Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series

Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series





bayesian inference, count data, excess zeros, garma(p, q), influenza


Many studies have used extensions of ARMA models for the analysis of non-Gaussian time series. One of them is the Generalized Autoregressive Moving Average, GARMA, enabling the modeling of count time series with distributions such as Poisson. The GARMA class is being expanded to accommodate other distributions, aiming to capture the typical characteristics of count data, including under or overdispersion and excess zeros. This study aims to propose an approach based on the GARMA class in order to analyze count time series with excess zeros, assuming a time-varying zero-adjusted Poisson distribution. This approach allows for capturing serial correlation, forecasting the future values, and estimating the future probability of zeros. For inference, a Bayesian analysis was adopted using the Hamiltonian Monte Carlo (HMC) algorithm for sampling from the joint posterior distribution. We conducted a simulation study and presented an application to influenza mortality reported in Brazil. Our findings demonstrated the usefulness of the model in estimating the probability of non-occurrence and the number of counts in future periods.


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Author Biographies

Luiz Otávio de Oliveira Pala, Universidade Federal de Lavras

Prof. Dr., Department of Statistics, UFLA, Lavras, MG, Brazil and PhD student, Statistics and Agricultural Experimentation, UFLA, Lavras, MG, Brazil. luizpala@ufla.br

Thelma Sáfadi, Universidade Federal de Lavras

Prof. Dr., Department of Statistics, UFLA, Lavras, MG, Brazil. safadi@ufla.br


Alqawba, M., Diawara, N., & Chaganty, N. (2019). Zero-inflated count time series models using Gaussian copula. Sequential analysis, 38(3), 342–357. DOI: https://doi.org/10.1080/07474946.2019.1648922

Andrade, B., Andrade, M., & Ehlers, R. (2015). Bayesian GARMA models for count data. Communications in statistics: case studies, data analysis and applications, 1(4), 192–205. DOI: https://doi.org/10.1080/23737484.2016.1190307

Aragaw, A., Azene, A., & Workie, M. (2022). Poisson logit hurdle model with associated factors of perinatal mortality in Ethiopia. Journal of big data, 9(16), 1–11. DOI: https://doi.org/10.1186/s40537-022-00567-6

Barreto-Souza, W. (2017). Mixed Poisson INAR(1) processes. Statistical papers, 60, 2119–2139. DOI: https://doi.org/10.1007/s00362-017-0912-x

Benjamin, M., Rigby, R., & Stasinopoulos, M. (2003). Generalized Autoregressive Moving Average Models. Journal of the American statistical association, 98(461), 214–223. DOI: https://doi.org/10.1198/016214503388619238

Bertoli, W., Conceição, K. S., Andrade, M. G., & Louzada, F. (2021). A New Regression Model for the Analysis of Overdispersed and Zero-Modified Count Data. Entropy, 23(646), 1–25. DOI: https://doi.org/10.3390/e23060646

Box, G., & Jenkins, G. (1976). Time series analysis, forecasting and control. Holden-Day.

Box, G., & Pierce, D. (1970). Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models. Journal of the American statistical association, 65(332), 1509–1526. DOI: https://doi.org/10.1080/01621459.1970.10481180

Briet, O., Amerasinghe, P., & Vounatsou, P. (2013). Generalized Seasonal Autoregressive Integrated Moving Average Models for Count Data with Application to Malaria Time Series with Low Case Numbers. Plos one, 8(6), 1–9. DOI: https://doi.org/10.1371/journal.pone.0065761

Broemeling, L. (2019). Bayesian analysis of time series. CRC Press. DOI: https://doi.org/10.1201/9780429488443

Burda, M., & Maheu, J. (2013). Bayesian adaptive Hamiltonian Monte Carlo with an application to highdimensional BEKK GARCH models. Studies in nonlinear dynamics and econometrics, 17, 345–372. DOI: https://doi.org/10.1515/snde-2013-0020

Canova, F., & Hansen, B. (1995). Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability. Journal of business & economic statistics, 13(3), 237–252. DOI: https://doi.org/10.1080/07350015.1995.10524598

Conceição, K., Suzuki, A., & Andrade, M. (2021). A Bayesian approach for zero-modified Skellam model with Hamiltonian MCMC. Statistical methods & applications, 30(2), 747–765. DOI: https://doi.org/10.1007/s10260-020-00541-7

Cox, D., & Stuart, A. (1955). Some Quick Sign Tests for Trend in Location and Dispersion. Biometrika, 42(1), 80–95. DOI: https://doi.org/10.1093/biomet/42.1-2.80

Davis, R., Fokianos, K., Holan, S., Joe, H., Livsey, J., Lund, R., Pipiras, V., & Ravishanker, N. (2021). Count time series: A methodological review. Journal of the American statistical association, 116(55), 1533–1547. DOI: https://doi.org/10.1080/01621459.2021.1904957

Duane, S., Kennedy, A., Pendleton, B., & Roweth, D. (1987). Hybrid Monte Carlo. Physics letters B, 195(2), 216–222. DOI: https://doi.org/10.1016/0370-2693(87)91197-X

Dunn, P., & Smyth, G. (1996). Randomized Quantile Residuals. Journal of computational and graphical statistics, 5(3), 236–244. DOI: https://doi.org/10.1080/10618600.1996.10474708

Ehlers, R. (2019). A Conway-Maxwell-Poisson GARMA Model for Count Data. Arxiv, 1–11.

Feng, C. (2021). A comparison of zero-inflated and hurdle models for modeling zero-inflated count data. Journal of statistical distributions and applications, 8(1), 1–19. DOI: https://doi.org/10.1186/s40488-021-00121-4

Gelman, A., Carlin, J., Stern, H., & Rubin, D. (2014). Bayesian data analysis. DOI: https://doi.org/10.1201/b16018

Ghahramani, M., & White, S. (2020). Time Series Regression for Zero-Inflated and Overdispersed Count Data: A Functional Response Model Approach. Journal of statistical theory and practice, 14(2), 1–18. DOI: https://doi.org/10.1007/s42519-020-00094-8

Hashim, L., Hashim, K., & Shiker, M. (2021). An Application Comparison of Two Poisson Models on Zero Count Data. Journal of physics, 1818. DOI: https://doi.org/10.1088/1742-6596/1818/1/012165

Hoffman, M., & Gelman, A. (2014). The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. Journal of machine learning research, 15(1), 1593–1623.

Khandelwal, I., Adhikari, R., & Verma, G. (2015). Time series forecasting using hybrid ARIMA and ANN models based on DWT decomposition. Procedia computer science, 48, 173–179. DOI: https://doi.org/10.1016/j.procs.2015.04.167

Maiti, R., Biswas, A., & Chakraborty, B. (2018). Modelling of low count heavy tailed time series data consisting large number of zeros and ones. Statistical Methods & Applications, 27(3), 407–435. DOI: https://doi.org/10.1007/s10260-017-0413-z

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Chapman & Hall. DOI: https://doi.org/10.1201/9780429029608

Melo, M., & Alencar, A. (2020). Conway–Maxwell–Poisson Autoregressive Moving Average Model for Equidispersed, Underdispersed, and Overdispersed Count Data. Journal of time series analysis, 41(6), 830–857. DOI: https://doi.org/10.1111/jtsa.12550

Mikis Stasinopoulos and Bob Rigby and Paul Eilers. (2016). Gamlss.util: Gamlss utilities [R package version 4.3-4].

Ministério da Saúde. (2022). Morbidade hospitalar do Sistema Único de Saúde.

Neal, R. (2011). MCMC using Hamiltonian dynamics. In S. Brooks, A. Gelman, G. Jones, & X. Meng (Eds.), Handbook of markov chain monte carlo (pp. 1–51). Chapman & Hall. DOI: https://doi.org/10.1201/b10905-6

Payne, E., Hardin, J., Egede, L., Ramakrishnan, V., Selassie, A., & Gebregziabher, M. (2017). Approaches for dealing with various sources of overdispersion in modeling count data: Scale adjustment versus modeling. Statistical methods in medical research, 26(4), 1802–1823. DOI: https://doi.org/10.1177/0962280215588569

R Core Team. (2022). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria.

Rigby, R., Stasinopoulos, M., Heller, G., & Bastiani, F. (2019). Distributions for modeling location, scale, and shape: using GAMLSS in R. Chapman & Hall. DOI: https://doi.org/10.1201/9780429298547

Sáfadi, T., & Morettin, P. (2003). A Bayesian analysis of autoregressive models with random normal coefficients. Journal of statistical computation and simulation, 73(8), 563–573. DOI: https://doi.org/10.1080/0094965031000136003

Sales, L., Alencar, A., & Ho, L. (2022). The BerG generalized autoregressive moving average model for count time series. Computers & industrial engineering, 168, 1–13. DOI: https://doi.org/10.1016/j.cie.2022.108104

Sathish, V., Mukhopadhyay, S., & Tiwari, R. (2021). Autoregressive and moving average models for zero-inflated count time series. Statistica Neerlandica, 76(2), 1–23. DOI: https://doi.org/10.1111/stan.12255

Silva, R. (2020). Generalized Autoregressive Neural Network Models. ArXiv, 1, -11.

Stan Development Team. (2022). RStan: the R interface to Stan [R package version 2.26.13].

Stasinopoulos, M., & Rigby, R. (2020). Distributions for Generalized Additive Models for Location Scale and Shape [R package version 5.1-6].

Tawiah, K., Iddrisu, W., & Asosega, K. (2021). Zero-Inflated Time Series Modelling of COVID-19 Deaths in Ghana. Journal of environmental and public health, 1–9. DOI: https://doi.org/10.1155/2021/5543977

Tian, G., Liu, Y., Tang, M., & Jiang, X. (2018). Type I multivariate zero-truncated/adjusted Poisson distributions with applications. Journal of computational and applied mathematics, 344, 132–153. DOI: https://doi.org/10.1016/j.cam.2018.05.014

Zuur, A., Ieno, E., Walker, N., Saveliev, A., & Smith, G. (2009). Mixed effects models and extensions in ecology with R. Springer. DOI: https://doi.org/10.1007/978-0-387-87458-6




How to Cite

Pala, L. O. de O., & Sáfadi, T. (2024). Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series. Semina: Ciências Exatas E Tecnológicas, 45, e–49943. https://doi.org/10.5433/1679-0375.2024.v45.49943