rrr {rrpack}

Multivariate reduced-rank linear regression

Description

Produce solution paths of reduced-rank estimators and adaptive nuclear norm penalized estimators; compute the degrees of freeom of the RRR estimators and select a solution via certain information criterion.

Usage

rrr(
  Y,
  X,
  penaltySVD = c("rank", "ann"),
  ic.type = c("GIC", "AIC", "BIC", "BICP", "GCV"),
  df.type = c("exact", "naive"),
  maxrank = min(dim(Y), dim(X)),
  modstr = list(),
  control = list()
)

Arguments

Y

a matrix of response (n by q)

X

a matrix of covariate (n by p)

penaltySVD

‘rank’: rank-constrainted estimation; ‘ann’: adaptive nuclear norm estimation.

ic.type

the information criterion to be used; currently supporting ‘AIC’, ‘BIC’, ‘BICP’, ‘GCV’, and ‘GIC’.

df.type

‘exact’: the exact degrees of freedoms based on SURE theory; ‘naive’: the naive degress of freedoms based on counting number of free parameters

maxrank

an integer of maximum desired rank.

modstr

a list of model parameters controlling the model fitting

control

a list of parameters for controlling the fitting process: ‘sv.tol’ controls the tolerence of singular values; ‘qr.tol’ controls the tolerence of QR decomposition for the LS fit

Details

Model parameters can be specified through argument modstr. The available include

• gamma: A scalar power parameter of the adaptive weights in penalty == "ann".

• nlambda: The number of lambda values; no effect if penalty == "count".

• lambda: A vector of user-specified rank values if penalty == "count" or a vector of penalty values if penalty == "ann".

The available elements for argument control include

• sv.tol: singular value tolerence.

• qr.tol: QR decomposition tolerence.

Value

S3 rrr object, a list consisting of

call

original function call

Y

input matrix of response

X

input matrix of covariate

A

right singular matri x of the least square fitted matrix

Ad

a vector of squared singular values of the least square fitted matrix

coef.ls

coefficient estimate from LS

Spath

a matrix, each column containing shrinkage factors of the singular values of a solution; the first four objects can be used to recover all reduced-rank solutions

df.exact

the exact degrees of freedom

df.naive

the naive degrees of freedom

penaltySVD

the method of low-rank estimation

sse

a vecotr of sum of squard errors

ic

a vector of information criterion

coef

estimated coefficient matrix

U

estimated left singular matrix such that XU/sqrtn is orthogonal

V

estimated right singular matrix that is orthogonal

D

estimated singular value matrix such that C = UDVt

rank

estimated rank

References

Chen, K., Dong, H. and Chan, K.-S. (2013) Reduced rank regression via adaptive nuclear norm penalization. Biometrika, 100, 901–920.

Examples

library(rrpack)
p <- 50; q <- 50; n <- 100; nrank <- 3
mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL,
                   rho_X = 0.5, rho_E = 0.3)
rfit <- with(mydata, rrr(Y, X, maxrank = 10))
summary(rfit)
coef(rfit)
plot(rfit)