  We discuss a general Bayesian framework on modeling multidimensional
function-valued processes by using a Gaussian process or a heavy-tailed process
as a prior, enabling us to handle nonseparable and/or nonstationary covariance
structure. The nonstationarity is introduced by a convolution-based approach
through a varying anisotropy matrix, whose parameters vary along the input
space and are estimated via a local empirical Bayesian method. For the varying
matrix, we propose to use a spherical parametrization, leading to unconstrained
and interpretable parameters. The unconstrained nature allows the parameters to
be modeled as a nonparametric function of time, spatial location or other
covariates. The interpretation of the parameters is based on closed-form
expressions, providing valuable insights into nonseparable covariance
structures. Furthermore, to extract important information in data with complex
covariance structure, the Bayesian framework can decompose the function-valued
processes using the eigenvalues and eigensurfaces calculated from the estimated
covariance structure. The results are demonstrated by simulation studies and by
an application to wind intensity data. Supplementary materials for this article
are available online.
