3 research outputs found
Graphical Gaussian Process Models for Highly Multivariate Spatial Data
For multivariate spatial Gaussian process (GP) models, customary
specifications of cross-covariance functions do not exploit relational
inter-variable graphs to ensure process-level conditional independence among
the variables. This is undesirable, especially for highly multivariate
settings, where popular cross-covariance functions such as the multivariate
Mat\'ern suffer from a "curse of dimensionality" as the number of parameters
and floating point operations scale up in quadratic and cubic order,
respectively, in the number of variables. We propose a class of multivariate
"Graphical Gaussian Processes" using a general construction called "stitching"
that crafts cross-covariance functions from graphs and ensures process-level
conditional independence among variables. For the Mat\'ern family of functions,
stitching yields a multivariate GP whose univariate components are Mat\'ern
GPs, and conforms to process-level conditional independence as specified by the
graphical model. For highly multivariate settings and decomposable graphical
models, stitching offers massive computational gains and parameter dimension
reduction. We demonstrate the utility of the graphical Mat\'ern GP to jointly
model highly multivariate spatial data using simulation examples and an
application to air-pollution modelling
Graph-constrained Analysis for Multivariate Functional Data
Functional Gaussian graphical models (GGM) used for analyzing multivariate
functional data customarily estimate an unknown graphical model representing
the conditional relationships between the functional variables. However, in
many applications of multivariate functional data, the graph is known and
existing functional GGM methods cannot preserve a given graphical constraint.
In this manuscript, we demonstrate how to conduct multivariate functional
analysis that exactly conforms to a given inter-variable graph. We first show
the equivalence between partially separable functional GGM and graphical
Gaussian processes (GP), proposed originally for constructing optimal
covariance functions for multivariate spatial data that retain the conditional
independence relations in a given graphical model. The theoretical connection
help design a new algorithm that leverages Dempster's covariance selection to
calculate the maximum likelihood estimate of the covariance function for
multivariate functional data under graphical constraints. We also show that the
finite term truncation of functional GGM basis expansion used in practice is
equivalent to a low-rank graphical GP, which is known to oversmooth marginal
distributions. To remedy this, we extend our algorithm to better preserve
marginal distributions while still respecting the graph and retaining
computational scalability. The insights obtained from the new results presented
in this manuscript will help practitioners better understand the relationship
between these graphical models and in deciding on the appropriate method for
their specific multivariate data analysis task. The benefits of the proposed
algorithms are illustrated using empirical experiments and an application to
functional modeling of neuroimaging data using the connectivity graph among
regions of the brain.Comment: 23 pages, 6 figure