Page 160 - Statistics and Data Analysis in Geology

Analysis of Multivariate Data

multivariate means of the two groups. In matrix notation, we must solve an equa-
tion of the form
SA = D (6.13)

where S is an m x m matrix of pooled variances and covariances of the m variables.
The coefficients of the discriminant equation are represented by a column vector of
the unknown lambdas. Lowercase lambdas (A) are used by convention to represent
the coefficients of the discriminant function. These are exactly the same as the
betas (p) used (also by convention) in regression equations. They should not be
confused with lambdas used to represent eigenvalues in principal component or
factor analyses.
The right-hand side of the equation consists of the column vector of m differ-
ences between the means of the two groups, which we will refer to as A and B. You
will recall from Chapter 3 that such an equation can be solved by inversion and
multiplication, as
A = S-lD (6.14)

where S-' is the inverse of the variance-covariance matrix formed by pooling the
matrices of the sums of squares and cross products of the two groups, A and B. To
compute the discriminant function, we must determine the various entries in the
matrix equation. The mean differences are found simply by

(6.15)

In this notation, aij is the ith observation on variable j in group A and Zj
is the mean of variable j in group A, which is the arithmetic average of the na
observations of variable j in group A. The same conventions apply to group B. The
multivariate means of groups A and B can be regarded as forming two vectors. The
difference between these multivariate means therefore also forms a vector

D=A-B
or, in expanded form,

To construct the matrix of pooled variances and covariances, we must compute
a matrix of sums of squares and cross products of all variables in group A and a
similar matrix for group B. For example, considering only group A,

Here, Uij denotes the ith observation of variable j in group A as before, and d.ik
denotes the ith Observation of variable k in the same group. Of course, this quantity
will be the sum of squares of variable k whenever j = k. Similarly, a matrix of sums
of squares and cross products can be found for group B:

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