anneal: 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 = "default",
pcindices = "first_k", initialsol=NULL, force=FALSE, H=NULL, r=0, 
tolval=1000*.Machine$double.eps,tolsym=1000*.Machine$double.eps)

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.

Arguments

mat

a covariance/correlation, information or sums of squares and products matrix of the variables from which the k-subset is to be selected. See the Details section below.

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, the "RM", "RV", "GCD", "Tau2", "Xi2", "Zeta2", "ccr12" and "Wald" criteria are supported (see the Details section, the References and the links rm.coef, rv.coef, gcd.coef, tau2.coef, xi2.coef, zeta2.coef and ccr12.coef for further details). The default criterion is "Rm" if parameter r is zero (exploratory and PCA problems), "Wald" if r is equal to one and mat has a "FisherI" attribute set to TRUE (generalized linear models), and "Tau2" otherwise (multivariate linear model framework).

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.

H

Effect description matrix. Not used with the RM, RV or GCD criteria, hence the NULL default value. See the Details section below.

r

Expected rank of the effects (H) matrix. Not used with the RM, RV or GCD criteria. See the Details section below.

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 activate a restricted-search for well conditioned subsets.

tolsym

the tolerance level for symmetry of the covariance/correlation/total matrix and for the effects (H) matrix. If corresponding matrix entries differ by more than this value, the input matrices will be considered asymmetric and execution will be aborted. If corresponding entries are different, but by less than this value, the input matrix will be replaced by its symmetric part, i.e., input matrix A becomes (A+t(A))/2.

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 $\exp^{\frac{ac-cc}{t}}$ 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, the search is restricted to its well-conditioned subsets. The function trim.matrix may be used to obtain a well-conditioned input matrix.

In a general descriptive (Principal Components Analysis) setting, the three criteria Rm, Rv and Gcd can be used to select good k-variable subsets. Arguments H and r are not used in this context. See references [1] and [2] and the Examples for a more detailed discussion.

In the setting of a multivariate linear model, $X = A \Psi + U$, criteria Ccr12, Tau2, Xi2 and Zeta2 can be used to select subsets according to their contribution to an effect characterized by the violation of a reference hypothesis, $C \Psi = 0$ (see reference [3] for further details). In this setting, arguments mat and H should be set respectively to the usual Total (Hypothesis + Error) and Hypothesis, Sum of Squares and Cross-Products (SSCP) matrices. Argument r should be set to the expected rank of H. Currently, for reasons of computational efficiency, criterion Ccr12 is available only when $\code{r} \leq 3$. Particular cases in this setting include Linear Discriminant Analyis (LDA), Linear Regression Analysis (LRA), Canonical Correlation Analysis (CCA) with one set of variables fixed and several extensions of these and other classical multivariate methodologies.

In the setting of a generalized linear model, criterion Wald can be used to select subsets according to the (lack of) significance of the discarded variables, as measured by the respective Wald's statistic (see reference [4] for further details). In this setting arguments mat and H should be set respectively to FI and FI %*% b %*% t(b) %*% FI, where b is a column vector of variable coefficient estimates and FI is an estimate of the corresponding Fisher information matrix.

The auxiliary functions lmHmat, ldaHmat glhHmat and glmHmat are provided to automatically create the matrices mat and H in all the cases considered.

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 and 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.

[3] Duarte Silva, A.P. (2001) Efficient Variable Screening for Multivariate Analysis, Journal of Multivariate Analysis, Vol. 76, 35-62.

[4] Lawless, J. and Singhal, K. (1978). Efficient Screening of Nonnormal Regression Models, Biometrics, Vol. 34, 318-327.

Examples

Run this code


## --------------------------------------------------------------------

##
## (1) For illustration of use, a small data set with very few iterations
## of the algorithm, using the RM criterion.
##

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")

## --------------------------------------------------------------------

##
## (2) An example 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))

## --------------------------------------------------------------------

## (3) An example 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)

## --------------------------------------------------------------------

## (4) An example of subset selection in the context of Multiple Linear
## Regression. Variable 5 (average car price) in the Cars93 MASS library 
## data set is regressed on 13 other variables.  A best subset of linear 
## predictors is sought, using the "TAU_2" criterion which, in the case 
## of a Linear Regression, is merely  the standard Coefficient of Determination, 
## R^2 (like the other three criteria for the multivariate linear hypothesis, 
## "XI_2", "CCR1_2" and "ZETA_2"). 

library(MASS)
data(Cars93)
CarsHmat <- lmHmat(Cars93[,c(7:8,12:15,17:22,25)],Cars93[,5])

names(Cars93[,5,drop=FALSE])
##  [1] "Price"

colnames(CarsHmat$mat)

##  [1] "MPG.city"           "MPG.highway"        "EngineSize"        
##  [4] "Horsepower"         "RPM"                "Rev.per.mile"      
##  [7] "Fuel.tank.capacity" "Passengers"         "Length"            
## [10] "Wheelbase"          "Width"              "Turn.circle"       
## [13] "Weight"            

anneal(CarsHmat$mat, kmin=4, kmax=6, H=CarsHmat$H, r=1, crit="tau2")

## $subsets
## , , Card.4
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     4     5    10    11     0     0
## 
## , , Card.5
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     4     5    10    11    12     0
## 
## , , Card.6
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     4     5     9    10    11    12
## 
## 
## $values
##               card.4    card.5   card.6
## Solution 1 0.7143794 0.7241457 0.731015
## 
## $bestvalues
##    Card.4    Card.5    Card.6 
## 0.7143794 0.7241457 0.7310150 
## 
## $bestsets
##        Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Card.4     4     5    10    11     0     0
## Card.5     4     5    10    11    12     0
## Card.6     4     5     9    10    11    12
## 
## $call
## anneal(mat = CarsHmat$mat, kmin = 4, kmax = 6, criterion = "xi2", 
##     H = CarsHmat$H, r = 1)
##

## --------------------------------------------------------------------

## (5) A Linear Discriminant Analysis example with a very small data set. 
## We consider the Iris data and three groups, defined by species (setosa, 
## versicolor and virginica). The goal is to select the 2- and 3-variable
## subsets that are optimal for the linear discrimination (as measured 
## by the "CCR1_2" criterion).

 
data(iris)
irisHmat <- ldaHmat(iris[1:4],iris$Species)
anneal(irisHmat$mat,kmin=2,kmax=3,H=irisHmat$H,r=2,crit="ccr12")

## $subsets
## , , Card.2
## 
##            Var.1 Var.2 Var.3
## Solution 1     1     3     0
## 
## , , Card.3
## 
##            Var.1 Var.2 Var.3
## Solution 1     2     3     4
## 
## 
## $values
##               card.2   card.3
## Solution 1 0.9589055 0.967897
## 
## $bestvalues
##    Card.2    Card.3 
## 0.9589055 0.9678971 
## 
## $bestsets
##        Var.1 Var.2 Var.3
## Card.2     1     3     0
## Card.3     2     3     4
## 
## $call
## anneal(irisHmat$mat,kmin=2,kmax=3,H=irisHmat$H,r=2,crit="ccr12")
## 

## --------------------------------------------------------------------

## (6) An example of subset selection in the context of a Canonical
## Correlation Analysis. Two groups of variables within the Cars93
## MASS library data set are compared. The goal is to select 4- to
## 6-variable subsets of the 13-variable 'X' group that are optimal in
## terms of preserving the canonical correlations, according to the
## "XI_2" criterion (Warning: the 3-variable 'Y' group is kept
## intact; subset selection is carried out in the 'X' 
## group only). The 'tolsym' parameter is used to relax the symmetry
## requirements on the effect matrix H which, for numerical reasons,
## is slightly asymmetric. Since corresponding off-diagonal entries of
## matrix H are different, but by less than tolsym, H is replaced  
## by its symmetric part: (H+t(H))/2.

library(MASS)
data(Cars93)
CarsHmat <- lmHmat(Cars93[,c(7:8,12:15,17:22,25)],Cars93[,4:6])

names(Cars93[,4:6])
## [1] "Min.Price" "Price"     "Max.Price"

colnames(CarsHmat$mat)

##  [1] "MPG.city"           "MPG.highway"        "EngineSize"        
##  [4] "Horsepower"         "RPM"                "Rev.per.mile"      
##  [7] "Fuel.tank.capacity" "Passengers"         "Length"            
## [10] "Wheelbase"          "Width"              "Turn.circle"       
## [13] "Weight"            

anneal(CarsHmat$mat, kmin=4, kmax=6, H=CarsHmat$H, r=CarsHmat$r,
crit="tau2" , tolsym=1e-9)

## $subsets
## , , Card.4
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     4     9    10    11     0     0
## 
## , , Card.5
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     3     4     9    10    11     0
## 
## , , Card.6
## 
##            Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Solution 1     3     4     5     9    10    11
## 
## 
## $values
##               card.4    card.5    card.6
## Solution 1 0.2818772 0.2943742 0.3057831
## 
## $bestvalues
##    Card.4    Card.5    Card.6 
## 0.2818772 0.2943742 0.3057831 
## 
## $bestsets
##        Var.1 Var.2 Var.3 Var.4 Var.5 Var.6
## Card.4     4     9    10    11     0     0
## Card.5     3     4     9    10    11     0
## Card.6     3     4     5     9    10    11
## 
## $call
## anneal(mat = CarsHmat$mat, kmin = 4, kmax = 6, criterion = "xi2", 
##     H = CarsHmat$H, r = CarsHmat$r, tolsym = 1e-09)
## 
## Warning message:
## 
##  The effect description matrix (H) supplied was slightly asymmetric: 
##  symmetric entries differed by up to 3.63797880709171e-12.
##  (less than the 'tolsym' parameter).
##  The H matrix has been replaced by its symmetric part.
##  in: validnovcrit(mat, criterion, H, r, p, tolval, tolsym) 

## --------------------------------------------------------------------

##  (7) An example of variable selection in the context of a logistic 
##  regression model. We consider the last 100 observations of
##  the iris data set (versicolor and verginica species) and try
##  to find the best variable subsets for the model that takes species
##  as response variable.

data(iris)
iris2sp <- iris[iris$Species != "setosa",]
logrfit <- glm(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width,
iris2sp,family=binomial)
Hmat <- glmHmat(logrfit)
anneal(Hmat$mat,1,3,H=Hmat$H,r=1,criterion="Wald")

## $subsets
## , , Card.1
##
##           Var.1 Var.2 Var.3
## Solution 1     4     0     0

## , , Card.2

##            Var.1 Var.2 Var.3
## Solution 1     1     3     0

## , , Card.3

##            Var.1 Var.2 Var.3
## Solution 1     2     3     4


## $values
##              card.1   card.2   card.3
## Solution 1 4.894554 3.522885 1.060121

## $bestvalues
##   Card.1   Card.2   Card.3 
## 4.894554 3.522885 1.060121 

## $bestsets
##        Var.1 Var.2 Var.3
## Card.1     4     0     0
## Card.2     1     3     0
## Card.3     2     3     4

## $call
## anneal(mat = Hmat$mat, kmin = 1, kmax = 3, criterion = "Wald", 
##     H = Hmat$H, r = 1)
## --------------------------------------------------------------------

## It should be stressed that, unlike other criteria in the
## subselect package, the Wald criterion is not bounded above by
## 1 and is a decreasing function of subset quality, so that the
## 3-variable subsets do, in fact, perform better than their smaller-sized
## counterparts.

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