Directed Canonical Analysis And the Performance of Classifiers under Its Associated Linear Transformation

Directed canonical analysis is presented as an extension of the general form of canonical analysis, which is a method for reducing the dimensionality of multivariate data sets with minimum loss of discriminatory variance. The reduction takes the form of a linear transformation, y = Cx, that condenses the discriminatory variance onto a relatively few, high-variance orthogonal discriminant axes. Canonical analysis is developed as an analog to the one-way MANOVA. The directed extension allows user-specified contrasts to define linear relationships that are known or suspected to exist within the data. The linear transformation C is defined by means of the symmetric canonical form of the matrix eigenproblem. Canonical and principal components transformations and various distance classifiers were applied to 3 representative remotely sensed MSS data sets. Results indicate that use of a piecewise maximum likelihood classifier with the directed canonical discriminant axes will give the best overall combination of classification accuracy and computational efficiency if adequate sample sizes are available to estimate category statistics. For small sample sizes, piecewise Euclidean distance with the general canonical axes is recommended. In canonically transformed space, Euclidean distance is equivalent to the Mahalanobis classifier.