Spatial Modelling Using a New Class of Nonstationary Covariance Functions
Christopher J. Paciorek and Mark J. Schervish
We introduce a new class of nonstationary covariance functions for
spatial modelling. Nonstationary covariance functions allow the model
to adapt to spatial surfaces whose variability changes with location.
The class includes a nonstationary version of the stationary
covariance, in which the differentiability of the spatial surface
is controlled by a parameter, freeing one from fixing the differentiability
in advance. The class allows one to knit together local covariance
parameters into a valid global nonstationary covariance, regardless
of how the local covariance structure is estimated. We employ this
new nonstationary covariance in a fully Bayesian model in which the
unknown spatial process has a Gaussian process (GP) distribution with
a nonstationary covariance function from the class. We model the nonstationary
structure in a computationally efficient way that creates nearly stationary
local behavior and for which stationarity is a special case. We also
suggest non-Bayesian approaches to nonstationary kriging.
To assess the method, we compare the Bayesian nonstationary GP model
with a Bayesian stationary GP model, various standard spatial smoothing
approaches, and nonstationary models that can adapt to function heterogeneity.
In simulations, the nonstationary GP model adapts to function heterogeneity,
unlike the stationary models, and also outperforms the other nonstationary
models. On a real dataset, GP models outperform the competitors, but
while the nonstationary GP gives qualitatively more sensible results,
it fails to outperform the stationary GP on held-out data, illustrating
the difficulty in fitting complex spatial functions with relatively
The nonstationary covariance model could also be used for non-Gaussian
data and embedded in additive models as well as in more complicated,
hierarchical spatial or spatio-temporal models. More complicated models
may require simpler parameterizations for computational efficiency.
Keywords: smoothing, Gaussian process, kriging, kernel convolution