Statistical models for timer series with discrete time index and discrete state index.

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.

## Other

Non-linear processes, arbitrary interaction dynamics, Turing machines, discretised continuous processes…

Todo: get a handle on Twitter’s Robust anomaly detection

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.

## Refs

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- Mcke03: (2003) Discrete variate time series. In Handbook of Statistics (Vol. 21, pp. 573–606). Elsevier
- DrAW09: (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), 467–485. DOI - Lato98: (1998) Existence and Stochastic Structure of a Non-negative Integer-valued Autoregressive Process.
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*The Annals of Statistics*, 23(5), 1779–1801. DOI - AlAl87: (1987) First-Order Integer-Valued Autoregressive (INAR(1)) Process.
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*Journal of the Royal Statistical Society. Series C (Applied Statistics)*, 44(2), 201–212. DOI - FeLO06: (2006) Integer-Valued GARCH Process.
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*Biometrika*, 84(3), 669–684. DOI - OgAk82: (1982) On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes.
*Journal of the Royal Statistical Society, Series B*, 44, 269–274. DOI - ScWZ16: (2016) Poisson-Gamma dynamical systems. In Advances In Neural Information Processing Systems (pp. 5006–5014).
- Foki11: (2011) Some recent progress in count time series.
*Statistics*, 45(1), 49–58. DOI - Weiß08: (2008) Thinning operations for modeling time series of counts—a survey.
*Advances in Statistical Analysis*, 92(3), 319–341. DOI - BHIK09: (2009) Time series analysis via mechanistic models.
*The Annals of Applied Statistics*, 3(1), 319–348. DOI - LiFF15: (2015) tscount: An R package for analysis of count time series following generalized linear models. DOI