# Convolutional Gaussian processes

March 1, 2021 — March 1, 2021

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 . 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. Journal of Agricultural, Biological, and Environmental Statistics.
Bolin, and Lindgren. 2011. The Annals of Applied Statistics.
Higdon, David. 1998. Environmental and Ecological Statistics.
Higdon, Dave. 2002. 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. Technometrics.
Lee, Herbert KH, Higdon, Calder, et al. 2005. Statistical Modelling.
Lee, Herbert K H, Holloman, Calder, et al. 2002b. “Flexible Gaussian Processes via Convolution.”
Lindgren, Rue, and Lindström. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Scharf, Hooten, Johnson, et al. 2017. 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.”