subselect (version 0.15.5)

trim.matrix: Given an ill-conditioned square matrix, deletes rows/columns until a well-conditioned submatrix is obtained.

Description

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.

Usage

trim.matrix(mat,tolval=10*.Machine$double.eps)

Value

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 tolval. This matrix can be used as input for the search algorithms in this package.

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.

Arguments

mat

a symmetric matrix, assumed positive semi-definite.

tolval

the tolerance value for the reciprocal condition number of matrix mat.

Details

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.

References

Jolliffe, I.T. (2002) Principal Component Analysis, second edition, Springer Series in Statistics.

Examples

# 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