The PredictiveLeastSquares command returns a list, P, and a vector, V that best satisfies the equation A[..,P] . x is approximately equal to B according to random trials using a subset of the data for fitting and the remaining data to test the goodness of the fit.

This command works best in situations where the problem is underspecified.  That is, the number of variables, or columns in A is of the same order of magnitude or less than the number of observations or rows of A and B.  The returned list, P contains the column index for the variables that have been tested to be most relevant, thus minimizing the effect of outliers and overfitting when using the model to predict new results.

A and B must contain numeric entries.  A is a m x n Matrix, and B is a m x 1 vector.

The options argument can contain one or more of the options shown below.  

numtrials= integer -- Specify how many random subsamples should be used to determine which variables to drop at each phase of regression.  After each sweep that causes one or more variables/columns to be removed, another phase consisting of the specified number of trials is performed.  The default is numtrials=15.

tolerance = realcons(nonnegative) -- Set the tolerance that determines whether a fit coefficient can be considered insignificant, and therefore should be removed. This is a relative tolerance, compared to the largest coefficient. The default is 1e-10.

samplesize= realcons(nonnegative) -- Provide the fraction of data that will be used for building the model. A setting of samplesize=.7, will cause snapshots using 70% of the data to be used for fitting, and the remaining 30% to be used for testing. This must be a number between 0 and 1.  The default is .55.

The underlying computation is done in floating-point; therefore, all data points must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.  For more information about numeric computation in the Statistics package, see the Statistics/Computation help page.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$A≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(100\,100\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\:$

$B≔\mathrm{Vector}\left(100\,i\mapsto A\left[i,1\right]+0.1\cdot A\left[i,2\right]+0.5\cdot A\left[i,3\right]-0.3\cdot A\left[i,4\right]\right)\:$

$p,\mathrm{LSP}≔\mathrm{PredictiveLeastSquares}\left(A\,B\right)$

$p,\mathrm{LSP}≔\left[1\,2\,3\,4\right],\left[\begin{array}{c}1.\\ 0.100000000000000\\ 0.500000000000000\\ \mathrm{-0.300000000000000}\end{array}\right]$

$\mathrm{numsamples}≔50\:$

$\mathrm{numvariables}≔100\:$

$Z≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples}\,10\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\:$

$A≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples}\,\mathrm{numvariables}\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\:$

$A\left[..,1..5\right]≔Z\left[..,1..5\right]\:$

$B≔\mathrm{Vector}\left(\mathrm{numsamples}\,i\mapsto \mathrm{add}\left(\frac{Z\left[i,j\right]}{j}\,j=1..10\right)\right)\:$

$p,\mathrm{LSP}≔\mathrm{PredictiveLeastSquares}\left(A\,B\right)\:$

$\mathrm{Correlation}\left(B\,A\left[..,p\right]·\mathrm{LSP}\right)$

$0.992800745975950$

$\mathrm{LS}≔\mathrm{LinearAlgebra}:-\mathrm{LeastSquares}\left(A\,B\right)\:$

$\mathrm{Correlation}\left(B\,A·\mathrm{LS}\right)$

$1.$

$\mathrm{Z2}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples}\,10\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\:$

$\mathrm{A2}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(\mathrm{numsamples}\,\mathrm{numvariables}\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\:$

$\mathrm{A2}\left[..,1..5\right]≔\mathrm{Z2}\left[..,1..5\right]\:$

$\mathrm{GuessLS}≔\mathrm{A2}·\mathrm{LS}\:$

$\mathrm{GuessLSP}≔\mathrm{A2}\left[..,p\right]·\mathrm{LSP}\:$

$\mathrm{Actual}≔\mathrm{Vector}\left(\mathrm{numsamples}\,i\mapsto \mathrm{add}\left(\frac{\mathrm{Z2}\left[i,j\right]}{j}\,j=1..10\right)\right)\:$

$\mathrm{Correlation}\left(\mathrm{GuessLS}\,\mathrm{Actual}\right)$

$0.727378412013161$

$\mathrm{Correlation}\left(\mathrm{GuessLSP}\,\mathrm{Actual}\right)$

$0.959754787645568$

The Statistics[PredictiveLeastSquares] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.