GloCovReg {frechet}

Global Fréchet regression of covariance matrices

Description

Global Fréchet regression of covariance matrices with Euclidean predictors.

Usage

GloCovReg(x, y = NULL, M = NULL, xout, optns = list())

Arguments

x

An n by p matrix of predictors.

y

An n by l matrix, each row corresponds to an observation, l is the length of time points where the responses are observed. See 'metric' option in 'Details' for more details.

M

A q by q by n array (resp. a list of q by q matrices) where M[,,i] (resp. M[[i]]) contains the i-th covariance matrix of dimension q by q. See 'metric' option in 'Details' for more details.

xout

An m by p matrix of output predictor levels.

optns

A list of options control parameters specified by list(name=value). See ‘Details’.

Details

Available control options are

corrOut

Boolean indicating if output is shown as correlation or covariance matrix. Default is FALSE and corresponds to a covariance matrix.

metric

Metric type choice, "frobenius", "power", "log_cholesky", "cholesky" - default: "frobenius" which corresponds to the power metric with alpha equal to 1. For power (and Frobenius) metrics, either y or M must be input; y would override M. For Cholesky and log-Cholesky metrics, M must be input and y does not apply.

alpha

The power parameter for the power metric. Default is 1 which corresponds to Frobenius metric.

Value

A covReg object — a list containing the following fields:

xout

An m by p matrix of output predictor levels.

Mout

A list of estimated conditional covariance or correlation matrices at xout.

optns

A list containing the optns parameters utilized.

References

• Petersen, A. and Müller, H.-G. (2019). Fréchet regression for random objects with Euclidean predictors. The Annals of Statistics, 47(2), 691–719.

• Petersen, A., Deoni, S. and Müller, H.-G. (2019). Fréchet estimation of time-varying covariance matrices from sparse data, with application to the regional co-evolution of myelination in the developing brain. The Annals of Applied Statistics, 13(1), 393–419.

• Lin, Z. (2019). Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition. Siam. J. Matrix. Anal, A. 40, 1353–1370.

Examples

#Example y input
n=50             # sample size
t=seq(0,1,length.out=100)       # length of data
x = matrix(runif(n),n)
theta1 = theta2 = array(0,n)
for(i in 1:n){
 theta1[i] = rnorm(1,x[i],x[i]^2)
 theta2[i] = rnorm(1,x[i]/2,(1-x[i])^2)
}
y = matrix(0,n,length(t))
phi1 = sqrt(3)*t
phi2 = sqrt(6/5)*(1-t/2)
y = theta1%*%t(phi1) + theta2 %*% t(phi2)
xout = matrix(c(0.25,0.5,0.75),3)
Cov_est=GloCovReg(x=x,y=y,xout=xout,optns=list(corrOut=FALSE,metric="power",alpha=3))
#Example M input
n=10 #sample size
m=5 # dimension of covariance matrices
M <- array(0,c(m,m,n))
for (i in 1:n){
 y0=rnorm(m)
 aux<-diag(m)+y0%*%t(y0)
 M[,,i]<-aux
}
x=cbind(matrix(rnorm(n),n),matrix(rnorm(n),n)) #vector of predictor values
xout=cbind(runif(3),runif(3)) #output predictor levels
Cov_est=GloCovReg(x=x,M=M,xout=xout,optns=list(corrOut=FALSE,metric="power",alpha=3))