Count time series models

June 4, 2015 — December 9, 2015

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.

1 Maximum processes

Series monotonic increasing at a decreasing rate? Perhaps you have a maximum process.

2 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.

3 GLM-type autoregressive

GLMs applied to time series. Fokianos et al.

4 Linear branching-type and self-decomposable

A.k.a. INAR(p), GINAR(p), INMA.

See Galton Watson processes and other branching processes.

5 Queeing models

See Queueing models.

6 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.

7 References

Al-Osh, and Alzaid. 1987. First-Order Integer-Valued Autoregressive (INAR(1)) Process.” Journal of Time Series Analysis.
Bretó, He, Ionides, et al. 2009. Time Series Analysis via Mechanistic Models.” The Annals of Applied Statistics.
Burridge. 2013a. Cascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [Math].
———. 2013b. Crossover Behavior in Driven Cascades.” Physical Review E.
Cui, and Lund. 2009. A New Look at Time Series of Counts.” Biometrika.
Drost, Akker, and 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).
Durbin, and Koopman. 1997. Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models.” Biometrika.
Ferland, Latour, and Oraichi. 2006. Integer-Valued GARCH Process.” Journal of Time Series Analysis.
Fokianos. 2011. Some Recent Progress in Count Time Series.” Statistics.
Freeland, and McCabe. 2004. Analysis of Low Count Time Series Data by Poisson Autoregression.” Journal of Time Series Analysis.
Karlis, and Mamode Khan. 2023. Models for Integer Data.” Annual Review of Statistics and Its Application.
Latour. 1998. Existence and Stochastic Structure of a Non-Negative Integer-Valued Autoregressive Process.” Journal of Time Series Analysis.
Liboschik, Fokianos, and Fried. 2015. Tscount: An R Package for Analysis of Count Time Series Following Generalized Linear Models.”
Lindsey. 1995. Fitting Parametric Counting Processes by Using Log-Linear Models.” Journal of the Royal Statistical Society. Series C (Applied Statistics).
McKenzie. 2003. Discrete Variate Time Series.” In Handbook of Statistics. Stochastic Processes: Modelling and Simulation.
Ogata, and Akaike. 1982. On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes.” Journal of the Royal Statistical Society, Series B.
Pnevmatikakis. 2017. Compressed Sensing and Optimal Denoising of Monotone Signals.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Schein, Wallach, and Zhou. 2016. Poisson-Gamma Dynamical Systems.” In Advances In Neural Information Processing Systems.
van de Geer. 1995. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.” The Annals of Statistics.
Weiß. 2008. Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis.
———. 2009. A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods.
Zeger. 1988. A Regression Model for Time Series of Counts.” Biometrika.