Convolutional Gaussian processes

March 1, 2021 — March 1, 2021

Figure 1

Gaussian processes by convolution of noise with smoothing kernels, which is a kind of dual to defining them through covariances.

This is especially interesting because it can be made computationally convenient (we can enforce locality) and non-stationarity.

1 Convolutions with respect to a non-stationary driving noise

H. K. Lee et al. (2005):

A convenient representation of a GP model uses process convolutions (Barry and Hoef 1996; Dave Higdon 2002; Thiebaux and Pedder 1987). One may construct a Gaussian process \(z(\mathbf{s})\) over a region \(\mathcal{S}\) by convolving a continuous, unit variance, white noise process \(x(\mathbf{s}),\) with a smoothing kernel \(k(\mathbf{s}):\) \[ z(\mathbf{s})=\int_{\mathcal{S}} k(\mathbf{u}-\mathbf{s}) x(\mathbf{u}) d \mathbf{u} \]

If we take \(x(\mathbf{s})\) to be an intrinsically stationary process with variogram \(\gamma_{x}(\mathbf{d})=\operatorname{Var}(x(\mathbf{s})-\) \(x(\mathbf{s}+\mathbf{d}))\) the resulting variogram of the process \(z(\mathbf{s})\) is given by \[ \gamma_{z}(\mathbf{d})=\gamma_{z}^{*}(\mathbf{d})-\gamma_{z}^{*}(\mathbf{0}) \text { where } \gamma_{z}^{*}(\mathbf{q})=\int_{\mathcal{S}} \int_{\mathcal{S}} k(\mathbf{v}-\mathbf{q}) k(\mathbf{u}-\mathbf{v}) \gamma_{x}(\mathbf{u}) d \mathbf{u} d \mathbf{v} \] …With this approach, one can fix the smoothing kernel \(k(\mathbf{s})\) and then modify the spatial dependence for \(z(\mathbf{s})\) by controlling \(\gamma_{x}(\mathbf{d}) .\)

2 Varying convolutions with respect to a stationary white noise

e.g. Dave Higdon, Swall, and Kern (1999);David Higdon (1998). Alternatively we can fix the driving noise and vary the smoothing kernel. TBC.

3 References

Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115.
Adler, Robert J, Taylor, and Worsley. 2016. Applications of Random Fields and Geometry Draft.
Barry, and Hoef. 1996. Blackbox Kriging: Spatial Prediction Without Specifying Variogram Models.” Journal of Agricultural, Biological, and Environmental Statistics.
Bolin, and Lindgren. 2011. Spatial Models Generated by Nested Stochastic Partial Differential Equations, with an Application to Global Ozone Mapping.” The Annals of Applied Statistics.
Higdon, David. 1998. A Process-Convolution Approach to Modelling Temperatures in the North Atlantic Ocean.” Environmental and Ecological Statistics.
Higdon, Dave. 2002. Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues.
Higdon, Dave, Swall, and Kern. 1999. “Non-Stationary Spatial Modeling.” Bayesian Statistics.
Lee, Herbert K. H., Higdon, Bi, et al. 2002a. Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media.” Technometrics.
Lee, Herbert KH, Higdon, Calder, et al. 2005. Efficient Models for Correlated Data via Convolutions of Intrinsic Processes.” Statistical Modelling.
Lee, Herbert K H, Holloman, Calder, et al. 2002b. “Flexible Gaussian Processes via Convolution.”
Lindgren, Rue, and Lindström. 2011. An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Scharf, Hooten, Johnson, et al. 2017. Process Convolution Approaches for Modeling Interacting Trajectories.” arXiv:1703.02112 [Stat].
Thiebaux, and Pedder. 1987. “Spatial Objective Analysis with Applications in Atmospheric Science.” London and Orlando, FL, Academic Press, 1987, 308.
Tobar, Bui, and Turner. 2015. “Learning Stationary Time Series Using Gaussian Processes with Nonparametric Kernels.”