Multivariate analysis of variance

Multivariate analysis of variance-covariance (MANOVA/MANCOVA) techniques. 

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). 

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. 

The simultaneous comparison of the mean values of a set of features, i.e. two vectors of means in the simplest case of two classes, we will call multivariate analysis of variance (MANOVA). 

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. 

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

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. 

Fig. 9-9 demonstrates the results of MVDA for the three investigated territories in the plane of the computed two discriminant functions. The separation line corresponds to the limits of discrimination for the highest probability. The results prove that good separation of the three territories with a similar geological background is possible by means of discriminant analysis. The misclassification rate amounts to 13.0%. The scattering radii of the 5% risk of error of the multivariate analysis of variance overlap considerably. They demonstrate also that the differences in the multivariate data structure of the three territories are only small. 

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