covKCD {covKCD}

Kronecker-core covariance decomposition

Description

Computes the Kronecker-core decomposition of a covariance matrix.

Usage

covKCD(S, p1, p2, tol = 1e-08)

Arguments

S

a covariance matrix of dimension (p1p2)*(p1p2).

p1

the row dimension.

p2

the column dimension.

tol

the convergence tolerance of the iterative algorithm.

Details

The Kronecker-core decomposition is a representation of an arbitrary covariance matrix S in terms of a separable Kronecker covariance matrix K and a complementary non-separable core covariance matrix C. The Kronecker covariance is the separable covariance matrix that is closest to S in terms of the divergence function

\log|K| + \mbox{trace}(K^{-1}S).

The core covariance matrix C is computed from S and K via

C = K^{-1/2} S K^{-1/2}.

Value

covKCD returns a list with the following elements:

K

the Kronecker covariance matrix;

C

the core covariance matrix;

K1

the row covariance matrix;

K2

the column covariance matrix;

div

the divergence between S and K across iterations of the algorithm.

Author(s)

Peter Hoff

Examples

p1<-4 ; p2<-3 ; n<-200

# create a matrix Y with separable covariance
A<-matrix(rnorm(p1*p1),p1,p1)  
B<-matrix(rnorm(p2*p2),p2,p2)/3
Y<-array(rnorm(n*p1*p2),dim=c(n,p1,p2))  
Y<-aperm( apply(Y,c(1,3),function(y){ A%*%y } ),c(2,1,3)) 
Y<-aperm( apply(Y,c(1,2),function(y){ B%*%y } ),c(2,3,1)) 

# covariance 
S<-mcov(Y) 

KCD<-covKCD(S,p1,p2) 

plot(A%*%t(A), KCD$K1)
plot(B%*%t(B), KCD$K2)