NRRR.ic: Select ranks with information criterion

In xliu-stat/NRRR: Multivariate Functional Regression via Nested Reduced-Rank Regularization

Description

This function selects the optimal ranks (r, rx, ry) using a user-specified information criterion. The degrees of freedom is estimated by the number of free parameters in the model. The blockwise coordinate descent algorithm is used to fit the model with any combinations of (r, rx, ry).

Usage

NRRR.ic(Y, X, Ag0 = NULL, Bg0 = NULL,
        jx, jy, p, d, n, maxiter = 300, conv = 1e-4,
        method = c('RRR','RRS')[1],
        lambda = 0, ic = c("BIC","BICP","AIC","GCV")[1],
        dimred = c(TRUE,TRUE,TRUE), rankfix = NULL,
        xrankfix = NULL, yrankfix = NULL,
        lang = c('R', 'Rcpp')[1])

Arguments

Y	response matrix of dimension n-by-jy*d.
X	design matrix of dimension n-by-jx*p.
Ag0	an initial estimator of matrix U. If Ag0 = NULL then generate it by NRRR.ini. Default is NULL. If lang = 'Rcpp', then Ag0 is automatically generated by NRRR.ini.
Bg0	an initial estimator of matrix V, if Bg0 = NULL then generate it by NRRR.ini. Default is NULL. If lang = 'Rcpp', then Bg0 is automatically generated by NRRR.ini.
jx	number of basis functions to expand the functional predictor.
jy	number of basis functions to expand the functional response.
p	number of predictors.
d	number of responses.
n	sample size.
maxiter	the maximum iteration number of the blockwise coordinate descent algorithm. Default is 300.
conv	the tolerance level used to control the convergence of the blockwise coordinate descent algorithm. Default is 1e-4.
method	'RRR' (default): no additional ridge penalty; 'RRS': add an additional ridge penalty.
lambda	the tuning parameter to control the amount of ridge penalization. It is only used when method = 'RRS'. Default is 0.
ic	the user-specified information criterion. Four options are available, i.e., BIC, BICP, AIC, GCV.
dimred	a vector of logical values to decide whether to use the selected information criterion do rank selection on certain dimensions. TRUE means the rank is selected by the selected information criterion. If dimred[1] = FALSE, r is provided by rankfix or min(jy*d, rank(X)); If dimred[2] = FALSE, rx equals to xrankfix or p; If dimred[3] = FALSE, ry equals to yrankfix or d. Default is c(TRUE, TRUE, TRUE).
rankfix	a user-provided value of r when dimred[1] = FALSE. Default is NULL which leads to r = min(jy*d, rank(X)).
xrankfix	a user-provided value of rx when dimred[2] = FALSE. Default is NULL which leads to rx = p.
yrankfix	a user-provided value of ry when dimred[3] = FALSE. Default is NULL which leads to ry = d.
lang	'R' (default): the R version function is used; 'Rcpp': the Rcpp version function is used.

Details

Denote \hat C(r, rx, ry) as the estimator of the coefficient matrix with the rank values fixed at some (r, rx, ry) and write the sum of squared errors as SSE(r, rx, ry)=||Y - X\hat C(r, rx, ry)||_F^2. We define

BIC(r, rx, ry) = n*d*jy*log(SSE(r, rx, ry)/(n*d*jy)) + log(n*d*jy)*df(r, rx, ry),

where df(r, rx, ry) is the effective degrees of freedom and is estimated by the number of free parameters

\hat df(r, rx, ry) = rx*(rank(X)/jx - rx) + ry*(d - ry) + (jy*ry + jx*rx - r)*r.

We can define other information criteria in a similar way. With the defined BIC, a three-dimensional grid search procedure of the rank values is performed, and the best model is chosen as the one with the smallest BIC value. Instead of a nested rank selection method, we apply a one-at-a-time selection approach. We first set rx = p, ry = d, and select the best local rank \hat r among the models with 1 ≤ r ≤ min(rank(X), jy*d). We then fix the local rank at \hat r and repeat a similar procedure to determine \hat rx and \hat ry, one at a time. Finally, with fixed \hat rx and \hat ry, we refine the estimation of r.

Value

The function returns a list:

Ag	the estimated U.
Bg	the estimated V.
Al	the estimated A.
Bl	the estimated B.
C	the estimated coefficient matrix C.
df	the estimated degrees of freedom of the NRRR model.
sse	the sum of squared errors of the selected model.
ic	a vector containing values of BIC, BICP, AIC, GCV of the selected model.
rank	the estimated r.
rx	the estimated rx.
ry	the estimated ry.

References

Liu, X., Ma, S., & Chen, K. (2020). Multivariate Functional Regression via Nested Reduced-Rank Regularization. arXiv: Methodology.

Examples

library(NRRR)
set.seed(3)
# Simulation setting 2 in NRRR paper
simDat <- NRRR.sim(n = 100, ns = 100, nt = 100, r = 3, rx = 3, ry = 3,
                   jx = 8, jy = 8, p = 20, d = 20, s2n = 1, rho_X = 0.5,
                   rho_E = 0, Sigma = "CorrAR")
# using R function
fit_R <- with(simDat, NRRR.ic(Yest, Xest, Ag0 = NULL, Bg0 = NULL,
              jx = 8, jy = 8, p = 20, d = 20, n = 100,
              maxiter = 300, conv = 1e-4, method = c("RRR", "RRS")[1],
              lambda = 0, ic = c("BIC", "BICP", "AIC", "GCV")[1],
              dimred = c(TRUE, TRUE, TRUE), rankfix = NULL,
              xrankfix = NULL, yrankfix = NULL,
              lang=c('R','Rcpp')[1]))
# using Rcpp function
fit_Rcpp <- with(simDat, NRRR.ic(Yest, Xest, Ag0 = NULL, Bg0 = NULL,
                 jx = 8, jy = 8, p = 20, d = 20, n = 100,
                 maxiter = 300, conv = 1e-4, method = c("RRR", "RRS")[1],
                 lambda = 0, ic = c("BIC", "BICP", "AIC", "GCV")[1],
                 dimred = c(TRUE, TRUE, TRUE), rankfix = NULL,
                 xrankfix = NULL, yrankfix = NULL,
                 lang=c('R','Rcpp')[2]))

xliu-stat/NRRR documentation built on Jan. 9, 2021, 3:23 p.m.