# Unified linear principal sufficient dimension reduction methods A function for a linear principal sufficient dimension reduction. ```r psdr( x, y, loss = "svm", h = 10, lambda = 1, eps = 1e-05, max.iter = 100, eta = 0.1, mtype = "m", plot = FALSE ) ``` ## 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. ## Returns 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 ## 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. ## Examples ```r ## ---------------------------- ## 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") ``` ## 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` ## Author(s) Jungmin Shin, jungminshin@korea.ac.kr , Seung Jun Shin, sjshin@korea.ac.kr , Andreas Artemiou artemiou@uol.ac.cy

psdr function