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

Processes without an arrow of time arise naturally, say, where you observe only snaphosts of the dynamics, (where you captured those koalas that 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 timescale you can model (locations of trees).

This is fiddly because without directed interaction over time, and where the points in your process interact in an undirected fashion, these processes a pain in the arse, 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. All the convenient features of point processes are dead to you.

In a sense, this is an artificial problem. 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 you sample is presumably sampled from a causal process. Indeed, e.g. Bayesian Monte Carlo sampling works by constructing a miniature physics with an artificial time arrow to sample from possible configurations of such random processes. Particles, at least in classical physics, don't just teleport themselves into a particular configuration with a probability inversely proportional to the exponential of some potential energy field. However, maybe the time-wise dynamics are even more complex and you wish to average over all possible such configurations or something. Welcome to statistical mechanics.

There is a broad and fairly deep introduction in the two-volume volume Daley and Jones epic. (DaVe03, DaVe08) By the way, DaVe03 Example 5.3(c) is a Gibbs interaction process and explains what the hell physicists mean with their “grand canonical ensembles” for proabilists in under two pages, which is much easier than sitting through a semester of thermodynamics. If you want to add phase transitions into the same formalism, see MøWa07. DaVe08 Ch 15 give an overarching view of Papengelou intensities and other tools for spatial point processes.

Anyway, practicalities.

## Statistical theory

Adrian Baddeley's course notes for spatial statistics using R are online.

### unconditional intensity estimation via pseudolikelihood

Lies11 discusses two major intensity estimators – KDEs and Delaunay tessellation-based local estimates. MøWa07 discuss intensity estimation questions generally. RaCr94 gives inhomogeneous Poisson asymptotics.

From BaTu00:

Originally Besag (Besa75, Besa77) defined the pseudolikelihood of a finite set of random variables as the product of the conditional likelihoods of each given the other variables . This was extended (Besa75, Besa77; 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 (BeTu92) which we adapted to pseudolikelihoods in Baddeley and Turner (BaTu00). […] 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 (Besa77; JeMø91; JeKü94_) relative to maximum likelihood estimators (MLE).

## Implementations of methods

Hmm.

## Refs

- DaVe03: Daryl J. Daley, David Vere-Jones (2003)
*An introduction to the theory of point processes*(Vol. 1. Elementary theory and methods). New York: Springer - DaVe08: Daryl J. Daley, David Vere-Jones (2008)
*An introduction to the theory of point processes*(Vol. 2. General theory and structure). New York: Springer - BeTu92: Mark Berman, T. Rolf Turner (1992) Approximating Point Process Likelihoods with GLIM.
*Journal of the Royal Statistical Society. Series C (Applied Statistics)*, 41(1), 31–38. DOI - BaVa95: A. J. Baddeley, Marie-Colette NM Van Lieshout (1995) Area-interaction point processes.
*Annals of the Institute of Statistical Mathematics*, 47(4), 601–619. DOI - RaCr94: Stephen L. Rathbun, Noel Cressie (1994) Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes.
*Advances in Applied Probability*, 26(1), 122–154. DOI - Rath96: Stephen L. Rathbun (1996) Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes.
*Journal of Statistical Planning and Inference*, 51(1), 55–74. DOI - CrLi16: O. Cronie, M. N. M. van Lieshout (2016) Bandwidth selection for kernel estimators of the spatial intensity function.
*ArXiv:1611.10221 [Stat]*. - BGMS06: Adrian Baddeley, Pablo Gregori, Jorge Mateu, Radu Stoica, Dietrich Stoyan (2006)
*Case studies in spatial point process modeling*(Vol. 185). Springer - HäLM99: Olle Häggström, Marie-Colette N. M. van Lieshout, Jesper Møller (1999) Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes.
*Bernoulli*, 5(4), 641–658. - Scho05: Frederic Paik Schoenberg (2005) Consistent parametric estimation of the intensity of a spatial–temporal point process.
*Journal of Statistical Planning and Inference*, 128(1), 79–93. DOI - Besa77: Julian Besag (1977) Efficiency of Pseudolikelihood Estimation for Simple Gaussian Fields.
*Biometrika*, 64(3), 616–618. DOI - BeDi89: Mark Berman, 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. - HuOg99: Fuchun Huang, Yosihiko Ogata (1999) Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models.
*Journal of Computational and Graphical Statistics*, 8(3), 510–530. DOI - Cuca08: Lionel Cucala (2008) Intensity Estimation for Spatial Point Processes Observed with Noise.
*Scandinavian Journal of Statistics*, 35(2), 322–334. DOI - BoTo07: Charles Bordenave, Giovanni Luca Torrisi (2007) Large Deviations of Poisson Cluster Processes.
*Stochastic Models*, 23(4), 593–625. DOI - RiKe77: B. D. Ripley, F. P. Kelly (1977) Markov Point Processes.
*Journal of the London Mathematical Society*, s2-15(1), 188–192. DOI - BaTu06: Adrian Baddeley, Rolf Turner (2006) Modelling Spatial Point Patterns in R. In Case Studies in Spatial Point Process Modeling (pp. 23–74). Springer New York
- MøWa07: Jesper Møller, Rasmus P. Waagepetersen (2007) Modern Statistics for Spatial Point Processes.
*Scandinavian Journal of Statistics*, 34(4), 643–684. DOI - BaMø89: Adrian Baddeley, Jesper Møller (1989) Nearest-Neighbour Markov Point Processes and Random Sets.
*International Statistical Review / Revue Internationale de Statistique*, 57(2), 89–121. DOI - BaMW00: A. J. Baddeley, J. Møller, R. Waagepetersen (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns.
*Statistica Neerlandica*, 54(3), 329–350. DOI - JeKü94: Jens Ledet Jensen, Hans R. Künsch (1994) On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes.
*Annals of the Institute of Statistical Mathematics*, 46(3), 475–486. DOI - Lies11: Marie-Colette N. M. van Lieshout (2011) On Estimation of the Intensity Function of a Point Process.
*Methodology and Computing in Applied Probability*, 14(3), 567–578. DOI - BaTu00: Adrian Baddeley, Rolf Turner (2000) Practical Maximum Pseudolikelihood for Spatial Point Patterns.
*Australian & New Zealand Journal of Statistics*, 42(3), 283–322. DOI - BaMP08: Adrian Baddeley, Jesper Møller, Anthony G. Pakes (2008) Properties of residuals for spatial point processes.
*Annals of the Institute of Statistical Mathematics*, 60(3), 627–649. - JeMø91: Jens Ledet Jensen, Jesper Møller (1991) Pseudolikelihood for Exponential Family Models of Spatial Point Processes.
*The Annals of Applied Probability*, 1(3), 445–461. DOI - BTMH05: A. Baddeley, R. Turner, J. Møller, M. Hazelton (2005) Residual analysis for spatial point processes (with discussion).
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 67(5), 617–666. DOI - GeMø94: Charles J. Geyer, Jesper Møller (1994) Simulation procedures and likelihood inference for spatial point processes.
*Scandinavian Journal of Statistics*, 359–373. - MøWa16: Jesper Møller, Rasmus Waagepetersen (2016) Some recent developments in statistics for spatial point patterns.
- ACCG00: Luc Anselin, Jacqueline Cohen, David Cook, Wilpen Gorr, George Tita (2000) Spatial analyses of crime.
- BaRT16: Adrian Baddeley, Ege Rubak, Rolf Turner (2016)
*Spatial point patterns: methodology and applications with R*. Boca Raton ; London ; New York: CRC Press, Taylor & Francis Group - Badd07: Adrian Baddeley (2007) Spatial Point Processes and their Applications. In Stochastic Geometry (pp. 1–75). Springer Berlin Heidelberg
- KrBo13: Dirk P. Kroese, Zdravko I. Botev (2013) Spatial process generation.
*ArXiv:1308.0399 [Stat]*. - Besa75: Julian Besag (1975) Statistical Analysis of Non-Lattice Data.
*Journal of the Royal Statistical Society. Series D (The Statistician)*, 24(3), 179–195. DOI - Scho04: Frederic Paik Schoenberg (2004) Testing Separability in Spatial-Temporal Marked Point Processes.
*Biometrics*, 60(2), 471–481. - BBVK02: ERVRL Brown, Riccardo Barbieri, Valérie Ventura, R. Kass, L. Frank (2002) The time-rescaling theorem and its application to neural spike train data analysis.
*Neural Computation*, 14(2), 325–346. DOI - Scho99: Frederic Schoenberg (1999) Transforming spatial point processes into Poisson processes.
*Stochastic Processes and Their Applications*, 81(2), 155–164. DOI - MøBe12: Jesper Møller, Kasper K. Berthelsen (2012) Transforming spatial point processes into Poisson processes using random superposition.
*Advances in Applied Probability*, 44(1), 42–62. DOI