abesspca {abess}

Adaptive best subset selection for principal component analysis

Description

Adaptive best subset selection for principal component analysis

Usage

abesspca(
  x,
  type = c("predictor", "gram"),
  sparse.type = c("fpc", "kpc"),
  cor = FALSE,
  kpc.num = NULL,
  support.size = NULL,
  gs.range = NULL,
  tune.path = c("sequence", "gsection"),
  tune.type = c("gic", "aic", "bic", "ebic", "cv"),
  nfolds = 5,
  foldid = NULL,
  ic.scale = 1,
  c.max = NULL,
  always.include = NULL,
  group.index = NULL,
  screening.num = NULL,
  splicing.type = 1,
  max.splicing.iter = 20,
  warm.start = TRUE,
  num.threads = 0,
  ...
)

Arguments

x

A matrix object. It can be either a predictor matrix where each row is an observation and each column is a predictor or a sample covariance/correlation matrix. If x is a predictor matrix, it can be in sparse matrix format (inherit from class "dgCMatrix" in package Matrix).

type

If type = "predictor", x is considered as the predictor matrix. If type = "gram", x is considered as a sample covariance or correlation matrix.

sparse.type

If sparse.type = "fpc", then best subset selection performs on the first principal component; If sparse.type = "kpc", then best subset selection would be sequentially performed on the first kpc.num number of principal components. If kpc.num is supplied, the default is sparse.type = "kpc"; otherwise, is sparse.type = "fpc".

cor

A logical value. If cor = TRUE, perform PCA on the correlation matrix; otherwise, the covariance matrix. This option is available only if type = "predictor". Default: cor = FALSE.

kpc.num

A integer decide the number of principal components to be sequentially considered.

support.size

It is a flexible input. If it is an integer vector. It represents the support sizes to be considered for each principal component. If it is a list object containing kpc.num number of integer vectors, the i-th principal component consider the support size specified in the i-th element in the list. Only used for tune.path = "sequence". The default is support.size = NULL, and some rules in details section are used to specify support.size.

gs.range

A integer vector with two elements. The first element is the minimum model size considered by golden-section, the later one is the maximum one. Default is gs.range = c(1, min(n, round(n/(log(log(n))log(p))))).

tune.path

The method to be used to select the optimal support size. For tune.path = "sequence", we solve the best subset selection problem for each size in support.size. For tune.path = "gsection", we solve the best subset selection problem with support size ranged in gs.range, where the specific support size to be considered is determined by golden section.

tune.type

The type of criterion for choosing the support size. Available options are "gic", "ebic", "bic", "aic" and "cv". Default is "gic". tune.type = "cv" is available only when type = "predictor".

nfolds

The number of folds in cross-validation. Default is nfolds = 5.

foldid

an optional integer vector of values between 1, ..., nfolds identifying what fold each observation is in. The default foldid = NULL would generate a random foldid.

ic.scale

A non-negative value used for multiplying the penalty term in information criterion. Default: ic.scale = 1.

c.max

an integer splicing size. The default of c.max is the maximum of 2 and max(support.size) / 2.

always.include

An integer vector containing the indexes of variables that should always be included in the model.

group.index

A vector of integers indicating the which group each variable is in. For variables in the same group, they should be located in adjacent columns of x and their corresponding index in group.index should be the same. Denote the first group as 1, the second 2, etc. If you do not fit a model with a group structure, please set group.index = NULL (the default).

screening.num

An integer number. Preserve screening.num number of predictors with the largest marginal maximum likelihood estimator before running algorithm.

splicing.type

Optional type for splicing. If splicing.type = 1, the number of variables to be spliced is c.max, ..., 1; if splicing.type = 2, the number of variables to be spliced is c.max, c.max/2, ..., 1. Default: splicing.type = 1.

max.splicing.iter

The maximum number of performing splicing algorithm. In most of the case, only a few times of splicing iteration can guarantee the convergence. Default is max.splicing.iter = 20.

warm.start

Whether to use the last solution as a warm start. Default is warm.start = TRUE.

num.threads

An integer decide the number of threads to be concurrently used for cross-validation (i.e., tune.type = "cv"). If num.threads = 0, then all of available cores will be used. Default: num.threads = 0.

...

further arguments to be passed to or from methods.

Details

Adaptive best subset selection for principal component analysis (abessPCA) aim to solve the non-convex optimization problem:

-\arg\min_{v} v^\top \Sigma v, s.t.\quad v^\top v=1, \|v\|_0 \leq s,

where s is support size. Here, \Sigma is covariance matrix, i.e.,

\Sigma = \frac{1}{n} X^{\top} X.

A generic splicing technique is implemented to solve this problem. By exploiting the warm-start initialization, the non-convex optimization problem at different support size (specified by support.size) can be efficiently solved.

The abessPCA can be conduct sequentially for each component. Please see the multiple principal components Section on the website for more details about this function. For abesspca function, the arguments kpc.num control the number of components to be consider.

When sparse.type = "fpc" but support.size is not supplied, it is set as support.size = 1:min(ncol(x), 100) if group.index = NULL; otherwise, support.size = 1:min(length(unique(group.index)), 100). When sparse.type = "kpc" but support.size is not supplied, then for 20\ it is set as min(ncol(x), 100) if group.index = NULL; otherwise, min(length(unique(group.index)), 100).

Value

A S3 abesspca class object, which is a list with the following components:

coef

A p-by-length(support.size) loading matrix of sparse principal components (PC), where each row is a variable and each column is a support size;

nvars

The number of variables.

sparse.type

The same as input.

support.size

The actual support.size values used. Note that it is not necessary the same as the input if the later have non-integer values or duplicated values.

ev

A vector with size length(support.size). It records the cumulative sums of explained variance at each support size.

tune.value

A value of tuning criterion of length length(support.size).

kpc.num

The number of principal component being considered.

var.pc

The variance of principal components obtained by performing standard PCA.

cum.var.pc

Cumulative sums of var.pc.

var.all

If sparse.type = "fpc", it is the total standard deviations of all principal components.

pev

A vector with the same length as ev. It records the percent of explained variance (compared to var.all) at each support size.

pev.pc

It records the percent of explained variance (compared to var.pc) at each support size.

tune.type

The criterion type for tuning parameters.

tune.path

The strategy for tuning parameters.

call

The original call to abess.

It is worthy to note that, if sparse.type == "kpc", the coef, support.size, ev, tune.value, pev and pev.pc in list are list objects.

Note

Some parameters not described in the Details Section is explained in the document for abess because the meaning of these parameters are very similar.

Author(s)

Jin Zhu, Junxian Zhu, Ruihuang Liu, Junhao Huang, Xueqin Wang

References

A polynomial algorithm for best-subset selection problem. Junxian Zhu, Canhong Wen, Jin Zhu, Heping Zhang, Xueqin Wang. Proceedings of the National Academy of Sciences Dec 2020, 117 (52) 33117-33123; doi:10.1073/pnas.2014241117

Sparse principal component analysis. Hui Zou, Hastie Trevor, and Tibshirani Robert. Journal of computational and graphical statistics 15.2 (2006): 265-286. doi:10.1198/106186006X113430

See Also

print.abesspca, coef.abesspca, plot.abesspca.

Examples

library(abess)
Sys.setenv("OMP_THREAD_LIMIT" = 2)

## predictor matrix input:
head(USArrests)
pca_fit <- abesspca(USArrests)
pca_fit
plot(pca_fit)

## covariance matrix input:
cov_mat <- stats::cov(USArrests) * (nrow(USArrests) - 1) / nrow(USArrests)
pca_fit <- abesspca(cov_mat, type = "gram")
pca_fit

## robust covariance matrix input:
rob_cov <- MASS::cov.rob(USArrests)[["cov"]]
rob_cov <- (rob_cov + t(rob_cov)) / 2
pca_fit <- abesspca(rob_cov, type = "gram")
pca_fit

## K-component principal component analysis
pca_fit <- abesspca(USArrests,
  sparse.type = "kpc",
  support.size = 1:4
)
coef(pca_fit)
plot(pca_fit)
plot(pca_fit, "coef")

## select support size via cross-validation ##
n <- 500
p <- 50
support_size <- 3
dataset <- generate.spc.matrix(n, p, support_size, snr = 20)
spca_fit <- abesspca(dataset[["x"]], tune.type = "cv", nfolds = 5)
plot(spca_fit, type = "tune")