Statistical models for time series with discrete time index and discrete state index, i.e. lists of non-negative whole numbers with a causal ordering.
C&c symbolic dynamics, nonlinear time series wizardry, random fields, branching processes and Galton Watson processes for some important special cases. If there is no serial dependence, you might want unadorned count models.
Maximum processes
Series monotonic increasing at a decreasing rate? Perhaps you have a maximum process.
Finite state Markov chains
Often fit as if non-parametric, although there exist parametric transition tables if you’d like, and if you have a large state space you probably would like.
GLM-type autoregressive
GLMs applied to time series. Fokianos et al.
Linear branching-type and self-decomposable
A.k.a. INAR(p), GINAR(p), INMA.
See Galton Watson processes and other branching processes.
Queeing models
See Queueing models.
Other
Non-linear processes, arbitrary interaction dynamics, Turing machines, discretised continuous processes…
Todo: get a handle on Twitter’s Robust anomaly detection
(Cui and Lund 2009):
This paper proposes a simple new model for stationary time series of integer counts. Previous work has focused on thinning methods and classical time series autoregressive moving-average difference equations; in contrast, our methods use a renewal process to generate a correlated sequence of Bernoulli trials. By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric or any other discrete marginal distribution can be readily constructed. The model class proposed is parsimonious, non-Markov and readily generates series with either short- or long-memory autocovariances.
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Bretó, Carles, Daihai He, Edward L. Ionides, and Aaron A. King. 2009. “Time Series Analysis via Mechanistic Models.” The Annals of Applied Statistics 3 (1): 319–48. https://doi.org/10.1214/08-AOAS201.
Burridge, James. 2013a. “Cascade Sizes in a Branching Process with Gamma Distributed Generations,” April. http://arxiv.org/abs/1304.3741.
———. 2013b. “Crossover Behavior in Driven Cascades.” Physical Review E 88 (3): 032124. https://doi.org/10.1103/PhysRevE.88.032124.
Cui, Yunwei, and Robert Lund. 2009. “A New Look at Time Series of Counts.” Biometrika 96 (4): 781–92. https://doi.org/10.1093/biomet/asp057.
Drost, Feike C., Ramon van den Akker, and Bas J. M. Werker. 2009. “Efficient Estimation of Auto-Regression Parameters and Innovation Distributions for Semiparametric Integer-Valued AR(p) Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71 (2): 467–85. https://doi.org/10.1111/j.1467-9868.2008.00687.x.
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Fokianos, Konstantinos. 2011. “Some Recent Progress in Count Time Series.” Statistics 45 (1): 49–58. https://doi.org/10.1080/02331888.2010.541250.
Freeland, R. K., and B. P. M. McCabe. 2004. “Analysis of Low Count Time Series Data by Poisson Autoregression.” Journal of Time Series Analysis 25 (5): 701–22. https://doi.org/10.1111/j.1467-9892.2004.01885.x.
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Latour, Alain. 1998. “Existence and Stochastic Structure of a Non-Negative Integer-Valued Autoregressive Process.” Journal of Time Series Analysis 19 (4): 439–55. https://doi.org/10.1111/1467-9892.00102.
Liboschik, Tobias, Konstantinos Fokianos, and Roland Fried. 2015. “Tscount: An R Package for Analysis of Count Time Series Following Generalized Linear Models.” https://doi.org/10.17877/DE290R-7239.
Lindsey, J. K. 1995. “Fitting Parametric Counting Processes by Using Log-Linear Models.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 44 (2): 201–12. https://doi.org/10.2307/2986345.
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Ogata, Yosihiko, and Hirotugu Akaike. 1982. “On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes.” Journal of the Royal Statistical Society, Series B 44: 269–74. https://doi.org/10.1007/978-1-4612-1694-0_20.
Pnevmatikakis, Eftychios A. 2017. “Compressed Sensing and Optimal Denoising of Monotone Signals.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4740–4. https://doi.org/10.1109/ICASSP.2017.7953056.
Schein, Aaron, Hanna Wallach, and Mingyuan Zhou. 2016. “Poisson-Gamma Dynamical Systems.” In Advances in Neural Information Processing Systems, 5006–14. http://papers.nips.cc/paper/6082-poisson-gamma-dynamical-systems.
Weiß, Christian H. 2008. “Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis 92 (3): 319–41. https://doi.org/10.1007/s10182-008-0072-3.
———. 2009. “A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods 38 (4): 447–60. https://doi.org/10.1080/03610920802233937.
Zeger, Scott L. 1988. “A Regression Model for Time Series of Counts.” Biometrika 75 (4): 621–29. https://doi.org/10.1093/biomet/75.4.621.