By A. J. Miller (auth.)

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**Example text**

For i = 1 to k - 1, set index(i) = i. Setp=k-1. Swap rows p and p + 1. Add 1 to index(p). U index(p)~k - 1, set index (p + 1) = index(p), add 1 to p, go to step 3. 5(b). Else, substract 1 from p, U p > 0, go to step 3. Else the end has been reached. A new subset is generated each time that two rows are swapped. Hence row i and (i + 1) are swapped (kC i - 1) times. Using the Hammarling algorithm, it requires 10 + 2(k - i) operations to perform the swapo This count comprises 8 operations to set up the rotation and calculate the new elements in columns i and (i + 1), 2 operations on the Q'Y-vector, 2 operations to calculate the residual sum of squares for the new subset of i variables, and 1 operation for each remaining element in rows i and (i + 1).

J' L... 2) If this expression is divided by L~= 1 y~, then we have the square of the eosine of the angle between veetors Xj and Y. If the mean has been subtraeted from eaeh variable, then the eosine is the eorrelation between variables X j and Y. 46 FINDING SUBSETS WHICH FIT WELL Let the first variable selected be denoted by X(l); this variable is then forced into all further subsets. (1)' The required sums of squares and products can be calculated directly from previous sums of squares and products without calculating these orthogonal components for each of the n observations, in fact the calculations are precisely those of a Gauss-Jordan pivoting out of the selected variable.

Set last = last - 1. Calculate new residual sums of squares for rows ipos to last. For i = p + 1 to k - 2, set nout(i) = nout(p) + 1. Simulate the deletion of variable number k - 1 which is in row last - 1. Go to step 3. As for the Garside algorithm, variable i is operated upon (2 i - 1) times, except that no calculations are required when i = k. In general, when variable number i is deleted, all of the higher-numbered variables are in the subset. Hence the variable must be rota ted past variables numbered i + 1, i + 2, ...