Convolutional Gaussian processes

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.

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 \(\gamma_{x}(\mathbf{d}) .\)

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.


Barry, Ronald Paul, and Jay M. Ver Hoef. 1996. “Blackbox Kriging: Spatial Prediction Without Specifying Variogram Models.” Journal of Agricultural, Biological, and Environmental Statistics 1 (3): 297–322.
Bolin, David, and Finn Lindgren. 2011. “Spatial Models Generated by Nested Stochastic Partial Differential Equations, with an Application to Global Ozone Mapping.” The Annals of Applied Statistics 5 (1): 523–50.
Higdon, Dave. 2002. “Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues, edited by Clive W. Anderson, Vic Barnett, Philip C. Chatwin, and Abdel H. El-Shaarawi, 37–56. London: Springer.
Higdon, Dave, Jenise Swall, and J. Kern. 1999. “Non-Stationary Spatial Modeling.” Bayesian Statistics 6 (1): 761–68.
Higdon, David. 1998. “A Process-Convolution Approach to Modelling Temperatures in the North Atlantic Ocean.” Environmental and Ecological Statistics 5 (2): 173–90.
Lee, Herbert K. H., Dave M. Higdon, Zhuoxin Bi, Marco A. R. Ferreira, and Mike West. 2002a. “Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media.” Technometrics 44 (3): 230–41.
Lee, Herbert K H, Christopher H Holloman, Catherine A Calder, and Dave M Higdon. 2002b. “Flexible Gaussian Processes via Convolution,” 12.
Lee, Herbert KH, Dave M Higdon, Catherine A Calder, and Christopher H Holloman. 2005. “Efficient Models for Correlated Data via Convolutions of Intrinsic Processes.” Statistical Modelling 5 (1): 53–74.
Lindgren, Finn, Håvard Rue, and Johan 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) 73 (4): 423–98.
Scharf, Henry R., Mevin B. Hooten, Devin S. Johnson, and John W. Durban. 2017. “Process Convolution Approaches for Modeling Interacting Trajectories.” November 21, 2017.
Thiebaux, HJ, and MA Pedder. 1987. “Spatial Objective Analysis with Applications in Atmospheric Science.” London and Orlando, FL, Academic Press, 1987, 308.

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