# Convolutional stochastic processes

Moving averages of noise

March 1, 2021 — August 16, 2021

Gaussian

geometry

Hilbert space

how do science

kernel tricks

machine learning

PDEs

physics

regression

signal processing

spatial

statistics

stochastic processes

time series

uncertainty

Stochastic processes generated by convolution of white noise with smoothing kernels, which is not unlike kernel smoothing where the “data” is random. Or, to put it another way, these are processes defined as *moving averages* of some stochastic noise.

For now, I am mostly interested in certain special cases Gaussian convolutions and subordinator convolutions.

## 1 References

Adler, Robert J. 2010.

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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*.
Bolin. 2014. “Spatial Matérn Fields Driven by Non-Gaussian Noise.”

*Scandinavian Journal of Statistics*.
Higdon, David. 1998. “A Process-Convolution Approach to Modelling Temperatures in the North Atlantic Ocean.”

*Environmental and Ecological Statistics*.
Higdon, Dave. 2002. “Space and Space-Time Modeling Using Process Convolutions.” In

*Quantitative Methods for Current Environmental Issues*.
Lee, Herbert K. H., Higdon, Bi, et al. 2002. “Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media.”

*Technometrics*.
Lee, Herbert KH, Higdon, Calder, et al. 2005. “Efficient Models for Correlated Data via Convolutions of Intrinsic Processes.”

*Statistical Modelling*.
Scharf, Hooten, Johnson, et al. 2017. “Process Convolution Approaches for Modeling Interacting Trajectories.”

*arXiv:1703.02112 [Stat]*.
Thiebaux, and Pedder. 1987. “Spatial Objective Analysis with Applications in Atmospheric Science.”

*London and Orlando, FL, Academic Press, 1987, 308*.
Wolpert, and Ickstadt. 1998. “Poisson/Gamma Random Field Models for Spatial Statistics.”

*Biometrika*.