secure.path {secure}

Sequential Co-Sparse Factor Regression

Description

Sequential factor extraction via co-sparse unit-rank estimation (SeCURE)

Usage

secure.path(Y, X = NULL, nrank = 3, nlambda = 100, U0 = NULL,
  V0 = NULL, D0 = NULL, orthXU = FALSE, orthV = FALSE,
  keepPath = TRUE, control = list(), ic = c("GIC", "BICP", "AIC")[1])

Arguments

Y

response matrix

X

covariate matrix; when X = NULL, the fucntion performs unsupervised learning

nrank

an integer specifying the desired rank/number of factors

nlambda

number of lambda values to be used along each path

U0

initial value of U

V0

initial value of V

D0

initial value of D

orthXU

if TRUE, orthogonality of XU is required

orthV

if TRUE, orthogonality of V is required

keepPath

if TRUE, th solution paths of U, V, D are reported

control

a list of internal parameters controlling the model fitting

ic

character specifying which information criterion to use for selecting the tuning parameter: "GIC"(default), "BICP", and "AIC"

Value

C.est

estimated coefficient matrix; based on modified BIC

U

estimated U matrix (factor weights)

D

estimated singular values

V

estimated V matrix (factor loadings)

ortX

if TRUE, X is treated as an orthogonal matrix in the computation

lam

selected lambda values based on the chosen information criterion

lampath

sequences of lambda values used in model fitting. In each sequential unit-rank estimation step, a sequence of length nlambda is first generated between (lamMax*lamMaxFac, lamMax*lamMaxFac*lamMinFac) equally spaced on the log scale, in which lamMax is estimated and the other parameters are specified in secure.control. The model fitting starts from the largest lambda and stops when the maximum proportion of nonzero elements is reached in either u or v, as specified by spU and spV in secure.control.

IC

values of information criteria

Upath

solution path of U

Dpath

solution path of D

Vpath

solution path of D

References

Mishra, A., Dey, D., Chen, K. (2017) Sequential Co-Sparse Factor Regression, To appear in Journal of Computational and Graphical Statistics (JCGS)

Examples

#require(secure)

# Simulate data from a sparse factor regression model
p <- 100; q <- 100; n <- 200
xrho <- 0.5; nlambda <- 100 
nrank <- 3 

U <- matrix(0,ncol=nrank ,nrow=p);  V <- matrix(0,ncol=nrank ,nrow=q)
U[,1]<-c(sample(c(1,-1),8,replace=TRUE),rep(0,p-8))
U[,2]<-c(rep(0,5),sample(c(1,-1),9,replace=TRUE),rep(0,p-14))
U[,3]<-c(rep(0,11),sample(c(1,-1),9,replace=TRUE),rep(0,p-20))
V[,1]<-c(sample(c(1,-1),5,replace=TRUE)*runif(5,0.3,1),rep(0,q-5))
V[,2]<-c(rep(0,5),sample(c(1,-1),5,replace=TRUE)*runif(5,0.3,1),rep(0,q-10))
V[,3]<-c(rep(0,10),sample(c(1,-1),5,replace=TRUE)*runif(5,0.3,1),rep(0,q-15))
U[,1:3]<- apply(U[,1:3],2,function(x)x/sqrt(sum(x^2)))
V[,1:3]<- apply(V[,1:3],2,function(x)x/sqrt(sum(x^2)))
D <- diag(c(20,15,10)) 
C <- U%*%D%*%t(V)

Xsigma <- xrho^abs(outer(1:p, 1:p,FUN="-"))
sim.sample <- secure.sim(U,D,V,n,snr = 0.25,Xsigma,rho=0.3)
Y <- sim.sample$Y; 
X <- sim.sample$X



# Fitting secure. Set maximum rank to be 4.
rank.ini <- 4

# Set largest model to about 25% sparsity
# See secure.control for setting other parameters
control <- secure.control(spU=0.25, spV=0.25)

# Complete data case. 
# Fit secure without orthogonality
fit.orthF <- secure.path(Y,X,nrank=rank.ini,nlambda = nlambda,
                        control=control)
# check orthogonality
crossprod(X%*%fit.orthF$U)/n
# check solution
# fit.orthF$U
# fit.orthF$V
# fit.orthF$D

# Fit secure with orthogonality if desired. It takes longer time.
# fit.orthT <- secure.path(Y,X,nrank=rank.ini,nlambda = nlambda,
#                                   orthXU=TRUE,orthV=TRUE,control=control)
# check orthogonality
# crossprod(X%*%fit.orthT$U)/n

  
# 15% missing case
miss <- 0.15
t.ind <- sample.int(n*q, size = miss*n*q)
y <- as.vector(Y); y[t.ind] <- NA;  Ym <- matrix(y,n,q)

fit.orthF.miss <- secure.path(Ym, X, nrank = rank.ini, nlambda = nlambda, 
                            control = control) 
# fit.orthT.miss <- secure.path(Ym, X, nrank = rank.ini, nlambda = nlambda,
#                           orthXU=TRUE,orthV=TRUE, control = control)