improve {subselect}R Documentation

Restricted Local Improvement search for an optimal k-variable subset

Description

Given a set of variables, a Restricted Local Improvement algorithm seeks a k-variable subset which is optimal, as a surrogate for the whole set, with respect to a given criterion.

Usage

improve( mat, kmin, kmax = kmin, nsol = 1, exclude = NULL,
include = NULL, setseed = FALSE, criterion = "RM", pcindices="first_k", initialsol=NULL)

Arguments

mat a covariance or correlation matrix of the variables from which the k-subset is to be selected.
kmin the cardinality of the smallest subset that is wanted.
kmax the cardinality of the largest subset that is wanted.
nsol the number of different subsets (runs of the algorithm) wanted.
exclude a vector of variables (referenced by their row/column numbers in matrix mat) that are to be forcibly excluded from the subsets.
include a vector of variables (referenced by their row/column numbers in matrix mat) that are to be forcibly included from the subsets.
setseed logical variable indicating whether to fix an initial seed for the random number generator, which will be re-used in future calls to this function whenever setseed is again set to TRUE.
criterion Character variable, which indicates which criterion is to be used in judging the quality of the subsets. Currently, only the RM, RV and GCD criteria are supported, and referenced as "RM", "RV" or "GCD" (see References, rm.coef, rv.coef and gcd.coef for further details).
pcindices either a vector of ranks of Principal Components that are to be used for comparison with the k-variable subsets (for the GCD criterion only, see gcd.coef) or the default text first_k. The latter will associate PCs 1 to k with each cardinality k that has been requested by the user.
initialsol vector, matrix or 3-d array of initial solutions for the restricted local improvement search. If a single cardinality is required, initialsol may be a vector of length k(accepted even if nsol > 1, in which case it is used as the initial solution for all nsol final solutions that are requested with a warning that the same initial solution necessarily produces the same final solution); a 1 x k matrix (as produced by the $bestsets output value of the algorithm functions anneal, genetic, or improve, or a 1 x k x 1 array (as produced by the $subsets output value), in which case it will be treated as the above k-vector; or an nsol x k matrix, or nsol x k x 1 3-d array, in which case each row (dimension 1) will be used as the initial solution for each of the nsol final solutions requested. If more than one cardinality is requested, initialsol can be a length(kmin:kmax) x kmax matrix (as produced by the $bestsets option of the algorithm functions) (even if nsol > 1, in which case each row will be replicated to produced the initial solution for all nsol final solutions requested in each cardinality, with a warning that a single initial solution necessarily produces identical final solutions), or a nsol x kmax x length(kmin:kmax) 3-d array (as produced by the $subsets output option), in which case each row (dimension 1) is interpreted as a different initial solution.
If the exclude and/or include options are used, initialsol must also respect those requirements.

Details

An initial k-variable subset (for k ranging from kmin to kmax) of a full set of p (p not exceeding 300) variables is randomly selected and the variables not belonging to this subset are placed in a queue. The possibility of replacing a variable in the current k-subset with a variable from the queue is then explored. More precisely, a variable is selected, removed from the queue, and the k values of the criterion which would result from swapping this selected variable with each variable in the current subset are computed. If the best of these values improves the current criterion value, the current subset is updated accordingly. In this case, the variable which leaves the subset is added to the queue, but only if it has not previously been in the queue (i.e., no variable can enter the queue twice). The algorithm proceeds until the queue is emptied.

The user may force variables to be included and/or excluded from the k-subsets, and may specify initial solutions.

For each cardinality k, the total number of calls to the procedure which computes the criterion values is O(nsol x k x p). These calls are the dominant computational effort in each iteration of the algorithm.

In order to improve computation times, the bulk of computations are carried out in a Fortran routine. Further details about the algorithm can be found in Reference 1 and in the comments to the Fortran code (in the src subdirectory for this package). For p > 300, it is necessary to change the declarative statements in the Fortran code.

Value

A list with five items:

subsets An nsol x kmax x length(kmin:kmax) 3-dimensional array, giving for each cardinality (dimension 3) and each solution (dimension 1) the list of variables (referenced by their row/column numbers in matrix mat) in the subset (dimension 2). (For cardinalities smaller than kmax, the extra final positions are set to zero).
values An nsol x length(kmin:kmax) matrix, giving for each cardinality (columns), the criterion values of the nsol (rows) solutions obtained.
bestvalues A length(kmin:kmax) vector giving the best values of the criterion obtained for each cardinality.
bestsets A length(kmin:kmax) x kmax matrix, giving, for each cardinality (rows), the variables (referenced by their row/column numbers in matrix mat) in the best k-subset that was found.
call The function call which generated the output.

References

1) Cadima, J., Cerdeira, J. Orestes and Minhoto, M. (2004) Computational aspects of algorithms for variable selection in the context of principal components. Accepted for publication in Computational Statistics & Data Analysis.

2) Cadima, J. and Jolliffe, I.T. (2001). Variable Selection and the Interpretation of Principal Subspaces, Journal of Agricultural, Biological and Environmental Statistics, Vol. 6, 62-79.

See Also

rm.coef, rv.coef, gcd.coef, genetic, anneal.

Examples

# For illustration of use, a small data set with very few iterations
# of the algorithm. 
#

data(swiss)
improve(cor(swiss),2,3,nsol=4,criterion="GCD")
## $subsets
## , , Card.2
##
##            Var.1 Var.2 Var.3
## Solution 1     3     6     0
## Solution 2     3     6     0
## Solution 3     3     6     0
## Solution 4     3     6     0
##
## , , Card.3
##
##            Var.1 Var.2 Var.3
## Solution 1     4     5     6
## Solution 2     4     5     6
## Solution 3     4     5     6
## Solution 4     4     5     6
##
##
## $values
##               card.2   card.3
## Solution 1 0.8487026 0.925372
## Solution 2 0.8487026 0.925372
## Solution 3 0.8487026 0.925372
## Solution 4 0.8487026 0.925372
##
## $bestvalues
##    Card.2    Card.3 
## 0.8487026 0.9253720 
##
## $bestsets
##        Var.1 Var.2 Var.3
## Card.2     3     6     0
## Card.3     4     5     6
##
##$call
##improve(cor(swiss), 2, 3, nsol = 4, criterion = "GCD")

#
#
# Forcing the inclusion of variable 1 in the subset
#

 improve(cor(swiss),2,3,nsol=4,criterion="GCD",include=c(1))

## $subsets
## , , Card.2
##
##            Var.1 Var.2 Var.3
## Solution 1     1     6     0
## Solution 2     1     6     0
## Solution 3     1     6     0
## Solution 4     1     6     0
##
## , , Card.3
##
##            Var.1 Var.2 Var.3
## Solution 1     1     5     6
## Solution 2     1     5     6
## Solution 3     1     5     6
## Solution 4     1     5     6
##
##
## $values
##               card.2    card.3
## Solution 1 0.7284477 0.8048528
## Solution 2 0.7284477 0.8048528
## Solution 3 0.7284477 0.8048528
## Solution 4 0.7284477 0.8048528
##
## $bestvalues
##    Card.2    Card.3 
## 0.7284477 0.8048528 
##
## $bestsets
##        Var.1 Var.2 Var.3
## Card.2     1     6     0
## Card.3     1     5     6
##
##$call
##improve(cor(swiss), 2, 3, nsol = 4, criterion = "GCD", include = c(1))

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