Spatial processes and statistics thereof

Why is there some kind of Catholic edict or something in the background? Image credit Jordan McDowall

Why is there some kind of Catholic edict or something in the background? Image credit Jordan McDowall

Statistics on fields with index sets of more than one dimension of support and, frequently, an implicit 2-norm. Sometimes they are also time-indexed. Especially, for processes on a continuous index set with continuous state and undirected interaction. Sometimes over fancy manifolds, although often you can get away with plain old euclidean space, unless you if you are doing spatial statistics over the entire planet, which turns out to be curved. Lattice models are frequently considered spatial statistics, but more arbitrary graph structures usually get filed under undirected graphical models/random fields. For spatial point processes I will make a new notebook. Often we mean some kind of Gaussian process regression to handle spatial statistics, although the use of these tool in the ML and spatial literatures is weirdly disjoint. There are many other random fields we might also wish to infer that relate to spatial index sets, and these can be taxonomised as I notice their existence. There are lots of interesting problem with statistics on such fields. Consider the illustrative problem of declustering.

I’m curious about how spatial statistics generalise to high-dimensional fields such as fitness landscapes, loss functions, and embedding of network processes in space…

Kriging

The spatial statistics name for Gaussian process regression.

Spatial point processes

A particular sub-case combining point processes with spatial statics, now with its own notebook

Implementations

All recommendations made to me and passed on here are offered unreviewed and unendorsed.

spatstat

Spatstat is the reference general-purpose spatial data analysis. based on R.

Pysal

PySAL Python. Library of statistical functions for continuous-state spatial processes.

PASSaGE

Passage is also Python. GUI full of statistical analyses.

Abrahamsen, Petter. 1997. “A Review of Gaussian Random Fields and Correlation Functions.” http://publications.nr.no/publications.nr.no/directdownload/publications.nr.no/rask/old/917_Rapport.pdf.

Anselin, Luc. 1995. “Local Indicators of Spatial Association - LISA.” Geographical Analysis 27 (2): 93–115. https://doi.org/10.1111/j.1538-4632.1995.tb00338.x.

Baddeley, Adrian, Ege Rubak, and Rolf Turner. 2016. Spatial Point Patterns: Methodology and Applications with R. Champan & Hall/CRC Interdisciplinary Statistics Series. Boca Raton ; London ; New York: CRC Press, Taylor & Francis Group.

Baddeley, A., R. Turner, J. Møller, and M. Hazelton. 2005. “Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67 (5): 617–66. https://doi.org/10.1111/j.1467-9868.2005.00519.x.

Besag, Julian. 1974. “Spatial Interaction and the Statistical Analysis of Lattice Systems.” Journal of the Royal Statistical Society. Series B (Methodological) 36 (2): 192–236. https://doi.org/10.1111/j.2517-6161.1974.tb00999.x.

———. 1986. “On the Statistical Analysis of Dirty Pictures.” Journal of the Royal Statistical Society. Series B (Methodological) 48 (3): 259–302.

Brémaud, Pierre, Laurent Massoulié, and Andrea Ridolfi. 2005. “Power Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability 37 (4): 1116–46. https://doi.org/10.1239/aap/1134587756.

Cressie, Noel, and Christopher K. Wikle. 2015. Statistics for Spatio-Temporal Data. John Wiley & Sons. http://books.google.com?id=4L_dCgAAQBAJ.

Donoho, David L., and Jain M. Johnstone. 1994. “Ideal Spatial Adaptation by Wavelet Shrinkage.” Biometrika 81 (3): 425–55. https://doi.org/10.1093/biomet/81.3.425.

Fuentes, Montserrat. 2006. “Testing for Separability of Spatial–Temporal Covariance Functions.” Journal of Statistical Planning and Inference 136 (2): 447–66. https://doi.org/10.1016/j.jspi.2004.07.004.

Huang, Fuchun, and Yosihiko Ogata. 1999. “Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models.” Journal of Computational and Graphical Statistics 8 (3): 510–30. https://doi.org/10.1080/10618600.1999.10474829.

Liu, Chong, Surajit Ray, and Giles Hooker. 2014. “Functional Principal Components Analysis of Spatially Correlated Data,” November. http://arxiv.org/abs/1411.4681.

Lovelace, Robin, Jakub Nowosad, and Jannes Münchow. 2019. Geocomputation with R. Boca Raton: Taylor & Francis, a CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa, plc.

Mackay, David J. C. 1995. “Probable Networks and Plausible Predictions — a Review of Practical Bayesian Methods for Supervised Neural Networks.” Network: Computation in Neural Systems 6 (3): 469–505. https://doi.org/10.1088/0954-898X_6_3_011.

Mardia, K. V., and R. J. Marshall. 1984. “Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression.” Biometrika 71 (1): 135–46. https://doi.org/10.1093/biomet/71.1.135.

Mohler, George. 2013. “Modeling and Estimation of Multi-Source Clustering in Crime and Security Data.” The Annals of Applied Statistics 7 (3): 1525–39. https://doi.org/10.1214/13-AOAS647.

Møller, Jesper, and Giovanni Luca Torrisi. 2007. “The Pair Correlation Function of Spatial Hawkes Processes.” Statistics & Probability Letters 77 (10): 995–1003. https://doi.org/10.1016/j.spl.2007.01.007.

Nowak, W., and A. Litvinenko. 2013. “Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques.” Mathematical Geosciences 45 (4): 411–35. https://doi.org/10.1007/s11004-013-9453-6.

Pollard, Dave. 2004. “Hammersley-Clifford Theorem for Markov Random Fields.”

Possolo, Antonio. 1986. “Estimation of Binary Markov Random Fields.” Department of StatisticsPreprints, University of Washington, Seattle. http://www.stat.washington.edu/research/reports/1986/tr77.pdf.

Rey, Sergio J., and Luc Anselin. 2010. “PySAL: A Python Library of Spatial Analytical Methods.” In Handbook of Applied Spatial Analysis, 175–93. Springer. http://link.springer.com/chapter/10.1007/978-3-642-03647-7_11.

Richardson, Matthew, and Pedro Domingos. 2006. “Markov Logic Networks.” Machine Learning 62 (1-2): 107–36. http://link.springer.com/article/10.1007/s10994-006-5833-1.

Ripley, B. D. 1977. “Modelling Spatial Patterns.” Journal of the Royal Statistical Society. Series B (Methodological) 39 (2): 172–212. http://www.stat.colostate.edu/~bradb/files/brad.pdf.

Ripley, Brian D. 1988. Statistical Inference for Spatial Processes. Cambridge [England]; New York: Cambridge University Press.

Rosenberg, Michael S., and Corey Devin Anderson. 2011. “PASSaGE: Pattern Analysis, Spatial Statistics and Geographic Exegesis. Version 2: PASSaGE.” Methods in Ecology and Evolution 2 (3): 229–32. https://doi.org/10.1111/j.2041-210X.2010.00081.x.

Saichev, A., and D. Sornette. 2006. “Power Law Distribution of Seismic Rates: Theory and Data.” The European Physical Journal B 49 (3): 377–401. https://doi.org/10.1140/epjb/e2006-00075-3.

Saparin, Peter I, Wolfgang Gowin, Jürgen Kurths, and Dieter Felsenberg. 1998. “Quantification of Cancellous Bone Structure Using Symbolic Dynamics and Measures of Complexity.” Physical Review E 58 (5): 6449–59. https://doi.org/10.1103/PhysRevE.58.6449.

Stein, Michael L. 2008. “A Modeling Approach for Large Spatial Datasets.” Journal of the Korean Statistical Society 37 (1): 3–10. https://doi.org/10.1016/j.jkss.2007.09.001.

Stein, Michael L. 2005. “Space-Time Covariance Functions.” Journal of the American Statistical Association 100 (469): 310–21. https://doi.org/10.1198/016214504000000854.

Stein, Michael L., Zhiyi Chi, and Leah J. Welty. 2004. “Approximating Likelihoods for Large Spatial Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 66 (2): 275–96. https://doi.org/10.1046/j.1369-7412.2003.05512.x.

Sun, Ying, and Michael L. Stein. 2016. “Statistically and Computationally Efficient Estimating Equations for Large Spatial Datasets.” Journal of Computational and Graphical Statistics 25 (1): 187–208. https://doi.org/10.1080/10618600.2014.975230.

Whittle, P. 1954. “On Stationary Processes in the Plane.” Biometrika 41 (3/4): 434–49.