psvmSDR (version 2.0.1)

psdr: Unified linear principal sufficient dimension reduction methods

Description

A function for a linear principal sufficient dimension reduction.

Usage

psdr(
  x,
  y,
  loss = "svm",
  h = 10,
  lambda = 1,
  eps = 1e-05,
  max.iter = 100,
  eta = 0.1,
  mtype = "m",
  plot = FALSE
)

Value

An object with S3 class "psdr". Details are listed below.

Mn

The estimated working matrix, which is obtained by the cumulative outer product of the estimated parameters over the slices. It will not print out, unless it is called manually.

evalues

Eigenvalues of the working matrix \(Mn\)

evectors

Eigenvectors of the \(Mn\), the first leading \(d\) eigenvectors consists the basis of the central subspace

Arguments

x

input matrix, of dimension nobs x nvars; each row is an observation vector. Requirement: nvars>1; in other words, x should have 2 or more columns.

y

response variable, either can be continuous variable or (+1,-1) coded binary response vector.

loss

pre-specified loss functions belongs to "svm", "logit", "l2svm", "wsvm", "qr", "asls", "wlogit", "wl2svm", "lssvm", "wlssvm", and user-defined loss function object also can be used formed by inside double (or single) quotation mark. Default is 'svm'.

h

the number of slices and probabilities equally spaced in \((0,1)\). Default value is 10.

lambda

the cost parameter for the svm loss function. The default value is 1.

eps

the threshold for stopping iteration with respect to the magnitude of the change of the derivative. The default value is 1.0e-5.

max.iter

maximum iteration number for the optimization process. default value is 100.

eta

learning rate for the gradient descent algorithm. The default value is 0.1.

mtype

a margin type, which is either margin ("m") or residual ("r") (See, Table 1 in the manuscript). Only need when user-defined loss is used. Default is "m".

plot

If TRUE then it produces scatter plots of \(Y\) versus \(\hat{B^{\top}}_{j}\mathbf{X}\). \(j\) can be specified by the user with \(j=1\) as a default. The default is FALSE.

Author

Jungmin Shin, , Seung Jun Shin, , Andreas Artemiou

Details

Two examples of the usage of user-defined losses are presented below (u represents a margin):

mylogit <- function(u, ...) log(1+exp(-u)),

myls <- function(u ...) u^2.

Argument u is a function variable (any character is possible) and the argument mtype for psdr() determines a type of a margin, either (type="m") or (type="r") method. type="m" is a default. Users have to change type="r", when applying residual type loss. Any additional parameters of the loss can be specified via ... argument.

References

Artemiou, A. and Dong, Y. (2016) Sufficient dimension reduction via principal lq support vector machine, Electronic Journal of Statistics 10: 783–805.
Artemiou, A., Dong, Y. and Shin, S. J. (2021) Real-time sufficient dimension reduction through principal least squares support vector machines, Pattern Recognition 112: 107768.
Kim, B. and Shin, S. J. (2019) Principal weighted logistic regression for sufficient dimension reduction in binary classification, Journal of the Korean Statistical Society 48(2): 194–206.
Li, B., Artemiou, A. and Li, L. (2011) Principal support vector machines for linear and nonlinear sufficient dimension reduction, Annals of Statistics 39(6): 3182–3210.
Soale, A.-N. and Dong, Y. (2022) On sufficient dimension reduction via principal asymmetric least squares, Journal of Nonparametric Statistics 34(1): 77–94.
Wang, C., Shin, S. J. and Wu, Y. (2018) Principal quantile regression for sufficient dimension reduction with heteroscedasticity, Electronic Journal of Statistics 12(2): 2114–2140.
Shin, S. J., Wu, Y., Zhang, H. H. and Liu, Y. (2017) Principal weighted support vector machines for sufficient dimension reduction in binary classification, Biometrika 104(1): 67–81.
Li, L. (2007) Sparse sufficient dimension reduction, Biometrika 94(3): 603–613.

See Also

psdr_bic, rtpsdr

Examples

## ----------------------------
## Linear PM
## ----------------------------
set.seed(1)
n <- 200; p <- 5;
x <- matrix(rnorm(n*p, 0, 2), n, p)
y <-  x[,1]/(0.5 + (x[,2] + 1)^2) + 0.2*rnorm(n)
y.tilde <- sign(y)
obj <- psdr(x, y)
print(obj)
plot(obj, d=2)

## --------------------------------
## Linear PM (Binary classification)
## --------------------------------
obj_wsvm <- psdr(x, y.tilde, loss="wsvm")
plot(obj_wsvm)

## ----------------------------
## User-defined loss function
## ----------------------------
mylogistic <- function(u) log(1+exp(-u))
psdr(x, y, loss="mylogistic")