Given an ill-conditioned square matrix, deletes rows/columns until a well-conditioned submatrix is obtained.
This function seeks to deal with ill-conditioned matrices, for which the search algorithms of optimal k-variable subsets could encounter numerical problems. Given a square matrix mat which is assumed positive semi-definite, the function checks whether it has reciprocal of the 2-norm condition number (i.e., the ratio of the smallest to the largest eigenvalue) smaller than tolval. If not, the matrix is considered well-conditioned and remains unchanged. If the ratio of the smallest to largest eigenvalue is smaller than tolval, an iterative process is begun, which deletes rows/columns (using Jolliffe's method for subset selections described on pg. 138 of the Reference below) until a principal submatrix with reciprocal of the condition number larger than tolval is obtained.
trim.matrix(mat,tolval=10*.Machine$double.eps)
mat |
a symmetric matrix, assumed positive semi-definite. |
tolval |
the tolerance value for the reciprocal condition number of matrix mat. |
For the given matrix mat, eigenvalues are computed. If the
ratio of the smallest to the largest eigenvalue is less than tolval, matrix mat remains unchanged and the function stops. Otherwise, an iterative process is begun, in which the eigenvector associated with the smallest eigenvalue is considered and its largest (in absolute value) element is identified. The corresponding row/column are deleted from matrix mat and the eigendecomposition of the resulting submatrix is computed. This iterative process stops when the ratio of the smallest to largest eigenvalue is not smaller than tolval.
The function checks whether the input matrix is square, but not whether it is positive semi-definite. This trim.matrix function can be used to delete rows/columns of square matrices, until only non-negative eigenvalues appear.
Output is a list with four items:
trimmedmat |
is a principal submatrix of the original matrix,
with the ratio of its smallest to largest eigenvalues no smaller than
|
numbers.discarded |
is a list of the integer numbers of the original variables that were discarded. |
names.discarded |
is a list of the original column numbers of the variables that were discarded. |
size |
is the size of the output matrix. |
When the trim.matrix function is used to produce a well-conditioned matrix for use with the anneal, genetic, improve or eleaps functions, care must be taken in interpreting the output of those functions. In those search functions, the selected variable subsets are specified by variable numbers, and those variable numbers indicate the position of the variables in the input matrix. Hence, if a trimmed matrix is supplied to functions anneal, genetic, improve or eleaps, variable numbers refer to the trimmed matrix.
Jolliffe, I.T. (2002) Principal Component Analysis, second edition, Springer Series in Statistics.
# a trivial example, for illustration of use: creating an extra column, # as the sum of columns in the "iris" data, and then using the function # trim.matrix to exclude it from the data's correlation matrix data(iris) lindepir<-cbind(apply(iris[,-5],1,sum),iris[,-5]) colnames(lindepir)[1]<-"Sum" cor(lindepir) ## Sum Sepal.Length Sepal.Width Petal.Length Petal.Width ##Sum 1.0000000 0.9409143 -0.2230928 0.9713793 0.9538850 ##Sepal.Length 0.9409143 1.0000000 -0.1175698 0.8717538 0.8179411 ##Sepal.Width -0.2230928 -0.1175698 1.0000000 -0.4284401 -0.3661259 ##Petal.Length 0.9713793 0.8717538 -0.4284401 1.0000000 0.9628654 ##Petal.Width 0.9538850 0.8179411 -0.3661259 0.9628654 1.0000000 trim.matrix(cor(lindepir)) ##$trimmedmat ## Sepal.Length Sepal.Width Petal.Length Petal.Width ##Sepal.Length 1.0000000 -0.1175698 0.8717538 0.8179411 ##Sepal.Width -0.1175698 1.0000000 -0.4284401 -0.3661259 ##Petal.Length 0.8717538 -0.4284401 1.0000000 0.9628654 ##Petal.Width 0.8179411 -0.3661259 0.9628654 1.0000000 ## ##$numbers.discarded ##[1] 1 ## ##$names.discarded ##[1] "Sum" ## ##$size ##[1] 4 data(swiss) lindepsw<-cbind(apply(swiss,1,sum),swiss) colnames(lindepsw)[1]<-"Sum" trim.matrix(cor(lindepsw)) ##$lowrankmat ## Fertility Agriculture examination Education Catholic ##Fertility 1.0000000 0.35307918 -0.6458827 -0.66378886 0.4636847 ##Agriculture 0.3530792 1.00000000 -0.6865422 -0.63952252 0.4010951 ##Examination -0.6458827 -0.68654221 1.0000000 0.69841530 -0.5727418 ##Education -0.6637889 -0.63952252 0.6984153 1.00000000 -0.1538589 ##Catholic 0.4636847 0.40109505 -0.5727418 -0.15385892 1.0000000 ##Infant.Mortality 0.4165560 -0.06085861 -0.1140216 -0.09932185 0.1754959 ## Infant.Mortality ##Fertility 0.41655603 ##Agriculture -0.06085861 ##Examination -0.11402160 ##Education -0.09932185 ##Catholic 0.17549591 ##Infant.Mortality 1.00000000 ## ##$numbers.discarded ##[1] 1 ## ##$names.discarded ##[1] "Sum" ## ##$size ##[1] 6
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