Another current obsession, tentatively placemarked. Discrete-state random fields/processes with a continuous index. In general I also assume they are non-lattice and simple, which terms I will define if I need them. For now, see DaVe03.
The most interesting class for me are the branching processes.
I’ve just spent 6 months thinking about nothing else, so I won’t write much here.
- What is the most general point process that we can represent by a non-Homogeneous Poisson process? (This isn’t a research question, this is something I need to collate)
There is a comprehensive introduction in the 2 volume Daley and Jones epic. (DaVe03, DaVe08)
A curious thing is that much point process estimation theory concerns estimating statistics from a single realisation of the point process. But in fact you may have many point process realisations. This is not news per se, just a new emphasis.
Sometimes including spatiotemporal point processes, depending on mood.
In these, one has an arrow of time which simplifies things because you know that you “only need to consider the past of a process to understand its future”, which potentially simplifies many calculations about the conditional intensity processes; We consider only interactions from the past to the future, rather than some kind of mutual interaction.
In particular, for nice processes you can do fairly cheap likelihood calculations to estimate process parameters etc. I do a lot of this, for example, over at the branching processes notebook, and I have no use at the moment for other types of process, so I won’t say much about other cases for the moment.
See also change of time.
Processes without an arrow of time arise naturally, say as processes where you observe only snaphosts of the dynamics, or where whatever dynamics that gave rise to the process being too slow to be considered as anything but static (forests).
- Baddeley, A. J., Møller, J., & Waagepetersen, R. (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54(3), 329–350. DOI.
- Baddeley, A., & Møller, J. (1989) Nearest-Neighbour Markov Point Processes and Random Sets. International Statistical Review / Revue Internationale de Statistique, 57(2), 89–121. DOI.
- Baddeley, A., Møller, J., & Pakes, A. G.(2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics, 60(3), 627–649.
- Baddeley, A., & Turner, R. (2000) Practical Maximum Pseudolikelihood for Spatial Point Patterns. Australian & New Zealand Journal of Statistics, 42(3), 283–322. DOI.
- Berman, M., & Diggle, P. (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.
- Chen, L. H. Y., & Xia, A. (2011) Poisson process approximation for dependent superposition of point processes. Bernoulli, 17(2), 530–544. DOI.
- Cox, D. R.(1965) On the Estimation of the Intensity Function of a Stationary Point Process. Journal of the Royal Statistical Society. Series B (Methodological), 27(2), 332–337.
- Daley, D. J., & Vere-Jones, D. (2003) An introduction to the theory of point processes. (2nd ed., Vol. 1. Elementary theory and methods). New York: Springer
- Daley, D. J., & Vere-Jones, D. (2008) An introduction to the theory of point processes. (2nd ed., Vol. 2. General theory and structure). New York: Springer
- Diggle, P. (1985) A Kernel Method for Smoothing Point Process Data. Journal of the Royal Statistical Society. Series C (Applied Statistics), 34(2), 138–147. DOI.
- Feigin, P. D.(1976) Maximum Likelihood Estimation for Continuous-Time Stochastic Processes. Advances in Applied Probability, 8(4), 712–736. DOI.
- Gallo, S., & Leonardi, F. G.(2014) Nonparametric statistical inference for the context tree of a stationary ergodic process. arXiv:1411.7650 [Math, Stat].
- Giesecke, K., Kakavand, H., & Mousavi, M. (2008) Simulating point processes by intensity projection. In Simulation Conference, 2008. WSC 2008. Winter (pp. 560–568). DOI.
- Giesecke, K., Kakavand, H., & Mousavi, M. (2011) Exact Simulation of Point Processes with Stochastic Intensities. Operations Research, 59(5), 1233–1245. DOI.
- Giesecke, K., & Schwenkler, G. (2011) Filtered Likelihood for Point Processes (SSRN Scholarly Paper No. ID 1898344). . Rochester, NY: Social Science Research Network
- Hawkes, A. G.(1971a) Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological), 33(3), 438–443.
- Hawkes, A. G.(1971b) Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90. DOI.
- Huang, F., & Ogata, Y. (1999) Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models. Journal of Computational and Graphical Statistics, 8(3), 510–530. DOI.
- Jensen, J. L., & Møller, J. (1991) Pseudolikelihood for Exponential Family Models of Spatial Point Processes. The Annals of Applied Probability, 1(3), 445–461. DOI.
- Kvitkovičová, A., & Panaretos, V. M.(2011) Asymptotic inference for partially observed branching processes. Advances in Applied Probability, 43(4), 1166–1190. DOI.
- Møller, J., & Berthelsen, K. K.(2012) Transforming spatial point processes into Poisson processes using random superposition. Advances in Applied Probability, 44(1), 42–62. DOI.
- Møller, J., & Waagepetersen, R. P.(2007) Modern Statistics for Spatial Point Processes. Scandinavian Journal of Statistics, 34(4), 643–684. DOI.
- Ogata, Y. (1978) The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics, 30(1), 243–261. DOI.
- Ogata, Y. (1999) Seismicity analysis through point-process modeling: a review. Pure and Applied Geophysics, 155(2–4), 471–507. DOI.
- Ogata, Y., Matsu’ura, R. S., & Katsura, K. (1993) Fast likelihood computation of epidemic type aftershock-sequence model. Geophysical Research Letters, 20(19), 2143–2146. DOI.
- Ozaki, T. (1979) Maximum likelihood estimation of Hawkes’ self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31(1), 145–155. DOI.
- Panaretos, V. M., & Zemel, Y. (2016) Separation of Amplitude and Phase Variation in Point Processes. The Annals of Statistics, 44(2), 771–812. DOI.
- Rasmussen, J. G.(2011, January) Temporal point processes the conditional intensity function.
- Rasmussen, J. G.(2013) Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 15(3), 623–642. DOI.
- Ripley, B. D., & Kelly, F. P.(1977) Markov Point Processes. Journal of the London Mathematical Society, s2-15(1), 188–192. DOI.
- Rubin, I. (1972) Regular point processes and their detection. IEEE Transactions on Information Theory, 18(5), 547–557. DOI.
- Schoenberg, F. (1999) Transforming spatial point processes into Poisson processes. Stochastic Processes and Their Applications, 81(2), 155–164. DOI.
- Schoenberg, F. P.(2002) On Rescaled Poisson Processes and the Brownian Bridge. Annals of the Institute of Statistical Mathematics, 54(2), 445–457. DOI.
- Schoenberg, F. P.(2004) Testing Separability in Spatial-Temporal Marked Point Processes. Biometrics, 60(2), 471–481.
- Schoenberg, F. P.(2005) Consistent parametric estimation of the intensity of a spatial–temporal point process. Journal of Statistical Planning and Inference, 128(1), 79–93. DOI.
- Smith, A., & Brown, E. (2003) Estimating a state-space model from point process observations. Neural Computation, 15(5), 965–991. DOI.
- van Lieshout, M.-C. N. M.(2011) On Estimation of the Intensity Function of a Point Process. Methodology and Computing in Applied Probability, 14(3), 567–578. DOI.
- Vere-Jones, D., & Schoenberg, F. P.(2004) Rescaling Marked Point Processes. Australian & New Zealand Journal of Statistics, 46(1), 133–143. DOI.
- Wu, S., Müller, H.-G., & Zhang, Z. (2013) Functional Data Analysis for Point Processes with Rare Events. Statistica Sinica, 23(1), 1–23.
- Wu, W., & Srivastava, A. (2012) Estimating summary statistics in the spike-train space. Journal of Computational Neuroscience, 34(3), 391–410. DOI.
- Wu, W., & Srivastava, A. (2014) Analysis of spike train data: Alignment and comparisons using the extended Fisher-Rao metric. Electronic Journal of Statistics, 8(2), 1776–1785. DOI.