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.

References

Adler, Robert J. 2010. The Geometry of Random Fields. SIAM ed. Philadelphia: Society for Industrial and Applied Mathematics.
Adler, Robert J., and Jonathan E. Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115. New York: Springer. https://doi.org/10.1007/978-0-387-48116-6.
Adler, Robert J, Jonathan E Taylor, and Keith J Worsley. 2016. Applications of Random Fields and Geometry Draft. https://robert.net.technion.ac.il/files/2016/08/hrf1.pdf.
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. https://doi.org/10.2307/1400521.
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. https://doi.org/10.1214/10-AOAS383.
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. https://doi.org/10.1007/978-1-4471-0657-9_2.
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. https://doi.org/10.1023/A:1009666805688.
Lee, Herbert K. H., Dave M. Higdon, Zhuoxin Bi, Marco A. R. Ferreira, and Mike West. 2002. “Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media.” Technometrics 44 (3): 230–41. https://doi.org/10.1198/004017002188618419.
Lee, Herbert K H, Christopher H Holloman, Catherine A Calder, and Dave M Higdon. 2002. “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. https://doi.org/10.1191/1471082X05st085oa.
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. https://doi.org/10.1111/j.1467-9868.2011.00777.x.
Scharf, Henry R., Mevin B. Hooten, Devin S. Johnson, and John W. Durban. 2017. “Process Convolution Approaches for Modeling Interacting Trajectories.” arXiv:1703.02112 [stat], November. http://arxiv.org/abs/1703.02112.
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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