The Living Thing / Notebooks : Count time series models

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

CuLu09:

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

AlAl87
Al-Osh, M. A., & Alzaid, A. A.(1987) First-Order Integer-Valued Autoregressive (INAR(1)) Process. Journal of Time Series Analysis, 8(3), 261–275. DOI.
BHIK09
Bretó, C., He, D., Ionides, E. L., & King, A. A.(2009) Time series analysis via mechanistic models. The Annals of Applied Statistics, 3(1), 319–348. DOI.
CuLu09
Cui, Y., & Lund, R. (2009) A new look at time series of counts. Biometrika, 96(4), 781–792. DOI.
DrAW09
Drost, F. C., Akker, R. van den, & Werker, B. J. M.(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.
DuKo97
Durbin, J., & Koopman, S. J.(1997) Monte Carlo maximum likelihood estimation for non-Gaussian state space models. Biometrika, 84(3), 669–684. DOI.
Foki11
Fokianos, K. (2011) Some recent progress in count time series. Statistics, 45(1), 49–58. DOI.
FrMc04
Freeland, R. K., & McCabe, B. P. M.(2004) Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis, 25(5), 701–722. DOI.
Free80
Freeman, G. H.(1980) Fitting Two-Parameter Discrete Distributions to Many Data Sets with One Common Parameter. Journal of the Royal Statistical Society. Series C (Applied Statistics), 29(3), 259–267. DOI.
FuBa02
Fukasawa, T., & Basawa, I. V.(2002) Estimation for a class of generalized state-space time series models. Statistics & Probability Letters, 60(4), 459–473. DOI.
Lato98
Latour, A. (1998) Existence and Stochastic Structure of a Non-negative Integer-valued Autoregressive Process. Journal of Time Series Analysis, 19(4), 439–455. DOI.
Lind95
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), 201–212. DOI.
Mcke03
McKenzie, E. (2003) Discrete variate time series. In B.-H. of Statistics (Ed.), Handbook of Statistics, C. Raoand D. Shanbhag, Eds., ElsevierScience, Amsterdam, 573–606. MR1973555 (Vol. 21, pp. 573–606). Elsevier
OgAk82
Ogata, Y., & Akaike, H. (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.
RiNB12
Ristić, M. M., Nastić, A. S., & Bakouch, H. S.(2012) Estimation in an Integer-Valued Autoregressive Process with Negative Binomial Marginals (NBINAR(1)). Communications in Statistics - Theory and Methods, 41(4), 606–618. DOI.
SiSi06
Silva, I., & Silva, M. E.(2006) Asymptotic distribution of the Yule–Walker estimator for INAR processes. Statistics & Probability Letters, 76(15), 1655–1663. DOI.
ViVS10
Villarini, G., Vecchi, G. A., & Smith, J. A.(2010) Modeling the Dependence of Tropical Storm Counts in the North Atlantic Basin on Climate Indices. Monthly Weather Review, 138(7), 2681–2705. DOI.
Weiß08
Weiß, C. H.(2008) Thinning operations for modeling time series of counts—a survey. AStA Advances in Statistical Analysis, 92(3), 319–341. DOI.
Weiß09
Weiß, C. H.(2009) A New Class of Autoregressive Models for Time Series of Binomial Counts. Communications in Statistics - Theory and Methods, 38(4), 447–460. DOI.
Zege88
Zeger, S. L.(1988) A regression model for time series of counts. Biometrika, 75(4), 621–629. DOI.
ZhBD06
Zheng, H., Basawa, I. V., & Datta, S. (2006) Inference for pth-order random coefficient integer-valued autoregressive processes. Journal of Time Series Analysis, 27(3), 411–440. DOI.