Package 'rqcanon'

Bootstrap

Description

Internal function to carry out the bootstrap ; It is a sub-module of summary.rqcan; not intended for general use.
The parameters may be passed by summary.rqcan (see below)

Usage

boot.can(
  a,
  Rep = 200,
  method = "Andrews",
  msub = 0.9,
  seed,
  nsing = 5,
  prb = FALSE
)

Arguments

a	output from rqcan
Rep	number of bootstrap replications (default=200)
method	"Andrews" (default) or ""xy"
msub	parameter defining the size of the bootstrap subsample for developmental work only: see the boot.can function
seed	a starting seed (default: missing: no new seed set)
nsing	number of consecutive singular replicatios to ignore (default = 5)
prb	if TRUE (default = FALSE), print every time 10 percent of the bootstrap samples are done

Details

See help(summary.rqcan) ; If errors occur or modification is wanted, see the routine boot.can

Value

Returns list(As, Bs, sdc): As (Bs) are N by dim(alpha) (dim(beta)) arrays of all bootstrap alphs and beta values; sdc = sqrt(m/n): SD adjustment for m-choose-n bootstrap, or 1 for "xy" bootstrap

---

Psychological data

Description

A dataset from UCLA Statistical Methods and Data Analytics about investigating the associations between psychological measures and academic achievement measures.

Usage

example_data

Format

## 'example_data' A data frame with 600 rows and 8 columns:

locus_of_control

Psychological Locus of Control

self_concept

Psychological Self Concept

motivation

Psychological Motivation

read

Academic Reading

write

Academic Writing

math

Academic Math

science

Academic Science

female

Binary flag for 1 being female.

Source

<https://stats.idre.ucla.edu/stat/data/mmreg.csv>

---

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)

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First Component

Description

Internal function to find the first component
It is not intended for general use, but the documentation may be helpful if errors occur or if one whishes to modify the algorithms

Usage

rqcan1(X, Y, tau = 0.5, a.pos = 1, wts = rep(1, nrow(X)))

Arguments

X	input X-matrix
Y	input Y-matrix
tau	probability for qualtile (default = .5)
a.pos	indices of Y-variable whose coefficient is constrained to be positive (default = 1)
wts	case weights (default = rep(1,nrow(X)) )

Details

The function finds the leading pair of indices. Notes: an intercept is added (X should not include 1st col = 1) ; ncol(Y) should be > 1 ; length(a.pos) should be at least 1 to specify coef signs (if tau = .5 and a.pos = NULL, coef and -coef give the same solution) ; for length(a.pos) < ncol(Y), the constraint sum(|alpha|) = 1 is set by setting Y1_j = s_j Y_j (j !in a.pos) where s_j = sgn(alpha_j) ; all sign choices are used and then constrained rq.fit.fnc( cbind(1,X,Y1),y0=0,R,r,tau) is applied
(R,r) contrains all alpha_j >= 0 and sum(alpha_j) >= 1 (makes sum = 1)

Value

Returns list(a,X,Y,a.pos,R,r,rho1): a = output from rq.fit.fnc(XY,y0,R,r,tau) ; X,Y,a.pos = input data ; R,r = constraint matrices for rq.fit.fnc ; rho1 = rq objective fct. ; if rq.fit.fnc generates a singular matrix, returns "sing"

---

Summary of rqcan function results.

Description

Uses one of two bootstrap methods to provide Standard Error and confidence intervals for the alpha and beta coefficients for all components.

Usage

## S3 method for class 'rqcan'
summary(object, pr = TRUE, ci = 1, fact = 1, ...)

Arguments

object	rqcan object returned by rqcan
pr	print tables if TRUE (default)
ci	type of 95 ci=1: use (adjusted) .025 and .975 percentiles of bootstrap distribution ci=2: use normal approx with adjusted SE based on interquartile range ci may be a vector to provide more than one type of interval; default=1
fact	a factor to adjust conf ints for components 2:na; used only for development
...	parameters that are sent to the bootstrap function: Rep: number of bootstrap replications (default=200) method: "Andrews" (default) or ""xy" msub: parameter defining the size of the bootstrap subsample for developmental work only: see the boot.can function seed: a starting seed (default: missing: no new seed set) nsing: number of consecutive singular replicatios to ignore (default = 5) prb: if TRUE (default = FALSE), print every time 10 percent of the bootstrap samples are done

Details

The Portnoy reference showed that a subsample bootstrap (as described by Andrews) gives consistent estimates of SE's and confidence intervals. The subsample size is m = ceiling( min(n, max(log(n)*(px+py+1), n^msub)) ) (where n = nrow(X), px = ncol(X), py = ncol(Y)), msub is as above). Some simulations and examples suggest that this is OK. The usual "xy" bootstrap (sampling rows independently with replacement) can be specified. It seems to give similar confidence intervals to "Andrews", but the SE estimates may be wrong; and no form of consistency has been proven. Note: as noted in help(rqcan), the quantreg function rq.fit.fnc may generate singular matrices even if the input design matrix is of full rank. In simulation examples, this can happen for some bootstrap replications (perhaps less than 1/1000 times). When this occurs, a new bootstrap replication is drawn. If more than nsing consecutive singularities are produced, the bootstrap function returns with those replications that it has already found (a number less than Rep), with a warning. If a singularity warning occurs, using "xy", or changing the seed or "jittering" the data (see jitter()) sometimes helps.

Value

Returns list(As,Bs,sdc): As and Bs are matrices with Rep rows giving alpha beta coefficients for each bootstrap replication; and sdc is a standard error adjustment based on the subsample bootstrap: sdc = sqrt(1 - m/n).

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