Computational Method

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The RSREG Procedure

Computational Method

Canonical Analysis

For each response variable, the model can be written in the form

$y_i = x_i^'A{x}_i + b^'x_i + c^'z_i + \epsilon_i$

where

y_i: is the ith observation of the response variable.
x_i: = (x_i1, x_i2, ... , x_ik)' are the k factor variables for the ith observation.
z_i: = (z_i1, z_i2, ... , z_iL)' are the L covariates, including the intercept term.
A: is the k ×k symmetrized matrix of quadratic parameters, with diagonal elements equal to the coefficients of the pure quadratic terms in the model and off-diagonal elements equal to half the coefficient of the corresponding cross product.
b: is the k ×1 vector of linear parameters.
c: is the L ×1 vector of covariate parameters, one of which is the intercept.
${\epsilon}_i$: is the error associated with the ith observation. Tests performed by PROC RSREG assume that errors are independently and normally distributed with mean zero and variance $\sigma^2$ .

The parameters in A, b, and c are estimated by least squares. To optimize y with respect to x, take partial derivatives, set them to zero, and solve:

$\frac{\partial y}{\partial x}=2{x}^'A + b^' = 0\Longrightarrow{x}=-\frac{1}2{A}^{-1}b$

You can determine if the solution is a maximum or minimum by looking at the eigenvalues of A:

If the eigenvalues...		then the solution is...
are all negative		a maximum
are all positive		a minimum
have mixed signs		a saddle point
contain zeros		in a flat area

Ridge Analysis

The eigenvector for the largest eigenvalue gives the direction of steepest ascent from the stationary point, if positive, or steepest descent, if negative. The eigenvectors corresponding to small or zero eigenvalues point in directions of relative flatness.

The point on the optimum response ridge at a given radius R from the ridge origin is found by optimizing

(x₀ + d)'A(x₀ + d) + b'(x₀ + d)

over d satisfying d'd = R², where x₀ is the k ×1 vector containing the ridge origin and A and b are as previously discussed. By the method of Lagrange multipliers, the optimal d has the form

$d = -(A - \mu I)^{-1}({Ax}_0 + 0.5 b)$

where I is the k ×k identity matrix and $\mu$ is chosen so that d'd = R². There may be several values of $\mu$ that satisfy this constraint; the right one depends on which sort of response ridge is of interest. If you are searching for the ridge of maximum response, then the appropriate $\mu$ is the unique one that satisfies the constraint and is greater than all the eigenvalues of A. Similarly, the appropriate $\mu$ for the ridge of minimum response satisfies the constraint and is less than all the eigenvalues of A. (Refer to Myers and Montgomery (1995) for details.)

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