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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 leastsquares, 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 betweenclass
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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 Fall '14

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