20-Discriminant2

# discriminant functions a discriminant function

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Unformatted text preview: Discriminant Functions A discriminant function, also called a canonical root, is a latent variable which is created as a linear combination of discriminating (independent) variables, such that L = b1 x1 + b2 x2 + ... + bp xp + c, where the b's are discriminant coefﬁcients, the x's are discriminating variables, and c is a constant. This is similar to multiple regression, but the b's are discriminant coefﬁcients which maximize the distance between the means of the criterion (dependent) variable. Note that the foregoing assumes the discriminant function is estimated using ordinary least-squares, the traditional method, but there is also a version involving maximum likelihood estimation. . . . . . . Least Squares Method of estimation of Discriminant Functions The variance covariances matrix can be decomposed into two parts: one is the variance within each class and the other the variability between clases, or we can decompose the sum of squares and cross products (the same up to a constant factor) T = B+W T = X′ (In − P1n)X B = X′ (Pg − P1n)X between-class W = X′ (In − Pg)X within In is the identity matrix. P1n is the orthogonal projection in the space 1n . (i.e. P1n = 1n 1′ /n ). Such that (In − P1n)X is n the matrix of centered cases. Pg is the matrix projecting onto the subspace generated by the columns of the binary discriminating matrix G. This matrix has g columns and a one on row i and column j if observaton i belongs to group j a′ Ta = a′...
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