The MAXQFORM subroutine computes the subsets of a matrix system that maximize the quadratic form.

If and are an matrix and an vector, respectively, then the MAXQFORM function computes the subsets of components such that is maximized.

The MAXQFORM subroutine returns the following values:

rc

is one of the following scalar return codes:

0	normal return
1	error: the number of elements of is too large to process
2	error: is not positive semidefinite

maxq

is an matrix, where is the total number of subsets computed and is the number of elements of b. The value of depends on the value of best and is equal to if best is not specified. Each row of maxq contains information for a selected subset of V and b. The first element of the row is the number of components in the subset. The second element is the value of the quadratic form. The following elements of the row are either 0 or 1, to indicate whether the corresponding components of V and b are included in the subset.

The input arguments to the MAXQFORM subroutine are as follows:

V

specifies an positive semidefinite matrix. Often this is generated as a crossproduct matrix, , where is a matrix.

b

specifies an vector. Often this arises as , where is a matrix, and is a vector.

best

specifies an optional scalar. If best is specified with the value , then the subsets with the largest value for the quadratic form are returned for each subset size.

The leaps and bounds algorithm by Furnival and Wilson (1974) computes the maximum value of quadratic forms for subsets of components. Many statistics computed as a quadratic form can then be used as the criterion for the method of subset selection. These include the regression sum of squares, Wald statistics, and score statistics.

Consider the following fitness data, which consists of observations with values for age measured in years, weight measured in kilograms, time to run 1.5 miles measured in minutes, heart rate while resting, heart rate while running, maximum heart rate recorded while running, and oxygen intake rate while running measured in milliliters per kilogram of body weight per minute.

fit = {
  44  89.47  11.37  62  178  182  44.609,
  40  75.07  10.07  62  185  185  45.313,
  44  85.84   8.65  45  156  168  54.297,
  42  68.15   8.17  40  166  172  59.571,
  38  89.02   9.22  55  178  180  49.874,
  47  77.45  11.63  58  176  176  44.811,
  40  75.98  11.95  70  176  180  45.681,
  43  81.19  10.85  64  162  170  49.091,
  44  81.42  13.08  63  174  176  39.442,
  38  81.87   8.63  48  170  186  60.055,
  44  73.03  10.13  45  168  168  50.541,
  45  87.66  14.03  56  186  192  37.388,
  45  66.45  11.12  51  176  176  44.754,
  47  79.15  10.60  47  162  164  47.273,
  54  83.12  10.33  50  166  170  51.855,
  49  81.42   8.95  44  180  185  49.156,
  51  69.63  10.95  57  168  172  40.836,
  51  77.91  10.00  48  162  168  46.672,
  48  91.63  10.25  48  162  164  46.774,
  49  73.37  10.08  67  168  168  50.388,
  57  73.37  12.63  58  174  176  39.407,
  54  79.38  11.17  62  156  165  46.080,
  52  76.32   9.63  48  164  166  45.441,
  50  70.87   8.92  48  146  155  54.625,
  51  67.25  11.08  48  172  172  45.118,
  54  91.63  12.88  44  168  172  39.203,
  51  73.71  10.47  59  186  188  45.790,
  57  59.08   9.93  49  148  155  50.545,
  49  76.32   9.40  56  186  188  48.673,
  48  61.24  11.50  52  170  176  47.920,
  52  82.78  10.50  53  170  172  47.467 };

Use the following statement to center the data:

fitc = fit - fit[:,];

Now compute the crossproduct matrices, as follows:

x = fitc[, 1:6];
y = fitc[, 7];
xpx = x`*x;
xpy = x`*y;

The following statements compute the best three regression sums of squares for each size of regressor set:

call maxqform(rc, maxq, xpx, xpy, 3);
print maxq;