LocCovReg {frechet}

Local Fréchet regression of covariance matrices

Description

Local Fréchet regression of covariance matrices with Euclidean predictors.

Usage

LocCovReg(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.

bwMean

A vector of length p holding the bandwidths for conditional mean estimation if y is provided. If bwMean is not provided, it is chosen by cross validation.

bwCov

A vector of length p holding the bandwidths for conditional covariance estimation. If bwCov is not provided, it is chosen by cross validation.

kernel

Name of the kernel function to be chosen from "rect", "gauss", "epan", "gausvar", "quar". Default is "gauss".

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=30             # 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=LocCovReg(x=x,y=y,xout=xout,optns=list(corrOut=FALSE,metric="power",alpha=3))

#Example M input
n=30 #sample size
m=30 #dimension of covariance matrices
M <- array(0,c(m,m,n))
for (i in 1:n){
 y0=rnorm(m)
 aux<-15*diag(m)+y0%*%t(y0)
 M[,,i]<-aux
}
x=matrix(rnorm(n),n)
xout = matrix(c(0.25,0.5,0.75),3) #output predictor levels
Cov_est=LocCovReg(x=x,M=M,xout=xout,optns=list(corrOut=FALSE,metric="power",alpha=0))