Restricted Local Improvement search for an optimal k-variable subset
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.
improve( mat, kmin, kmax = kmin, nsol = 1, exclude = NULL, include = NULL, setseed = FALSE, criterion = "default", pcindices="first_k", initialsol = NULL, force = FALSE, H=NULL, r=0, tolval=1000*.Machine$double.eps,tolsym=1000*.Machine$double.eps)
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 |
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 |
include |
a vector of variables (referenced by their row/column
numbers in matrix |
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,
the "Rm", "Rv", "Gcd", "Tau2", "Xi2", "Zeta2", "ccr12" and "Wald" criteria
are supported (see the |
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 |
initialsol |
vector, matrix or 3-d array of initial solutions
for the restricted local improvement search. If a single
cardinality is
required, If the |
force |
a logical variable indicating whether, for large data
sets (currently |
H |
Effect description matrix. Not used with the Rm, Rv or Gcd
criteria, hence the NULL default value. See the |
r |
Expected rank of the effects ( |
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
|
tolsym |
the tolerance level for symmetry of the
covariance/correlation/total matrix and for the effects ( |
An initial k-variable subset (for k ranging from kmin
to kmax) of a full set of p 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 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 B + 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, CB = 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}<=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.
A list with five items:
subsets |
An |
values |
An |
bestvalues |
A length( |
bestsets |
A length( |
call |
The function call which generated the output. |
[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.
[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.
rm.coef, rv.coef,
gcd.coef, tau2.coef, xi2.coef,
zeta2.coef, ccr12.coef, genetic,
anneal, eleaps, trim.matrix,
lmHmat, ldaHmat, glhHmat,
glmHmat.
## -------------------------------------------------------------------- ## ## 1) For illustration of use, a small data set with very few iterations ## of the algorithm. ## Subsets of 2 and of 3 variables are sought using the RM criterion. ## 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") ## -------------------------------------------------------------------- ## ## 2) 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)) ## -------------------------------------------------------------------- ## 3) 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. Three variable subsets of ## cardinalities 4, 5 and 6 are requested, using the "XI_2" criterion which, ## in the case of a Linear Regression, is merely the standard Coefficient of ## Determination, R^2 (as are the other three criteria for the ## multivariate linear hypothesis, "TAU_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$mat) ## [1] "MPG.city" "MPG.highway" "EngineSize" ## [4] "Horsepower" "RPM" "Rev.per.mile" ## [7] "Fuel.tank.capacity" "Passengers" "Length" ## [10] "Wheelbase" "Width" "Turn.circle" ## [13] "Weight" improve(CarsHmat$mat, kmin=4, kmax=6, H=CarsHmat$H, r=1, crit="xi2", nsol=3) ## $subsets ## , , Card.4 ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 3 4 11 13 0 0 ## Solution 2 3 4 11 13 0 0 ## Solution 3 4 5 10 11 0 0 ## ## , , Card.5 ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 3 4 8 11 13 0 ## Solution 2 4 5 10 11 12 0 ## Solution 3 4 5 10 11 12 0 ## ## , , Card.6 ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 4 5 6 10 11 12 ## Solution 2 4 5 8 10 11 12 ## Solution 3 4 5 9 10 11 12 ## ## ## $values ## card.4 card.5 card.6 ## Solution 1 0.6880773 0.6899182 0.7270257 ## Solution 2 0.6880773 0.7241457 0.7271056 ## Solution 3 0.7143794 0.7241457 0.7310150 ## ## $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 ## ## $call ## improve(mat = CarsHmat$mat, kmin = 4, kmax = 6, nsol = 3, criterion = "xi2", ## H = CarsHmat$H, r = 1) ## -------------------------------------------------------------------- ## 4) A Linear Discriminant Analysis example with a very small data set. ## We consider the Iris data and three groups, defined by species (setosa, ## versicolor and virginica). The goal is to select the 2- and 3-variable ## subsets that are optimal for the linear discrimination (as measured ## by the "TAU_2" criterion). data(iris) irisHmat <- ldaHmat(iris[1:4],iris$Species) improve(irisHmat$mat,kmin=2,kmax=3,H=irisHmat$H,r=2,crit="ccr12") ## $subsets ## , , Card.2 ## ## Var.1 Var.2 Var.3 ## Solution 1 2 3 0 ## ## , , Card.3 ## ## Var.1 Var.2 Var.3 ## Solution 1 2 3 4 ## ## ## $values ## card.2 card.3 ## Solution 1 0.8079476 0.8419635 ## ## $bestvalues ## Card.2 Card.3 ## 0.8079476 0.8419635 ## ## $bestsets ## Var.1 Var.2 Var.3 ## Card.2 2 3 0 ## Card.3 2 3 4 ## ## $call ## improve(mat = irisHmat$mat, kmin = 2, kmax = 3, ## criterion = "tau2", H = irisHmat$H, r = 2) ## ## -------------------------------------------------------------------- ## 5) An example of subset selection in the context of a Canonical ## Correlation Analysis. Two groups of variables within the Cars93 ## MASS library data set are compared. The goal is to select 4- to ## 6-variable subsets of the 13-variable 'X' group that are optimal in ## terms of preserving the canonical correlations, according to the ## "ZETA_2" criterion (Warning: the 3-variable 'Y' group is kept ## intact; subset selection is carried out in the 'X' ## group only). The 'tolsym' parameter is used to relax the symmetry ## requirements on the effect matrix H which, for numerical reasons, ## is slightly asymmetric. Since corresponding off-diagonal entries of ## matrix H are different, but by less than tolsym, H is replaced ## by its symmetric part: (H+t(H))/2. library(MASS) data(Cars93) CarsHmat <- lmHmat(Cars93[,c(7:8,12:15,17:22,25)],Cars93[,4:6]) names(Cars93[,4:6]) ## [1] "Min.Price" "Price" "Max.Price" colnames(CarsHmat$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" improve(CarsHmat$mat, kmin=4, kmax=6, H=CarsHmat$H, r=3, crit="zeta2", tolsym=1e-9) ## $subsets ## , , Card.4 ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 3 4 11 13 0 0 ## ## , , Card.5 ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 3 4 9 11 13 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 0.4626035 0.4875495 0.5071096 ## ## $bestvalues ## Card.4 Card.5 Card.6 ## 0.4626035 0.4875495 0.5071096 ## ## $bestsets ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Card.4 3 4 11 13 0 0 ## Card.5 3 4 9 11 13 0 ## Card.6 3 4 5 9 10 11 ## ## $call ## improve(mat = CarsHmat$mat, kmin = 4, kmax = 6, criterion = "zeta2", ## H = CarsHmat$H, r = 3, tolsym = 1e-09) ## ## Warning message: ## ## The effect description matrix (H) supplied was slightly asymmetric: ## symmetric entries differed by up to 3.63797880709171e-12. ## (less than the 'tolsym' parameter). ## The H matrix has been replaced by its symmetric part. ## in: validnovcrit(mat, criterion, H, r, p, tolval, tolsym) ## -------------------------------------------------------------------- ## 6) An example of variable selection in the context of a logistic ## regression model. We consider the last 100 observations of ## the iris data set (versicolor and verginica species) and try ## to find the best variable subsets for the model that takes species ## as response variable. data(iris) iris2sp <- iris[iris$Species != "setosa",] logrfit <- glm(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, iris2sp,family=binomial) Hmat <- glmHmat(logrfit) improve(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 ## $call ## improve(mat = Hmat$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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