# Spatial point process and their statistics

The intersection of point processes and spatial processes. Popular in, e.g, earthquake modelling.

The references here are (Daley and Vere-Jones 2003, 2008; ???), especially the latter.

Here is a non-exhaustive taxonomy of ones of interest to me.

## Gibbs point processes

Processes where the points themselves notionally interact.

Processes with undirected interactions and no time index. Such processes without an arrow of time arise naturally, say, where you observe only snapshots of the dynamics, (where you captured those koalas that one time you got fieldwork funding) aggregates of the dynamics (as with classical statmech) or where whatever dynamics that gave rise to the process were too slow to be considered as anything but static on the time-scale you can model (locations of trees).

This is fiddly because with only undirected interactions, inference for these processes is painful, because you have to, “regress everything against everything else” Gibbs ensemble-style, which is slow and boring, which is the general problem with Markov random fields.

It is, however an interesting introduction to statistical mechanics]. In the Gibbs interaction process example (Daley and Vere-Jones 2003 Example 5.3c) they explains to probabilists what physicists mean by “grand canonical ensembles” in under two pages, which is something I have not seen a physicist do yet. (Daley and Vere-Jones 2008 Ch 15) gives an overarching view of the tools of this trade such Papengelou intensities and pseudolikelihoods as does (Møller and Waagepetersen 2007) with some other connections to physics.

## Determinantal point processes

Determinantal point processes are a tractable special case of Gibbs point processes. They have their own notebook.

## Cox processes

A.k.a. doubly stochastic point processes. Processes where the latent Poisson intensity is not dependent upon interactions between the points but around some latent field which is itself stochastic.

Typically this a latent gaussian field transformed by some link function into a positive Poisson intensity,generalized linear models-style. I am sure other models are possible but I have not seen them.

### Log Gaussian point process

Cox process using the classic exponential link function.

## Permanental point processes

Permanental point processes assume a squared gaussian latent intensity, supposedly?

## Spatio-temporal point processes

Some of the problems with Gibbs point processes seem to me artificial. Everything we witness is the result of dynamics occurring in time, or we’d still be at the big bang. So any non-causal Markov random field we sample is presumably sampled from a causal process, which means that the interactions are directed over time. Indeed, e.g. Markov Chain Monte Carlo sampling works makes it easier to sample from weird random fields by constructing a miniature physics with an artificial time arrow to sample from possible configurations of random processes. Intuitively this makes sense. Particles, at least in classical physics, don’t teleport themselves into a particular configuration with a probability inversely proportional to the exponential of some potential energy field; rather they have have particular trajectory that got them there. However, maybe the time-wise dynamics are even more complex and we wish to average over all possible such configurations or something. Welcome to statistical mechanics!

Anyway, practicalities.

## Statistical theory

### Unconditional intensity estimation via pseudolikelihood

(van Lieshout 2011) discusses two major intensity estimators – KDEs and Delaunay tessellation-based local estimates. (Møller and Waagepetersen 2007) discuss intensity estimation questions generally. (Rathbun and Cressie 1994) gives inhomogeneous Poisson asymptotics.

From (Adrian Baddeley and Turner 2000):

Originally Besag (Besag 1977, 1975) defined the pseudolikelihood of a finite set of random variables $$X_1, \dots , X_n$$ as the product of the conditional likelihoods of each $$X_i$$ given the other variables $$\{X_j , j \neq i\}$$. This was extended ([(Besag 1977, 1975); Besag et al., 1982) to point processes, for which it can be viewed as an infinite product of infinitesimal conditional probabilities.

### Model estimation via GLM

Berman and Turner show how how to use quadrature and generalised linear models to fit models without a form for the integrated influence kernel (which I don’t care about) using a linear model (which I do).

From the spatstat documentation:

Models are currently fitted by the method of maximum pseudolikelihood, using a computational device developed by Berman and Turner (Berman and Turner 1992) which we adapted to pseudolikelihoods in (Adrian Baddeley and Turner 2000). it has the virtue that we can implement it in software with great generality. Disadvantages of maximum pseudolikelihood (MPL) include its small-sample bias and inefficiency (Jensen and Møller 1991; Besag 1975; Jensen and Künsch 1994) relative to maximum likelihood estimators (MLE).

## Implementations of methods

Hmm.

Presumably other spatial statistical software?

Baddeley, Adrian. 2007. “Spatial Point Processes and Their Applications.” In Stochastic Geometry, edited by Wolfgang Weil, 1–75. Lecture Notes in Mathematics 1892. Springer Berlin Heidelberg. http://ahvaz.ist.unomaha.edu/azad/temp/sac/07-baddeley-point-process-poisson-coverage-sensor-simulation.pdf.

Baddeley, Adrian, Pablo Gregori, Jorge Mateu, Radu Stoica, and Dietrich Stoyan. 2006. Case Studies in Spatial Point Process Modeling. Vol. 185. Springer. http://link.springer.com/content/pdf/10.1007/0-387-31144-0.pdf.

Baddeley, Adrian J, Jesper Møller, and Rasmus Plenge Waagepetersen. 2000. “Non- and Semi-Parametric Estimation of Interaction in Inhomogeneous Point Patterns.” Statistica Neerlandica 54 (3): 329–50. https://doi.org/10.1111/1467-9574.00144.

Baddeley, Adrian, and Jesper Møller. 1989. “Nearest-Neighbour Markov Point Processes and Random Sets.” International Statistical Review / Revue Internationale de Statistique 57 (2): 89–121. https://doi.org/10.2307/1403381.

Baddeley, Adrian, Jesper Møller, and Anthony G. Pakes. 2008. “Properties of Residuals for Spatial Point Processes.” Annals of the Institute of Statistical Mathematics 60 (3): 627–49. http://link.springer.com/article/10.1007/s10463-007-0116-6.

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, Adrian, and Rolf Turner. 2000. “Practical Maximum Pseudolikelihood for Spatial Point Patterns.” Australian & New Zealand Journal of Statistics 42 (3): 283–322. https://doi.org/10.1111/1467-842X.00128.

———. 2006. “Modelling Spatial Point Patterns in R.” In Case Studies in Spatial Point Process Modeling, edited by Adrian Baddeley, Pablo Gregori, Jorge Mateu, Radu Stoica, and Dietrich Stoyan, 23–74. Lecture Notes in Statistics 185. Springer New York. http://link.springer.com/chapter/10.1007/0-387-31144-0_2.

Baddeley, A. J., and Marie-Colette NM Van Lieshout. 1995. “Area-Interaction Point Processes.” Annals of the Institute of Statistical Mathematics 47 (4): 601–19. https://doi.org/10.1007/BF01856536.

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.

Berman, Mark, and Peter Diggle. 1989. “Estimating Weighted Integrals of the Second-Order Intensity of a Spatial Point Process.” Journal of the Royal Statistical Society. Series B (Methodological) 51 (1): 81–92. https://publications.csiro.au/rpr/pub?list=BRO&pid=procite:d5b7ecd7-435c-4dab-9063-f1cf2fbdf4cb.

Berman, Mark, and T. Rolf Turner. 1992. “Approximating Point Process Likelihoods with GLIM.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 41 (1): 31–38. https://doi.org/10.2307/2347614.

Besag, Julian. 1975. “Statistical Analysis of Non-Lattice Data.” Journal of the Royal Statistical Society. Series D (the Statistician) 24 (3): 179–95. https://doi.org/10.2307/2987782.

———. 1977. “Efficiency of Pseudolikelihood Estimation for Simple Gaussian Fields.” Biometrika 64 (3): 616–18. https://doi.org/10.2307/2345341.

Bordenave, Charles, and Giovanni Luca Torrisi. 2007. “Large Deviations of Poisson Cluster Processes.” Stochastic Models 23 (4): 593–625. https://doi.org/10.1080/15326340701645959.

Cronie, O., and M. N. M. van Lieshout. 2016. “Bandwidth Selection for Kernel Estimators of the Spatial Intensity Function,” November. http://arxiv.org/abs/1611.10221.

Cucala, Lionel. 2008. “Intensity Estimation for Spatial Point Processes Observed with Noise.” Scandinavian Journal of Statistics 35 (2): 322–34. https://doi.org/10.1111/j.1467-9469.2007.00583.x.

Daley, Daryl J., and David Vere-Jones. 2003. An Introduction to the Theory of Point Processes. 2nd ed. Vol. 1. Elementary theory and methods. New York: Springer. http://ebooks.springerlink.com/UrlApi.aspx?action=summary&v=1&bookid=108085.

———. 2008. An Introduction to the Theory of Point Processes. 2nd ed. Vol. 2. General theory and structure. Probability and Its Applications. New York: Springer. http://link.springer.com/chapter/10.1007/978-0-387-49835-5_7.

Geyer, Charles J., and Jesper Møller. 1994. “Simulation Procedures and Likelihood Inference for Spatial Point Processes.” Scandinavian Journal of Statistics, 359–73.

Häggström, Olle, Marie-Colette N. M. van Lieshout, and Jesper Møller. 1999. “Characterization Results and Markov Chain Monte Carlo Algorithms Including Exact Simulation for Some Spatial Point Processes.” Bernoulli 5 (4): 641–58. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1330.

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.

Jensen, Jens Ledet, and Hans R. Künsch. 1994. “On Asymptotic Normality of Pseudo Likelihood Estimates for Pairwise Interaction Processes.” Annals of the Institute of Statistical Mathematics 46 (3): 475–86. https://doi.org/10.1007/BF00773511.

Jensen, Jens Ledet, and Jesper Møller. 1991. “Pseudolikelihood for Exponential Family Models of Spatial Point Processes.” The Annals of Applied Probability 1 (3): 445–61. https://doi.org/10.1214/aoap/1177005877.

Kroese, Dirk P., and Zdravko I. Botev. 2013. “Spatial Process Generation,” August. http://arxiv.org/abs/1308.0399.

Lavancier, Frédéric, Jesper Møller, and Ege Rubak. 2015. “Determinantal Point Process Models and Statistical Inference.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (4): 853–77. https://doi.org/10.1111/rssb.12096.

Lieshout, Marie-Colette N. M. van. 2011. “On Estimation of the Intensity Function of a Point Process.” Methodology and Computing in Applied Probability 14 (3): 567–78. https://doi.org/10.1007/s11009-011-9244-9.

McCullagh, Peter, and Jesper Møller. 2006. “The Permanental Process.” Advances in Applied Probability 38 (4): 873–88. https://doi.org/10.1017/S0001867800001361.

Møller, Jesper, and Kasper K. Berthelsen. 2012. “Transforming Spatial Point Processes into Poisson Processes Using Random Superposition.” Advances in Applied Probability 44 (1): 42–62. https://doi.org/10.1239/aap/1331216644.

Møller, Jesper, and Rasmus Waagepetersen. 2017. “Some Recent Developments in Statistics for Spatial Point Patterns.” Annual Review of Statistics and Its Application 4 (1): 317–42. https://doi.org/10.1146/annurev-statistics-060116-054055.

Møller, Jesper, and Rasmus Plenge Waagepetersen. 2003. Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC. https://doi.org/10.1201/9780203496930.

———. 2007. “Modern Statistics for Spatial Point Processes.” Scandinavian Journal of Statistics 34 (4): 643–84. https://doi.org/10.1111/j.1467-9469.2007.00569.x.

Rathbun, Stephen L. 1996. “Asymptotic Properties of the Maximum Likelihood Estimator for Spatio-Temporal Point Processes.” Journal of Statistical Planning and Inference 51 (1): 55–74. https://doi.org/10.1016/0378-3758(95)00070-4.

Rathbun, Stephen L., and Noel Cressie. 1994. “Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes.” Advances in Applied Probability 26 (1): 122–54. https://doi.org/10.2307/1427583.

Ripley, B. D., and F. P. Kelly. 1977. “Markov Point Processes.” Journal of the London Mathematical Society s2-15 (1): 188–92. https://doi.org/10.1112/jlms/s2-15.1.188.

Schoenberg, Frederic. 1999. “Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications 81 (2): 155–64. https://doi.org/10.1016/S0304-4149(98)00098-2.

Schoenberg, Frederic Paik. 2004. “Testing Separability in Spatial-Temporal Marked Point Processes.” Biometrics 60 (2): 471–81. http://www.stat.ucla.edu/~frederic/papers/sep133.pdf.

———. 2005. “Consistent Parametric Estimation of the Intensity of a Spatial–Temporal Point Process.” Journal of Statistical Planning and Inference 128 (1): 79–93. https://doi.org/10.1016/j.jspi.2003.09.027.

Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press. http://www.sparseprocesses.org/sparseprocesses-123456.pdf.