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 “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.

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. 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.
Bolin, David. 2014. “Spatial Matérn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics 41 (3): 557–79. https://doi.org/10.1111/sjos.12046.
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, 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 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.
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.
Wolpert, R. 1998. “Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika 85 (2): 251–67. https://doi.org/10.1093/biomet/85.2.251.
Wolpert, Robert L. 2006. “Stationary Gamma Processes,” 13.

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