rqcan {rqcanon}

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))
)

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)

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.

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

Fields

list

A list.

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)