Discriminant & TableOfReal: To ClassificationTable...

A command to use the selected Discriminant to classify each row from the selected TableOfReal. The newly created ClassificationTable will then contain the posterior probabilities of group membership.

Settings

Pool covariance matrices
when on, all group covariance matrices are pooled and distances will be determined on the basis of only this pooled covariance matrix (see below).

Details

The posterior probabilities of group membership pj for a vector x are defined as:

 pj = p(j|x) = exp (–dj2(x) / 2) / ∑k=1..numberOfGroups exp (–dk2(x) / 2),

where di2 is the generalized squared distance function:

 di2(x) = ((x–μi)′ Σi-1 (x–μi) + ln determinant (Σi)) / 2 – ln aprioriProbabilityi

that depends on the individual covariance matrix Σi and the mean μi for group i.

When the covariances matrices are pooled, the squared distance function can be reduced to:

 di2(x) = ((x–μi)′ Σ-1 (x–μi) – ln aprioriProbabilityi,

and Σ is now the pooled covariance matrix.

The a priori probabilities normally will have values that are related to the number of training vectors ni in each group:

 aprioriProbabilityi = ni / Σk=1..numberOfGroups nk