# 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 . 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.

## References

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Higdon, David. 1998. Environmental and Ecological Statistics 5 (2): 173–90.
Lee, Herbert K H, Christopher H Holloman, Catherine A Calder, and Dave M Higdon. 2002a. “Flexible Gaussian Processes via Convolution,” 12.
Lee, Herbert K. H., Dave M. Higdon, Zhuoxin Bi, Marco A. R. Ferreira, and Mike West. 2002b. Technometrics 44 (3): 230–41.
Lee, Herbert KH, Dave M Higdon, Catherine A Calder, and Christopher H Holloman. 2005. Statistical Modelling 5 (1): 53–74.
Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (4): 423–98.
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Thiebaux, Hj, and Ma Pedder. 1987. “Spatial Objective Analysis with Applications in Atmospheric Science.” London and Orlando, FL, Academic Press, 1987, 308.
Tobar, Felipe, Thang D Bui, and Richard E Turner. 2015. “Learning Stationary Time Series Using Gaussian Processes with Nonparametric Kernels,” 9.

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