anneal {subselect}R Documentation

Simulated Annealing Search for an optimal k-variable subset

Description

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

Usage

anneal( mat, kmin, kmax = kmin, nsol = 1, niter = 1000, exclude
= NULL, include = NULL, improvement = TRUE, setseed = FALSE,
cooling = 0.05,  temp = 1, coolfreq = 1, criterion = "RM",
pcindices = "first_k", initialsol=NULL, force=FALSE, tolval=10*.Machine$double.eps)

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 initial/final subsets (runs of the algorithm).
niter the number of iterations of the algorithm for each initial subset.
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 in the subsets.
improvement a logical variable indicating whether or not the best final subset (for each cardinality) is to be passed as input to a local improvement algorithm (see function improve).
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.
cooling variable in the ]0,1[ interval indicating the rate of geometric cooling for the Simulated Annealing algorithm.
temp positive variable indicating the initial temperature for the Simulated Annealing algorithm.
coolfreq positive integer indicating the number of iterations of the algorithm between coolings of the temperature. By default, the temperature is cooled at every iteration.
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 simulated annealing search. If a single cardinality is required, initialsol may be a vector of length k, in which case it is used as the initial solution for all nsol final solutions that are requested; 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), in which case each row will be replicated to produced the initial solution for all nsol final solutions requested in each cardinality, 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.
force a logical variable indicating whether, for large data sets (currently p > 400) the algorithm should proceed anyways, regardless of possible memory problems which may crash the R session.
tolval the tolerance level for the reciprocal of the 2-norm condition number of the correlation/covariance matrix, i.e., for the ratio of the smallest to the largest eigenvalue of the input matrix. Matrices with a reciprocal of the condition number smaller than tolval will abort the search algorithm.

Details

An initial k-variable subset (for k ranging from kmin to kmax) of a full set of p variables is randomly selected and passed on to a Simulated Annealing algorithm. The algorithm then selects a random subset in the neighbourhood of the current subset (neighbourhood of a subset S being defined as the family of all k-variable subsets which differ from S by a single variable), and decides whether to replace the current subset according to the Simulated Annealing rule, i.e., either (i) always, if the alternative subset's value of the criterion is higher; or (ii) with probability exp((ac-cc)/t) if the alternative subset's value of the criterion (ac) is lower than that of the current solution (cc), where the parameter t (temperature) decreases throughout the iterations of the algorithm. For each cardinality k, the stopping criterion for the algorithm is the number of iterations (niter) which is controlled by the user. Also controlled by the user are the initial temperature (temp) the rate of geometric cooling of the temperature (cooling) and the frequency with which the temperature is cooled, as measured by coolfreq, the number of iterations after which the temperature is multiplied by 1-cooling.

Optionally, the best k-variable subset produced by Simulated Annealing may be passed as input to a restricted local search algorithm, for possible further improvement.

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 nsol x (niter + 1). These calls are the dominant computational effort in each iteration of the algorithm.

In order to improve computation times, the bulk of computations is carried out by a Fortran routine. Further details about the Simulated Annealing algorithm can be found in Reference 1 and in the comments to the Fortran code (in the src subdirectory for this package). For datasets with a very large number of variables (currently p > 400), it is necessary to set the force argument to TRUE for the function to run, but this may cause a session crash if there is not enough memory available.

The function checks for ill-conditioning of the input matrix (specifically, it checks whether the ratio of the input matrix's smallest and largest eigenvalues is less than tolval). For an ill-conditioned input matrix, execution is aborted. The function trim.matrix may be used to obtain a well-conditioned input matrix.

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) subsets obtained.
bestvalues A length(kmin:kmax) vector giving the best values of the criterion obtained for each cardinality. If improvement is TRUE, these values result from the final restricted local search algorithm (and may therefore exceed the largest value for that cardinality in values).
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. Computational Statistics & Data Analysis, 47, 225-236.

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, improve, leaps, trim.matrix.

Examples


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

data(swiss)
anneal(cor(swiss),2,3,nsol=4,niter=10,criterion="RM")

##$subsets
##, , Card.2
##
##           Var.1 Var.2 Var.3
##Solution 1     3     6     0
##Solution 2     4     5     0
##Solution 3     1     2     0
##Solution 4     3     6     0
##
##, , Card.3
##
##           Var.1 Var.2 Var.3
##Solution 1     4     5     6
##Solution 2     3     5     6
##Solution 3     3     4     6
##Solution 4     4     5     6
##
##
##$values
##              card.2    card.3
##Solution 1 0.8016409 0.9043760
##Solution 2 0.7982296 0.8769672
##Solution 3 0.7945390 0.8777509
##Solution 4 0.8016409 0.9043760
##
##$bestvalues
##   Card.2    Card.3 
##0.8016409 0.9043760 
##
##$bestsets
##       Var.1 Var.2 Var.3
##Card.2     3     6     0
##Card.3     4     5     6
##
##$call
##anneal(cor(swiss), 2, 3, nsol = 4, niter = 10, criterion = "RM")

#
#
# Excluding variable number 6 from the subsets.
# 

data(swiss)
anneal(cor(swiss),2,3,nsol=4,niter=10,criterion="RM",exclude=c(6))

##$subsets
##, , Card.2
##
##           Var.1 Var.2 Var.3
##Solution 1     4     5     0
##Solution 2     4     5     0
##Solution 3     4     5     0
##Solution 4     4     5     0
##
##, , Card.3
##
##           Var.1 Var.2 Var.3
##Solution 1     1     2     5
##Solution 2     1     2     5
##Solution 3     1     2     5
##Solution 4     1     4     5
##
##
##$values
##              card.2    card.3
##Solution 1 0.7982296 0.8791856
##Solution 2 0.7982296 0.8791856
##Solution 3 0.7982296 0.8791856
##Solution 4 0.7982296 0.8686515
##
##$bestvalues
##   Card.2    Card.3 
##0.7982296 0.8791856 
##
##$bestsets
##       Var.1 Var.2 Var.3
##Card.2     4     5     0
##Card.3     1     2     5
##
##$call
##anneal(cor(swiss), 2, 3, nsol = 4, niter = 10, criterion = "RM",
##     exclude=c(6))

# specifying initial solutions: using the subsets produced by
# simulated annealing for one criterion (RM, by default) as initial
# solutions for the simulated annealing search with a different criterion.

data(swiss)
rmresults<-anneal(cor(swiss),2,3,nsol=4,niter=10, setseed=TRUE)
anneal(cor(swiss),2,3,nsol=4,niter=10,criterion="gcd", initialsol=rmresults$subsets)

##$subsets
##, , Card.2
##
##           Var.1 Var.2 Var.3
##Solution 1     3     6     0
##Solution 2     3     6     0
##Solution 3     3     6     0
##Solution 4     3     6     0
##
##, , Card.3
##
##           Var.1 Var.2 Var.3
##Solution 1     4     5     6
##Solution 2     4     5     6
##Solution 3     3     4     6
##Solution 4     4     5     6
##
##
##$values
##              card.2   card.3
##Solution 1 0.8487026 0.925372
##Solution 2 0.8487026 0.925372
##Solution 3 0.8487026 0.798864
##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
##
##$call
##anneal(cor(swiss), 2, 3, nsol = 4, niter = 10, criterion = "gcd", 
##    initialsol = rmresults$subsets)

[Package subselect version 0.9-1 Index]