Genetic Algorithm searching for an optimal k-variable subset
Given a set of variables, a Genetic Algorithm algorithm seeks a k-variable subset which is optimal, as a surrogate for the whole set, with respect to a given criterion.
genetic( mat, kmin, kmax = kmin, popsize = max(100,2*ncol(mat)), nger = 100, mutate = FALSE, mutprob = 0.01, maxclone = 5, exclude = NULL, include = NULL, improvement = TRUE, setseed= FALSE, criterion = "default", pcindices = "first_k", initialpop = 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. |
popsize |
integer variable indicating the size of the population. |
nger |
integer variable giving the number of generations for which the genetic algorithm will run. |
mutate |
logical variable indicating whether each child
undergoes a mutation, with probability |
mutprob |
variable giving the probability of each child
undergoing a mutation, if |
maxclone |
integer variable specifying the maximum number of identical replicates (clones) of individuals that is acceptable in the population. Serves to ensure that the population has sufficient genetic diversity, which is necessary to enable the algorithm to complete the specified number of generations. However, even maxclone=0 does not guarantee that there are no repetitions: only the offspring of couples are tested for clones. If any such clones are rejected, they are replaced by a k-variable subset chosen at random, without any further clone tests. |
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 |
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 |
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 |
initialpop |
vector, matrix or 3-d array of initial population
for the genetic algorithm. 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 ( |
For each cardinality k (with k ranging from kmin to kmax),
an initial population of popsize k-variable subsets is randomly
selected from a full set of p variables.
In each iteration, popsize/2 couples
are formed from among the population and each couple generates a child
(a new k-variable subset)
which inherits properties of its parents (specifically, it inherits
all variables common to both parents and a random selection of
variables in the symmetric difference of its parents' genetic makeup).
Each offspring may optionally undergo a mutation (in the form of a
local improvement algorithm – see function improve),
with a user-specified probability. The parents
and offspring are ranked according to their criterion value, and the
best popsize of these k-subsets will make up the next
generation, which is used as the current population in the subsequent
iteration.
The stopping rule for the algorithm is the number of generations (nger).
Optionally, the best k-variable subset produced by the Genetic
Algorithm may be passed as input to a restricted local improvement
algorithm, for possible further improvement (see function
improve).
The user may force variables to be included and/or excluded from the k-subsets, and may specify an initial population.
For each cardinality k, the total number of calls to the procedure which computes the criterion values is popsize + nger x popsize/2. 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 by a Fortran routine. Further details about the Genetic
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 |
A |
values |
A |
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. Escoufier's 'RV' criterion is used to select variable ## subsets of size 3 and 4. ## data(swiss) genetic(cor(swiss),3,4,popsize=10,nger=5,criterion="Rv") ## For cardinality k= ##[1] 4 ## there is not enough genetic diversity in generation number ##[1] 3 ## for acceptable levels of consanguinity (couples differing by at least 2 genes). ## Try reducing the maximum acceptable number of clones (maxclone) or ## increasing the population size (popsize) ## Best criterion value found so far: ##[1] 0.9557145 ##$subsets ##, , Card.3 ## ## Var.1 Var.2 Var.3 Var.4 ##Solution 1 1 2 3 0 ##Solution 2 1 2 3 0 ##Solution 3 1 2 3 0 ##Solution 4 3 4 6 0 ##Solution 5 3 4 6 0 ##Solution 6 3 4 5 0 ##Solution 7 3 4 5 0 ##Solution 8 1 3 6 0 ##Solution 9 1 3 6 0 ##Solution 10 1 3 6 0 ## ##, , Card.4 ## ## Var.1 Var.2 Var.3 Var.4 ##Solution 1 2 4 5 6 ##Solution 2 1 2 5 6 ##Solution 3 1 2 3 5 ##Solution 4 1 2 4 5 ##Solution 5 1 2 4 5 ##Solution 6 1 4 5 6 ##Solution 7 1 4 5 6 ##Solution 8 1 4 5 6 ##Solution 9 1 3 4 5 ##Solution 10 1 3 4 5 ## ## ##$values ## card.3 card.4 ##Solution 1 0.9141995 0.9557145 ##Solution 2 0.9141995 0.9485699 ##Solution 3 0.9141995 0.9455508 ##Solution 4 0.9034868 0.9433203 ##Solution 5 0.9034868 0.9433203 ##Solution 6 0.9020271 0.9428967 ##Solution 7 0.9020271 0.9428967 ##Solution 8 0.8988192 0.9428967 ##Solution 9 0.8988192 0.9357982 ##Solution 10 0.8988192 0.9357982 ## ##$bestvalues ## Card.3 Card.4 ##0.9141995 0.9557145 ## ##$bestsets ## Var.1 Var.2 Var.3 Var.4 ##Card.3 1 2 3 0 ##Card.4 2 4 5 6 ## ##$call ##genetic(mat = cor(swiss), kmin = 3, kmax = 4, popsize = 10, nger = 5, ## criterion = "Rv") ## -------------------------------------------------------------------- ## ## 2) An example of subset selection in the context of Multiple Linear ## Regression. Variable 5 (average car price) in the Cars93 MASS library ## data set is regressed on 13 other variables. The six-variable subsets ## of linear predictors are chosen using the "CCR1_2" criterion which, ## in the case of a Linear Regression, is merely the standard Coefficient ## of Determination, R^2 (as are the other three criteria for the ## multivariate linear hypothesis, "XI_2", "TAU_2" and "ZETA_2"). ## library(MASS) data(Cars93) CarsHmat <- lmHmat(Cars93[,c(7:8,12:15,17:22,25)],Cars93[,5]) names(Cars93[,5,drop=FALSE]) ## [1] "Price" colnames(CarsHmat) ## [1] "MPG.city" "MPG.highway" "EngineSize" ## [4] "Horsepower" "RPM" "Rev.per.mile" ## [7] "Fuel.tank.capacity" "Passengers" "Length" ## [10] "Wheelbase" "Width" "Turn.circle" ## [13] "Weight" genetic(CarsHmat$mat, kmin=6, H=CarsHmat$H, r=1, crit="CCR12") ## ## (Partial results only) ## ## $subsets ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 4 5 9 10 11 12 ## Solution 2 4 5 9 10 11 12 ## Solution 3 4 5 9 10 11 12 ## Solution 4 4 5 9 10 11 12 ## Solution 5 4 5 9 10 11 12 ## Solution 6 4 5 9 10 11 12 ## Solution 7 4 5 8 10 11 12 ## ## (...) ## ## Solution 94 1 4 5 6 10 11 ## Solution 95 1 4 5 6 10 11 ## Solution 96 1 4 5 6 10 11 ## Solution 97 1 4 5 6 10 11 ## Solution 98 1 4 5 6 10 11 ## Solution 99 1 4 5 6 10 11 ## Solution 100 1 4 5 6 10 11 ## ## $values ## Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 ## 0.7310150 0.7310150 0.7310150 0.7310150 0.7310150 0.7310150 ## Solution 7 Solution 8 Solution 9 Solution 10 Solution 11 Solution 12 ## 0.7310150 0.7271056 0.7271056 0.7271056 0.7271056 0.7271056 ## Solution 13 Solution 14 Solution 15 Solution 16 Solution 17 Solution 18 ## 0.7271056 0.7270257 0.7270257 0.7270257 0.7270257 0.7270257 ## ## (...) ## ## Solution 85 Solution 86 Solution 87 Solution 88 Solution 89 Solution 90 ## 0.7228800 0.7228800 0.7228800 0.7228800 0.7228800 0.7228800 ## Solution 91 Solution 92 Solution 93 Solution 94 Solution 95 Solution 96 ## 0.7228463 0.7228463 0.7228463 0.7228463 0.7228463 0.7228463 ## Solution 97 Solution 98 Solution 99 Solution 100 ## 0.7228463 0.7228463 0.7228463 0.7228463 ## ## $bestvalues ## Card.6 ## 0.731015 ## ## $bestsets ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## 4 5 9 10 11 12 ## ## $call ## genetic(mat = CarsHmat$mat, kmin = 6, criterion = "CCR12", H = CarsHmat$H, ## r = 1) ## -------------------------------------------------------------------- ## 3) An example of subset selection in the context of a Canonical ## Correlation Analysis. Two groups of variables within the Cars93 ## MASS library data set are compared. The goal is to select 4- to ## 6-variable subsets of the 13-variable 'X' group that are optimal in ## terms of preserving the canonical correlations, according to the ## "ZETA_2" criterion (Warning: the 3-variable 'Y' group is kept ## intact; subset selection is carried out in the 'X' ## group only). The 'tolsym' parameter is used to relax the symmetry ## requirements on the effect matrix H which, for numerical reasons, ## is slightly asymmetric. Since corresponding off-diagonal entries of ## matrix H are different, but by less than tolsym, H is replaced ## by its symmetric part: (H+t(H))/2. library(MASS) data(Cars93) CarsHmat <- lmHmat(Cars93[,c(7:8,12:15,17:22,25)],Cars93[,4:6]) names(Cars93[,4:6]) ## [1] "Min.Price" "Price" "Max.Price" colnames(CarsHmat$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" genetic(CarsHmat$mat, kmin=5, kmax=6, H=CarsHmat$H, r=3, crit="zeta2", tolsym=1e-9) ## (PARTIAL RESULTS ONLY) ## ## $subsets ## ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Solution 1 4 5 9 10 11 0 ## Solution 2 4 5 9 10 11 0 ## Solution 3 4 5 9 10 11 0 ## Solution 4 4 5 9 10 11 0 ## Solution 5 4 5 9 10 11 0 ## Solution 6 4 5 9 10 11 0 ## Solution 7 4 5 9 10 11 0 ## Solution 8 3 4 9 10 11 0 ## Solution 9 3 4 9 10 11 0 ## Solution 10 3 4 9 10 11 0 ## ## (...) ## ## Solution 87 3 4 6 9 10 11 ## Solution 88 3 4 6 9 10 11 ## Solution 89 3 4 6 9 10 11 ## Solution 90 2 3 4 10 11 12 ## Solution 91 2 3 4 10 11 12 ## Solution 92 2 3 4 10 11 12 ## Solution 93 2 3 4 10 11 12 ## Solution 94 2 3 4 10 11 12 ## Solution 95 2 3 4 10 11 12 ## Solution 96 2 3 4 10 11 12 ## Solution 97 1 3 4 6 10 11 ## Solution 98 1 3 4 6 10 11 ## Solution 99 1 3 4 6 10 11 ## Solution 100 1 3 4 6 10 11 ## ## ## $values ## ## card.5 card.6 ## Solution 1 0.5018922 0.5168627 ## Solution 2 0.5018922 0.5168627 ## Solution 3 0.5018922 0.5168627 ## Solution 4 0.5018922 0.5168627 ## Solution 5 0.5018922 0.5168627 ## Solution 6 0.5018922 0.5168627 ## Solution 7 0.5018922 0.5096500 ## Solution 8 0.4966191 0.5096500 ## Solution 9 0.4966191 0.5096500 ## Solution 10 0.4966191 0.5096500 ## ## (...) ## ## Solution 87 0.4893824 0.5038649 ## Solution 88 0.4893824 0.5038649 ## Solution 89 0.4893824 0.5038649 ## Solution 90 0.4893824 0.5035489 ## Solution 91 0.4893824 0.5035489 ## Solution 92 0.4893824 0.5035489 ## Solution 93 0.4893824 0.5035489 ## Solution 94 0.4893824 0.5035489 ## Solution 95 0.4893824 0.5035489 ## Solution 96 0.4893824 0.5035489 ## Solution 97 0.4890986 0.5035386 ## Solution 98 0.4890986 0.5035386 ## Solution 99 0.4890986 0.5035386 ## Solution 100 0.4890986 0.5035386 ## ## $bestvalues ## Card.5 Card.6 ## 0.5018922 0.5168627 ## ## $bestsets ## Var.1 Var.2 Var.3 Var.4 Var.5 Var.6 ## Card.5 4 5 9 10 11 0 ## Card.6 4 5 9 10 11 12 ## ## $call ## genetic(mat = CarsHmat$mat, kmin = 5, kmax = 6, criterion = "zeta2", ## H = CarsHmat$H, r = 3, tolsym = 1e-09) ## ## Warning message: ## ## The effect description matrix (H) supplied was slightly asymmetric: ## symmetric entries differed by up to 3.63797880709171e-12. ## (less than the 'tolsym' parameter). ## The H matrix has been replaced by its symmetric part. ## in: validnovcrit(mat, criterion, H, r, p, tolval, tolsym) ## ## The selected best variable subsets colnames(CarsHmat$mat)[c(4,5,9,10,11)] ## [1] "Horsepower" "RPM" "Length" "Wheelbase" "Width" colnames(CarsHmat$mat)[c(4,5,9,10,11,12)] ## [1] "Horsepower" "RPM" "Length" "Wheelbase" "Width" ## [6] "Turn.circle" ## --------------------------------------------------------------------
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