2 = wtx = (xi - x2)T S /x which maximizes the ratio

wTSpiw wTSp(W

over all possible coefficient vectors w where d = (xi — x2). The maximum of the ratio in Eq. 8.27 is T2 = (xi - x2)TSpi1(xi - x2) [262]. For two populations with equal covariances, FDA corresponds to the particular case of the minimum ECM rule discussed in Section 8.2.1. The first terms in Eqs. 8.18 and 8.19 are the linear function obtained by FDA that maximizes the univariate between-class scatter relative to the within-class scatter (Eq. 8.26) [262],

The allocation rule of a new observation xo to classes 7Ti or 7:2 based on FDA is [262]

Allocate xo to 7Ti if

(X! - x2)TSpi1xo > i(x! - x2)TSpi1(x1 + x2) (8.28)

Allocate xo to 7r2 otherwise.

Separation of Many Classes (g > 2)

The generalization of the within-class scatter matrix Sw for g classes is

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