCA is recommended as an initial step to other multivariate analyses techniques, to help identify outliers and delineate classes. The data are randomly divided into a training set and a test set. Once the principal components model has been calculated on the training set, the test set may be applied to check the validity of the model. PCA differs most obviously from PLS in that it is optimized with respect to the variance of the descriptors. 

The cumulative variance is the variance explained by a principal component model constructed using factors 1 through j. 

Malinowski and others have observed that the indicator function often reaches a minimum value when the correct number of factors is used in a principal component model. We finish this section by giving a MAILAB function in Example 4.6 for calculating eigenvalues, variance, cumulative variance, Malinowski s RE, and Malinowski s REV and F (described in Section 4.6.2). Note that the function uses the SVD to determine the eigenvalues. 

A sample can be classified by calculating the sum of the squares of the difference between its measured spectrum vector and the same spectrum reproduced using a principal component model. The residual variance, s, of a data vector i fitted to the training set for class q indicates how similar the spectrum is to class q. For data vectors from the training set, the residual variance of a sample is given by Equation 4.44. 

In Equation 4.44, rtj is the residual absorbance of the ith sample at the jth variable, m is the number of wavelengths, and k is the number of principal components used in constructing the principal component model. If mean correction is used, then the denominator in Equation 4.44 should be changed to m-k - 1. For unknown data vectors (vectors not used in the training set), Equation 4.45 is used to calculate the residual variance, where r, is a residual absorbance datum for the ith sample s spectrum when fit to class q. 

The variation in the data not explained by the principal component model is called the residual variance. Classification in SIMCA is made by comparing the residual variance of a sample with the average residual variance of those samples that make up the class. This comparison provides a direct measure of the similarity of a sample to a particular class and can be considered as a measure of the goodness of fit of a sample for a particular class model. To provide a quantitative basis for this comparison, an / -statistic is used to compare the residual variance of the sample with the mean residual variance of the class [72], The F-statistic can also be used to compute an upper limit for the residual variance of those samples that belong to the class, with the final result being a set of probabilities of class membership for each sample. 

This partitioning of "unexplained variance" offers a means of determining the relevance of each descriptior in the principal components model. Descriptor... 

Objects that do not fit the estimated principal component model can be eliminated by testing the total residual variance of a class q against the residual variance of that object. The two variances are calculated as follows. 

The close link between lakes and their catchments was evident in a study of spatial variability in surface sediment composition in a small northern Swedish lake (Korsman et al., 1999). In this study, the information in the near-infrared spectra of surface sediment samples was used to determine how sediment composition varied over the lake bottom. The study showed that the NIR spectra per se provide information that can be used to study sediment characteristics as well as sediment focusing in a qualitative way. The variance in the NIR spectra (Fig. 7) was only to a minor extent explained by the variation in water depth or sediment organic content. More importantly, the spatial evaluation of the spectral data suggested that NIR analysis of lake sediments mainly reflects sediment properties that cannot be simply explained by water depth or amount of organic matter. Principal component modelling of NIR spectra from 165 coring sites, established along a 50m x 50m... 

We have seen that PLS regression (covariance criterion) forms a compromise between ordinary least squares regression (OLS, correlation criterion) and principal components regression (variance criterion). This has inspired Stone and Brooks [15] to devise a method in such a way that a continuum of models can be generated embracing OLS, PLS and PCR. To this end the PLS covariance criterion, cov(t,y) = s, s. r, is modified into a criterion T = r. (For... 

To construct the reference model, the interpretation system required routine process data collected over a period of several months. Cross-validation was applied to detect and remove outliers. Only data corresponding to normal process operations (that is, when top-grade product is made) were used in the model development. As stated earlier, the system ultimately involved two analysis approaches, both reduced-order models that capture dominant directions of variability in the data. A PLS analysis using two loadings explained about 60% of the variance in the measurements. A subsequent PCA analysis on the residuals showed that five principal components explain 90% of the residual variability. 

Figure 38 shows the variance explained by the two principal component (PC) model as a percentage of each of the two indices batch number and time. The lower set of bars in Fig. 38a are the explained variances for the first PC, while the upper set of bars reflects the additional contribution of the second PC. The lower line in Fig. 38b is the explained variance over time for the first PC and the upper line is the combination of PC 1 and 2. Figure 38a indicates, for example, that batch numbers 13 and 30 have very small explained variances, while batch numbers 12 and 33 have variances that are captured very well by the reference model after two PCs. It is impossible to conclude from this plot alone, however, that batches 13 and 30 are poorly represented by the reference model. 

Stanimirova I. Michalik K. Drzazga Z. Trzeciak H. Wentzell P.D. Walczak B. Interpretation of analysis of variance models using principal component analysis to assess the effect of a maternal anticancer treatment on the mineralization of rat bones. Analytica Chimica Acta, 2011,689 (1), 1-7. 

One must consider the number of product terms that should be included in a model. For chromatography data obtained from similar samples, it can be expected that the data will contain a high degree of correlation. In our experiments, two- or three-component models usually accounted for >90% of the variance in the data for a class of similar samples. Results from crossvalidation should be considered as the primary criteria in selecting the number of principal components to be extracted from a given data set (34). 

For the styrene-butadiene example, the use of the PCR method to develop a calibration for di-butadiene is summarized in Table 12.6. It should be mentioned that the data were mean-centered before application of the PCR method. Figure 12.12 shows the percentage of explained variance in both x (the spectral data) andy (the c/i-butadiene concentration data) after each principal component. After four principal components, it does not appear that the use of any additional PCs results in a large increase in the explained variance of X or y. If a PCR regression model using four PCs is built and applied to the calibration data, a fit RMSEE of 1.26 is obtained. 

The principal component space does not have the redundancy issue discussed above, because the PCs are orthogonal to one another. In addition, because each PC explains the most remaining variance in the x data, it is often the case that fewer PCs than original x variables are needed to capture the relevant information in the x data. This leads to simpler classification models, less susceptibility to overfitting through the use of too many dimensions in the model space, and less noise in the model. 

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