Convolutional Gaussian processes

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 ${\gamma }_x(\mathbf{d})=\mathrm{Var}(x(\mathbf{s})-$ $x(\mathbf{s}+\mathbf{d}))$ the resulting variogram of the process $z(\mathbf{s})$ is given by $${\gamma }_z(\mathbf{d})={\gamma }_z^{\ast }(\mathbf{d})-{\gamma }_z^{\ast }(\mathbf{0})\text{ where }{\gamma }_z^{\ast }(\mathbf{q})={\int }_{\mathcal{S}}{\int }_{\mathcal{S}}k(\mathbf{v}-\mathbf{q})k(\mathbf{u}-\mathbf{v}){\gamma }_x(\mathbf{u})d\mathbf{u}d\mathbf{v}$$ …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