NRRR.est: Nested reduced-rank regression with given ranks

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

Description

This function implements the blockwise coordinate descent algorithm to get the nested reduced-rank regression estimator with given rank values (r, rx, ry).

Usage

NRRR.est(Y, X, Ag0 = NULL, Bg0 = NULL,
         rini, r, rx, ry, jx, jy, p, d, n,
         maxiter = 100, conv = 1e-4,
         method = c("RRR", "RRS")[1], lambda = 0)

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.
Bg0	an initial estimator of matrix V. If Bg0 = NULL then generate it by NRRR.ini. Default is NULL.
rini, r	rank of the local reduced-rank structure. rini is used in NRRR.ini to get the initial estimators of U and V.
rx	number of latent predictors.
ry	number of latent responses.
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 100.
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' and a positive value should be provided. Default is 0.

Details

The nested reduced-rank regression (NRRR) is proposed to solve a multivariate functional linear regression problem where both the response and predictor are multivariate and functional (i.e., Y(t)=(y_1(t),...,y_d(t))^T and X(s)=(x_1(s),...,y_p(s))^T). To control the complexity of the problem, NRRR imposes a nested reduced-rank structure on the regression surface C(s,t). Specifically, a global dimension reduction makes use of the correlation within the components of multivariate response and multivariate predictor. Matrices U (d-by-ry) and V (p-by-rx) provide weights to form ry latent functional responses and rx latent functional predictors, respectively. Dimension reduction is achieved once ry < d or rx < p. Then, a local dimension reduction is conducted by restricting the latent regression surface C^*(s,t) to be of low-rank. After basis expansion and truncation, also by applying proper rearrangement to columns and rows of the resulting data matrices and coefficient matrices, we have the nested reduced-rank problem:

\min_{C} || Y - XC ||_F^2, s.t., C = (I_{jx} \otimes V) BA^T (I_{jy} \otimes U)^T,

where BA^T is a full-rank decomposition to control the local rank and jx, jy are the number of basis functions. Beyond the functional setup, this structure can also be applied to multiple scenarios, including multivariate time series autoregression analysis and tensor-on-tensor regression. This problem is non-convex and has no explicit solution, thus we use a blockwise coordinate descent algorithm to find a local solution. The convergence is decided by the change in objective values between two neighboring iterations.

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	degrees of freedom of the model.
sse	sum of squared errors.
ic	a vector containing values of BIC, BICP, AIC, GCV.
iter	the number of iterations needed to converge.

References

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

Examples

library(NRRR)
simDat <- NRRR.sim(n = 100, ns = 200, nt = 200, r = 5, rx = 3, ry = 3,
                   jx = 15, jy = 15, p = 10, d = 6, s2n = 1, rho_X = 0.5,
                   rho_E = 0, Sigma = "CorrAR")
fit_init <- with(simDat, NRRR.est(Y = Yest, X = Xest,
                 Ag0 = NULL, Bg0 = NULL, rini = 5, r = 5,
                 rx = 3, ry = 3, jx = 15, jy = 15,
                 p = 10, d = 6, n = 100))
fit_init$Ag

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