Convolutional stochastic processes

Moving averages of noise



Stochastic processes generated by convolution of white noise with smoothing kernels, which is not unlike kernel smoothing where the β€œdata” 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.

C&C Karhunen-Loeve expansion.

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.
Adler, Robert J, Jonathan E Taylor, and Keith J Worsley. 2016. Applications of Random Fields and Geometry Draft.
Bolin, David. 2014. β€œSpatial MatΓ©rn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics 41 (3): 557–79.
Higdon, Dave. 2002. β€œSpace 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.
Higdon, David. 1998. β€œA Process-Convolution Approach to Modelling Temperatures in the North Atlantic Ocean.” Environmental and Ecological Statistics 5 (2): 173–90.
Lee, Herbert K. H., Dave M. Higdon, Zhuoxin Bi, Marco A. R. Ferreira, and Mike West. 2002. β€œMarkov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media.” Technometrics 44 (3): 230–41.
Lee, Herbert KH, Dave M Higdon, Catherine A Calder, and Christopher H Holloman. 2005. β€œEfficient Models for Correlated Data via Convolutions of Intrinsic Processes.” Statistical Modelling 5 (1): 53–74.
Scharf, Henry R., Mevin B. Hooten, Devin S. Johnson, and John W. Durban. 2017. β€œProcess Convolution Approaches for Modeling Interacting Trajectories.” arXiv:1703.02112 [Stat], November.
Thiebaux, HJ, and MA Pedder. 1987. β€œSpatial Objective Analysis with Applications in Atmospheric Science.” London and Orlando, FL, Academic Press, 1987, 308.
Wolpert, R., and Katja Ickstadt. 1998. β€œPoisson/Gamma Random Field Models for Spatial Statistics.” Biometrika 85 (2): 251–67.

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