  Capturing the conditional covariances or correlations among the elements of a
multivariate response vector based on covariates is important to various fields
including neuroscience, epidemiology and biomedicine. We propose a new method
called Covariance Regression with Random Forests (CovRegRF) to estimate the
covariance matrix of a multivariate response given a set of covariates, using a
random forest framework. Random forest trees are built with a splitting rule
specially designed to maximize the difference between the sample covariance
matrix estimates of the child nodes. We also propose a significance test for
the partial effect of a subset of covariates. We evaluate the performance of
the proposed method and significance test through a simulation study which
shows that the proposed method provides accurate covariance matrix estimates
and that the Type-1 error is well controlled. An application of the proposed
method to thyroid disease data is also presented. CovRegRF is implemented in a
freely available R package on CRAN.
