Multivariate variance and discriminant analysis

One of the powerful classification methods is multivariate variance and discriminant analysis (MVDA) (Dillon and Goldstein [1984] Ahrens and Lauter [1974] Danzer et al. [1984]). 

Because multivariate mean comparison by variance analysis is strongly related to the task of discriminating means, or the respective classes of objects, in applications both aspects are seen at the same time. Therefore the name multivariate variance and discriminant analysis (MVDA) is commonly used. 

Supervised learning methods - multivariate analysis of variance and discriminant analysis (MVDA) - k nearest neighbors (kNN) - linear learning machine (LLM) - BAYES classification - soft independent modeling of class analogy (SIMCA) - UNEQ classification Quantitative demarcation of a priori classes, relationships between class properties and variables... 

The visual estimation of differences between groups of data has to be proved using multivariate statistical methods, as for example with multivariate analysis of variance and discriminant analysis (see Section 5.6). 

One has to keep in mind that groups of objects found by any clustering procedure are not statistical samples from a certain distribution of data. Nevertheless the groups or clusters are sometimes analyzed for their distinctness using statistical methods, e.g. by multivariate analysis of variance and discriminant analysis, see Section 5.6. As a result one could then discuss only those clusters which are statistically different from others. 

In the previous example the data situation was adequately investigated by multivariate analysis of variance and discriminant analysis. 

Multivariate Analysis of Variance and Discriminant Analysis, and PLS Modeling... 

The application of methods of multivariate statistics (here demonstrated with examples of cluster analysis, multivariate analysis of variance and discriminant analysis, and principal components analysis) enables clarification of the lateral structure of the types of feature change within a test area. 

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. 

As we mentioned in the preceding section, multivariate analysis of variance, like discriminant analysis, uses the scatter matrices B and W. 

Because analyses of water quality usually involve the collection of data on several variables, multivariate statistical analyses often are relevant. For example, multivariate analysis of variance (MANOVA) and discriminant analysis were used by Alden (1997) to investigate water quality trends in Chesapeake Bay. 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

Female adult budworm dry weights and the number of survivors of budworm were analyzed by multivariate analysis of variance to test for the effects of site and sex. Stepwise discriminant analysis was used to determine If tree chemical and physical parameters differed between sites (17). 

Multivariate statistical analysis using classes of variables and calculating discriminant functions as linear combinations of the variables that maximize the inter-class variance and minimize the intra-class variance. Volume 2(2). 

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. 

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