cv.SiER {SiER}

Cross-validation for high-dimensional multivariate regression

Description

Conduct the cross-validation and build the final model for the following high dimensional multivariate regression model:

Y= \mu+X\beta+\epsilon,

where Y is the n\times q response matrix with q\ge 1, X is the n\times p predictor matrix, and \epsilon is the noise matrix. The coefficient matrix \beta is p\times q and \mu is the intercept. The number of predictors p can be much larger than the sample size n. The response is univariate if q=1 and multivariate if q>1.

Usage

cv.SiER(X, Y, K.cv = 5, upper.comp = 10, thresh = 0.01)

Arguments

X

the n\times p predictor matrix.

Y

the n\times q response matrix, where q\ge 1 is the number of response variables.

K.cv

the number of CV sets. Default is 5.

upper.comp

the upper bound for the maximum number of components to be calculated. Default is 10.

thresh

a number between 0 and 1 specifying the minimum proportion of variation to be explained by each selected component relative to all the selected components. It is used to determine the maximum number of components to be calculated in the CV procedure. The optimal number of components will be selected from the integers from 1 to the minimum of upper.comp and this maximum number. A smaller thresh leads to a larger maximum number of components and a longer running time. A larger thresh value leads to a smaller running time, but may miss some important components and lead to a larger prediction error. Default is 0.01.

Details

Based on the best rank K approximation to X\beta, the coefficient matrix has decomposition \beta=\sum \alpha_k w_k ^T, where \alpha_k is the vector so that X\alpha_k has the maximum correlation with Y under the restriction that X\alpha_k has unit variance and is uncorrelated with X\alpha_1,..., X\alpha_{k-1}. We estimate \alpha_k by solving a penalized generalized eigenvalue problem with penalty \tau||\alpha_k||_{\lambda}^2 where ||\alpha_k||_{\lambda}^2=(1-\lambda)||\alpha_k||_2^2+\lambda||\alpha_k||_1^2 is a mixture of the squared l_2 and squared l_1 norms. The w_k is estimated by regressing Y on X\alpha_k.

Value

A fitted CV-object, which is used in the function pred.SiER() for prediction and getcoef.SiER() for extracting the estimated coefficient matrix.

mu

the estimated intercept vector.

beta

the estimated slope coefficient matrix.

min.error

minimum CV error.

scale.x

the maximum absolute value of X used to scale X.

X

the input X.

Y

the input Y.

params.set

a 9*2 matrix specifying the set of values of tau and lambda used in CV.

error

a list for CV errors.

opt.K

optimal number of components to be selected.

opt.tau

optimal value for tau.

opt.lambda

optimal value for lambda.

Author(s)

Ruiyan Luo and Xin Qi

References

Ruiyan Luo and Xin Qi (2017) Signal extraction approach for sparse multivariate response regression, Journal of Multivariate Statistics. 153: 83-97.

Examples

# q=1
library(MASS)
nvar=100
nvarq <- 1
sigmaY <- 0.1
sigmaX=0.1
nvar.eff=15
rho <- 0.3
Sigma=matrix(0,nvar.eff,nvar.eff)
for(i in 1:nvar.eff){
    for(j in 1:nvar.eff){
        Sigma[i,j]=rho^(abs(i-j))
    }
}

betas.true <- matrix(0, nvar, 1)
betas.true[1:15,1]=rep(1,15)/sqrt(15)

ntest <- 100
ntrain <- 90
ntot <- ntest+ntrain
X <- matrix(0,ntot,nvar)
X[,1:nvar.eff] <- mvrnorm(n=ntot, rep(0, nvar.eff), Sigma)
X[,-(1:nvar.eff)] <- matrix(sigmaX*rnorm((nvar-nvar.eff)*dim(X)[1]),
                            dim(X)[1],(nvar-nvar.eff))
Y <- X%*%betas.true
Y <- Y+rnorm(ntot, 0, sigmaY)

X.train <- X[1:ntrain,]
Y.train <- Y[1:ntrain,]
X.test <- X[-(1:ntrain),]
Y.test <- Y[-(1:ntrain),]

cv.fit <- cv.SiER(X.train,Y.train, K.cv=5)

Y.pred=pred.SiER(cv.fit, X.test)
error=sum((Y.pred-Y.test)^2)/ntest
print(c("predict error=", error))
coefs=getcoef.SiER(cv.fit)


#q>1
library(MASS)
total.noise <- 0.1
rho <- 0.3
rho.e <- 0.2
nvar=500
nvarq <- 3
sigma2 <- total.noise/nvarq
sigmaX=0.1
nvar.eff=150

Sigma=matrix(0,nvar.eff,nvar.eff)
for(i in 1:nvar.eff){
    for(j in 1:nvar.eff){
        Sigma[i,j]=rho^(abs(i-j))
    }
}
Sigma2.y <- matrix(sigma2*rho.e,nvarq, nvarq)
diag(Sigma2.y) <- sigma2

betas.true <- matrix(0, nvar, 3)
betas.true[1:15,1]=rep(1,15)/sqrt(15)
betas.true[16:45,2]=rep(0.5,30)/sqrt(30)
betas.true[46:105,3]=rep(0.25,60)/sqrt(60)

ntest <- 500
ntrain <- 90
ntot <- ntest+ntrain
X <- matrix(0,ntot,nvar)
X[,1:nvar.eff] <- mvrnorm(n=ntot, rep(0, nvar.eff), Sigma)
X[,-(1:nvar.eff)] <- matrix(sigmaX*rnorm((nvar-nvar.eff)*dim(X)[1]),
                           dim(X)[1],(nvar-nvar.eff))
Y <- X%*%betas.true
Y <- Y+mvrnorm(n=ntot, rep(0,nvarq), Sigma2.y)

X.train <- X[1:ntrain,]
Y.train <- Y[1:ntrain,]
X.test <- X[-(1:ntrain),]
Y.test <- Y[-(1:ntrain),]

cv.fit <- cv.SiER(X.train,Y.train, K.cv=5)

Y.pred=pred.SiER(cv.fit, X.test)
error=sum((Y.pred-Y.test)^2)/ntest
print(c("predict error=", error))