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.
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.
References
Al-Osh, M. A., and A. A. Alzaid. 1987.
“First-Order Integer-Valued Autoregressive (INAR(1)) Process.” Journal of Time Series Analysis 8 (3): 261–75.
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.
———. 2013b.
“Crossover Behavior in Driven Cascades.” Physical Review E 88 (3): 032124.
Cui, Yunwei, and Robert Lund. 2009.
“A New Look at Time Series of Counts.” Biometrika 96 (4): 781–92.
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.
Ferland, René, Alain Latour, and Driss Oraichi. 2006.
“Integer-Valued GARCH Process.” Journal of Time Series Analysis 27 (6): 923–42.
Fokianos, Konstantinos. 2011.
“Some Recent Progress in Count Time Series.” Statistics 45 (1): 49–58.
Freeland, R. K., and B. P. M. McCabe. 2004.
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Lindsey, J. K. 1995.
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McKenzie, Eddie. 2003.
“Discrete Variate Time Series.” In
Handbook of Statistics, edited by c Raoand and d Shanbhag, 21:573–606. Stochastic Processes: Modelling and Simulation. Elsevier.
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.
Pnevmatikakis, Eftychios A. 2017.
“Compressed Sensing and Optimal Denoising of Monotone Signals.” In
2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4740–44.
Schein, Aaron, Hanna Wallach, and Mingyuan Zhou. 2016.
“Poisson-Gamma Dynamical Systems.” In
Advances In Neural Information Processing Systems, 5006–14.
Weiß, Christian H. 2008.
“Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis 92 (3): 319–41.
———. 2009.
“A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods 38 (4): 447–60.
Zeger, Scott L. 1988.
“A Regression Model for Time Series of Counts.” Biometrika 75 (4): 621–29.
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