Barycentric Discriminant Analysis

Barycentric discriminant analysis (BADIA) generalizes discriminant analysis, and like discriminant analysis, it is performed when measurements made on some observations are combined to assign these observations, or new observations, to a priori defined categories. For example, BADIA can be used (a) to assign people to a given diagnostic group (e.g., patients with Alzheimer's disease, patients with other dementia, or people aging without dementia) on the basis of brain imaging data or psychological tests (here the a priori categories are the clinical groups), (b) to assign wines to a region of production on the basis of several physical and chemical measurements (here the a priori categories are the regions of production), (c) to use brain scans taken on a given participant to determine what type of object (e.g., a face, a cat, a chair) was watched by the participant when the scans were taken (here the a priori categories are the types of object), or (d) to use DNA measurements to predict whether a person is at risk for a given health problem (here the a priori categories are the types of health problem).

BADIA is more general than standard discriminant analysis because it can be used in cases for which discriminant analysis cannot be used. This is the case, for example, when there are more variables than observations or when the measurements are categorical.

BADIA is a class of methods that all rely on the same principle: Each category of interest is represented by the barycenter of its observations (i.e., the weighted average; the barycenter is also called the center of gravity of the observations of a given category), and a generalized principal components analysis (GPCA) is performed on the category by variable matrix. This analysis gives a set of discriminant factor scores for the categories and another set of factor scores for the variables. The original observations are then projected onto the category factor space, providing a set of factor scores for the observations. The distance of each observation to the set of categories is computed from the factor scores, and each observation is assigned to the closest category. The comparison between the a priori and a posteriori category assignments is used to assess the quality of the discriminant procedure. The prediction for the observations that were used to compute the barycenters is called the fixed-effect prediction. Fixed-effect performance is evaluated by counting the number of correct and incorrect assignments and storing these numbers in a confusion matrix. Another index of the performance of the fixed-effect model—equivalent to a squared coefficien [Page 65]of correlation—is the ratio of category variance to the sum of category variance plus variance of the observations within each category. This coefficient is denoted R2 and is interpreted as the proportion of variance of the observations explained by the categories or as the proportion of the variance explained by the discriminant model. The performance of the fixed-effect model can also be represented graphically as a tolerance ellipsoid that encompasses a given proportion (say 95%) of the observations. The overlap between the tolerance ellipsoids of two categories is proportional to the number of misclassifications between these two categories.

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