# 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

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. ; . Alternatively we can fix the driving noise and vary the smoothing kernel. TBC.

## References

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