Within-class variance

When we consider the multivariate situation, it is again evident that the discriminating power of the combined variables will be good when the centroids of the two sets of objects are sufficiently distant from each other and when the clusters are tight or dense. In mathematical terms this means that the between-class variance is large compared with the within-class variances. 

Although the development of a SIMCA model can be rather cumbersome, because it involves the development and optimization of J PCA models, the SIMCA method has several distinct advantages over other classification methods. First, it can be more robust in cases where the different classes involve discretely different analytical responses, or where the class responses are not linearly separable. Second, the treatment of each class separately allows SIMCA to better handle cases where the within-class variance structure is... 

Matrix B expresses the variance between the means of the classes, matrix expresses the pooled within-classes variance of all classes. The two matrices B and W are the starting point both for multivariate analysis of variance and for discriminant analysis. 

However, contrary to PCA, it is a supervised method that uses the information of which data point belongs to which class. The discriminants are linear combinations of the measured variables (e.g., sensor response). A discriminant function is found that maximizes the ratio of between-class variance to within-class variance. 

The selection of variables could separate relevant information from unwanted variability and at the same time allows data compression, that is more parsimonious models, simplification or improvement of model interpretation, and so on. Although many approaches can be used for features selection, in this work, a wavelet-based supervised feature selection/classification algorithm, WPTER [12], was applied. The best performing model was obtained using a daubechies 10 wavelet, a maximum decomposition level equal to 10, between-class/within-class variance ratio criterion for the thresholding operation and the percentage of selected coefficients equal to 2%. Six wavelet coefficients were selected, belonging to the 4th, 5th, 6th, 8th, and 9th levels of decomposition. 

The expected residual class variance for class q is calculated by using the residual data vectors for all samples in the training set. The resulting residual matrix is used to calculate the residual variance within class q. This value is an indication of how tight a class cluster is in multidimensional space. It is calculated according to Equation 4.46, where s02 is the residual variance in class q and n is the number of samples in class q. 

These observations may be summarized conveniently in an analysis-of-variance table-. Table 26-7 illustrates this type of table for the above case. The overall variance (total mean square) Sj(N — 1) contains contributions due to variances within as well as between classes. The variation between classes contains both variation within classes and a variation associated with the classes themselves and is given by the expected mean square aj + not. Whether not is significant can be determined by the F test. Under the null hypothesis, = 0. Whether the ratio... 

That is, a is the direction that maximizes the separation between the classes, both by having compact classes (a small within-groups variance) and by having the class centers far apart (a large between-groups variance). Large values in a indicate which variables are important in the discrimination. Another formulation is to calculate the Mahalanobis distance of a new sample x to the class centers... 

Of the four different methods of cluster analysis applied, the method of Ward described in the Clustan User Manual (10), worked best when compared to the single-, complete-, or average-linkage methods. Using Ward s method, two clusters, Gn and Gm, are fused when by pooling the variance within two existing clusters the variance of the so formed clusters increases minimally. The variance or the sum of squares within the classes will be chosen as the index h of a partition. 

Table 15.5 lists concentrations of the major photooxidants in surface waters, diurnally averaged over 24 hours. Note that, even if kox(i) values are measured or estimated accurately (within a factor two or three), oxidant concentrations in the environment vary widely, and averaged values have a variance of five- to tenfold for any given location. In extreme locations, such as pristine marine waters, or heavily polluted surface waters, oxidant concentrations may be 100 times smaller or larger than the values Table 15.5 lists. Table 15.6 lists rate constants (kox) for various photooxidants in their reaction with major classes of organic compounds. To estimate the rate of an indirect photoreaction for chemical C (Equation (18)), either a measured or estimated value of kox is required, specific for each oxidant and for each class of organic compounds. Methods for estimating kox from molecular structure with structure-activity relationships (SARs) have been developed for many photooxidants and are discussed below. 

With some exceptions, i.e. non-planar systems, variances in geometry are small within the class of benzenoid (alternant) hydrocarbons and thus can be neglected. Hence the individual representatives of this class of compounds differ only with regard to their molecular topologies. Provided the influence of the non-topological structural characteristics (kind of atoms, geometry, additional electronic interactions that are not referred to in the constitutional formulae) on the physical and chemical properties... 

In discriminant analysis, in a manner similar to factor analysis, new synthetic features have to be created as linear combinations of the original features which should best indicate the differences between the classes, in contrast with the variances within the classes. These new features are called discriminant functions. Discriminant analysis is based on the same matrices B and W as above. The above tested groups or classes of data are modeled with the aim of reclassifying the given objects with a low error risk and of classifying ( discriminating ) another objects using the model functions. 

The result from cluster analysis presented in Fig. 9-2 is subjected to MVDA (for mathematical fundamentals see Section 5.6 or [AHRENS and LAUTER, 1981]). The principle of MVDA is the separation of predicted classes of objects (sampling points). In simultaneous consideration of all the features observed (heavy metal content), the variance of the discriminant functions is maximized between the classes and minimized within them. The classification of new objects into a priori classes or the reclassification of the learning data set is carried out using the values of the discriminant function. These values represent linear combinations of the optimum separation set of the original features. The result of the reclassification is presented as follows ... 

The principle of multivariate analysis of variance and discriminant analysis (MVDA) consists in testing the differences between a priori classes (MANOVA) and their maximum separation by modeling (MDA). The variance between the classes will be maximized and the variance within the classes will be minimized by simultaneous consideration of all observed features. The classification of new objects into the a priori classes, i.e. the reclassification of the learning data set of the objects, takes place according to the values of discriminant functions. These discriminant functions are linear combinations of the optimum set of the original features for class separation. The mathematical fundamentals of the MVDA are explained in Section 5.6. 

The relationship between age and pharmacokinetics were assessed by an analysis of variance (ANOVA) on AUCs, MRT and Cmax with adjustments for treatment, period, sequence and subject within sequence effects by age class using the natural log transformed values to compare treatments within age class. Point estimates and 95 % confidence intervals were calculated for me treatment ratios per age class. 

---