rqcanon (version 0.1.0)

rqcan: Canonical Quantile Regression.

Description

Given multivariate data matrices X (explanatory variables) and Y (response variables), the function fits coefficients of the Y-variables that are best fit by a quantile regression on X. These are analogous to the coefficients given by a classical canonical correlations analysis, but replace the implicit L2 norm by an L1 norm. See: "Method" and "Reference" below.

This is a simple S3 class for display formatting purposes.

Usage

rqcan(
  X,
  Y,
  tau = 0.5,
  a.pos = 1,
  ap = rep(1, na),
  na = ncol(Y),
  wts = rep(1, nrow(X))
)

Value

object of class "rqcan"; a list of matrices of the alpha and beta coefficients: the j-th row of each matrix is the coefficients for the j-th component; input data and the constraint matrices R an r are also returned in the list

Arguments

X

input design matrix of explanatory variables (without intercept)

Y

input matrix of response variables (nrow(Y) = nrow(X))

tau

desired quantile, default = .5

a.pos

for first component: non-empty vector of indices of Y-variables (Y-columns) whose alpha coefficients are constrained to be positive (to provide the direction of increasing responses); default = 1

ap

for subsequent components j = 2, 3, . . . , na: vector whose (j-1)-th element is the Y-variable (column) index whose alpha coefficients are constrained to be positive; default = rep(1,na-1)

na

number of components desired (1 <= na <= ncol(Y))

wts

used only for use with the bootstrap methods. If weighting is desired for the sample observations, rqcan will multiply the columns of X and Y by the vector wts; but the bootstrap methods will apply to the unweighted data, and so will be incorrect; default = rep(1,nrow(X)) (unweighted analysis)

Fields

list

A list.

Details

Finds orthogonal alpha coefficients and corresponding best-fitting beta coefficients to minimize sum|x_i' beta - y_i' alpha| subject to sum|alpha| = 1 (where x_i and y_i are the i-th rows of X and Y). The intercept is included (X should not include intercept). Need ncol(Y) > 1. For first component: if length(a.pos) < ncol(Y), sum(|alpha|) = 1 is constrained by going through all sign choices (s_j = sign(alpha_j)) and setting Y1_j = s_j Y_j (j not in a.pos). A constrained regression quantile fit is applied from quantreg: rq.fit.fnc(cbind(1,X,Y1),y0=0,R,r,tau). where (R,r) constrains all alpha_j >= 0 and sum(alpha_j) >= 1 (sum = 1 at min). Note: rq.fit.fnc solves by generating a sequence of quadratic approximations. The matrix defining one quadratic problem may be singular (and stop the computation) even if he input design matrices are of full rank. If a singularity stop occurs, jittering the data (see jitter()) sometimes helps.For the subsequent j-th component, only the index given by ap(j-1) is constrained to be positive. Alpha coefficients for subsequent components are constrained to be orthogonal to previous alpha coefficients.

References

S. Portnoy, 2022. Canonical quantile regression, J. Multivar. Anal., 192, 105071.

See Also

See summary.rqcan for a description of the summary function.

Examples

X <- as.matrix(example_data[,1:3])
Y <- as.matrix(example_data[,4:7])

a <- rqcan(X,Y,tau=.75,a.pos=2)
summary(a)