# 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

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

*Journal of Time Series Analysis*.

*The Annals of Applied Statistics*.

*arXiv:1304.3741 [Math]*.

*Physical Review E*.

*Biometrika*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Biometrika*.

*Journal of Time Series Analysis*.

*Statistics*.

*Journal of Time Series Analysis*.

*Annual Review of Statistics and Its Application*.

*Journal of Time Series Analysis*.

*Journal of the Royal Statistical Society. Series C (Applied Statistics)*.

*Handbook of Statistics*. Stochastic Processes: Modelling and Simulation.

*Journal of the Royal Statistical Society, Series B*.

*2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*.

*Advances In Neural Information Processing Systems*.

*The Annals of Statistics*.

*Advances in Statistical Analysis*.

*Communications in Statistics - Theory and Methods*.

*Biometrika*.