Count time series with excess zeros: A Bayesian perspective using zero-adjusted distributions
DOI:
https://doi.org/10.5433/1679-0375.2022v43n2p147Keywords:
Processo ARMA(p, q), Dados de contagem, Metropolis-hastingsAbstract
Models for count data which are temporally correlated have been studied using many conditional distributions, such as the Poisson distribution, and the insertion of different dependence structures. Nonetheless, excess of zeros and over dispersion may be observed during the counting process and need to be considered when modelling and choosing a conditional distribution. In this paper, we propose models for counting time series using zero-adjusted distributions by inserting a dependence structure following the ARMA(p, q) process on a Bayesian framework. We perform a simulation study using the proposed Bayesian analysis and analyse the monthly time series of the number of deaths due to dengue haemorrhagic fever (ICD-A91) in Brazil.
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ANDRADE, B.; ANDRADE, M.; EHLERS, R. Bayesian GARMA models for count data. Communications in statistics: case studies, data analysis and applications, Philadelphia, v. 1, n. 4, p. 192-205, 2015. DOI: https://doi.org/10.1080/23737484.2016.1190307. DOI: https://doi.org/10.1080/23737484.2016.1190307
ALQAWBA, M.; DIAWARA, N.; CHAGANTY, N. Zero-inflated count time series models using Gaussian copula. Sequential Analysis, New York, v. 38, n. 3, p. 342-357, 2019. DOI: https://doi.org/10.1080/07474946.2019.1648922. DOI: https://doi.org/10.1080/07474946.2019.1648922
BARRETO-SOUZA, W. Mixed Poisson INAR(1) processes. Statistical papers, [London], v. 60, n. 6, p. 2119-2139, 2017. DOI: https://doi.org/ 10.1007/s00362-017-0912-x. DOI: https://doi.org/10.1007/s00362-017-0912-x
BENJAMIN, M.; RIGBY, R.; STASINOPOULOS, M. Generalized autoregressive moving average models. Journal of the American Statistical Association, Washington, v. 98, n. 461, p. 214-223, 2003. DOI: https://doi.org/10.1198/016214503388619238. DOI: https://doi.org/10.1198/016214503388619238
BOX, G.; PIERCE, D. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, Washington, v. 65, n. 332, p. 1509-1526, 1970. DOI: https://doi.org/10.2307/2284333. DOI: https://doi.org/10.1080/01621459.1970.10481180
BRASIL. Ministério da Saúde. Mortalidade - Brasil. Brasília: MS, [2022]. Available from: http://tabnet.datasus.gov.br/. Access in: Jan. 2022.
BROEK, J. A score test for zero inflation in a Poisson distribution. Biometrics, New York, n. 1, p. 738-743, 1995. DOI: https://doi.org/10.2307/2532959. DOI: https://doi.org/10.2307/2532959
BROEMELING, L. Bayesian analysis of time series. New York: CRC Press, 2019. DOI: https://doi.org/10.1201/9780429488443
CANOVA, F.; HANSEN, B. Are seasonal patterns constant over time? a test for seasonal stability. Journal of Business & Economic Statistics, Washington, v. 13, n. 3, p. 237-252, 1995. DOI: https://doi.org/10.2307/1392184. DOI: https://doi.org/10.1080/07350015.1995.10524598
CHIVERS, C. MHadaptive: General Markov chain Monte Carlo for Bayesian inference using adaptive Metropolis-Hastings sampling. 2015. Available from: https://CRAN.R-project.org/package=MHadaptive. Access in: Jan. 2022.
COX, D.; STUART, A. Some quick sign tests for trend in location and dispersion. Biometrika, Oxford, v. 42, n. 1/2, p. 80-95, 1955. DOI: https://doi.org/10.2307/2333424. DOI: https://doi.org/10.1093/biomet/42.1-2.80
DUANE, S.; KENNEDY, A.; PENDLETON, B.; ROWETH, D. Hybrid Monte Carlo. Physics letters B, Amsterdam, v. 195, n. 2, p. 216-222, 1987. DOI: https://doi.org/10.1016/0370-2693(87)91197-X. DOI: https://doi.org/10.1016/0370-2693(87)91197-X
FENG, C. A comparison of zero-inflated and hurdle models for modeling zero-inflated count data. Journal of Statistical Distributions and Applications, Heidelberg, v. 8, n. 1, p. 1-19, 2021. DOI: https://doi.org/10.1186/s40488-021-00121-4. DOI: https://doi.org/10.1186/s40488-021-00121-4
GEWEKE, J. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Staff Report, [s. l.], n. 148, p. 1-29, 1991. DOI: https://doi.org/10.21034/sr.148
GHAHRAMANI, M.; WHITE, S. Time series regression for zero-inflated and over dispersed count data: a functional response model approach. Journal of statistical theory and practice, Greensboro, v. 14, n. 2, p. 1-18, 2020. DOI: https://doi.org/10.1007/s42519-020-00094-8. DOI: https://doi.org/10.1007/s42519-020-00094-8
GONÇALVES, J.; BARRETO-SOUZA, W. Flexible regression models for counts with high-inflation of zeros. Metron, [s. l.], v. 78, n. 1, p. 71-95, 2020. DOI: https://doi.org/10.1007/s40300-020-00163-9. DOI: https://doi.org/10.1007/s40300-020-00163-9
HASHIM, L.; HASHIM, K.; SHIKER, M. An application comparison of two Poisson models on zero count data. Journal of Physics, [Bristol], v. 1818, n. 1, 2021. DOI: https://doi.org/10.1088/1742- 6596/1818/1/012165. DOI: https://doi.org/10.1088/1742-6596/1818/1/012165
HEIDELBERGER, P.; WELCH, P. Simulation run length control in the presence of an initial transient. Operations Research, Switzerland, v. 31, n. 6, p. 1109-1144, 1983. DOI: https://doi.org/0030- 364X/83/3106-1109 $01.25. DOI: https://doi.org/10.1287/opre.31.6.1109
HILBE, J. Modeling count data. Cambridge: Cambridge University Press, 2014. DOI: https://doi.org/10.1017/CBO9781139236065
KORENEV, B. Bessel functions and their applications. New York: CRC Press, 2002. DOI: https://doi.org/10.1201/b12551
LAMBERT, D. Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, Richmond v. 34, n. 1, p. 1-14, 1992. DOI: https://doi.org/10.2307/1269547. DOI: https://doi.org/10.2307/1269547
LÜDECKE, D.; BEN-SHACHAR, M.; PATIL, I.; WAGGONER, P.; MAKOWSKI, D. An R package for assessment, comparison and testing of statistical models. Journal of Open Source Software, Chicago, v. 6, n. 60, p. 31-39, 2021. DOI 10.21105/joss.03139. DOI: https://doi.org/10.21105/joss.03139
MULLAHY, J. Specification and testing of some modified count data models. Journal of Econometrics, Amsterdam, v. 33. n. 3, p. 341-365, 1986. DOI: https://doi.org/10.1016/0304-4076(86)90002-3. DOI: https://doi.org/10.1016/0304-4076(86)90002-3
NELDER, J.; WEDDERBURN, R. Generalized linear models. Journal of the Royal Statistical Society, London, v. 135, p. 370-384, 1972. DOI: https://doi.org/10.2307/2344614. DOI: https://doi.org/10.2307/2344614
PAYNE, E.; HARDIN, J.; EGEDE, L.; RAMAKRISH- NAN, V.;SELASSIE, A.; GEBREGZIABHER, M. Approaches for dealing with various sources of over dispersion in modeling count data: Scale adjustment versus modeling. Statistical Methods in Medical Research, Singapore v. 26, n. 4, p. 1802-1823, 2017. DOI: https://doi.org/10.1177/0962280215588569. DOI: https://doi.org/10.1177/0962280215588569
RAFTERY, A.; LEWIS, S. Comment: One long run with diagnostics: implementation strategies for Markov chain Monte Carlo. Statistical Science, [Commack], v. 7, n. 4, p. 493-497, 1992. DOI: https://doi.org/10.1214/ss/1177011143. DOI: https://doi.org/10.1214/ss/1177011143
RIGBY, R.; STASINOPOULOS, M.; HELLER, G.; DE BASTIANI, F. Distributions for modeling location, scale, and shape. New York: Chapman and Hall: CRC, 2019. DOI: https://doi.org/10.1201/9780429298547
ROBERT, C. Modified Bessel functions and their applications in probability and statistics. Statistics & Probability Letters, Amsterdam, v. 9, n. 2, p. 155-161, 1990. DOI: https://doi.org/10.1016/0167- 7152(92)90011-S. DOI: https://doi.org/10.1016/0167-7152(92)90011-S
ROCHA, A.; CRIBARI-NETO, F. Beta autoregressive moving average models. Test, [London], v. 18, n. 3, p. 529-545, 2009. DOI: https://doi.org/10.1007/s11749-008-0112-z. DOI: https://doi.org/10.1007/s11749-008-0112-z
SÁFADI, T.; MORETTIN, P. A Bayesian analysis of autoregressive models with random normal coefficients. Journal of Statistical Computation and Simulation, Philadelphia, v. 73, n. 8, p. 563-573, 2003. DOI: https://doi.org/10.1080/0094965031000136003. DOI: https://doi.org/10.1080/0094965031000136003
SATHISH, V.; MUKHOPADHYAY, S.; TIWARI, R. Autoregressive and moving average models for zero-inflated count time series. Statistica Neerlandica, Gravenhage, v. 76, n. 2, p. 190-218, 2021. DOI: https://doi.org/10.1111/stan.12255. DOI: https://doi.org/10.1111/stan.12255
STASINOPOULOS, M.; RIGBY, R. Gamlss dist: distributions for generalized additive models for location scale and shape. 2020. Available from: https://cran.r-project.org/web/packages/gamlss.dist/gamlss.dist.pdf. Access in: Jan. 2022.
TAWIAH, K.; IDDRISU, W.; ASOSEGA, K. Zero inflated time series modelling of COVID-19 deaths in Ghana. Journal of Environmental and Public health, New York, v. 2021, p. 1-9, 2021. DOI 10.1155/2021/5543977. DOI: https://doi.org/10.1155/2021/5543977
WORLD HEALTH ORGANIZATION. ICD-10: International statistical classification of dis- eases and related health problems. 2. nd. Geneva: WHO, 2004. Available from: https://apps.who.int/iris/handle/10665/42980. Access in: Nov. 2022.
YANG, M.; ZAMBA, G.; CAVANAUGH, J. Markov regression models for count time series with excess zeros: a partial likelihood approach. Statistical Methodology, [London], v. 14, p. 26-38, 2013. DOI: https://doi.org/10.1016/j.stamet.2013.02.001. DOI: https://doi.org/10.1016/j.stamet.2013.02.001
ZUO, G.; FU, K.; DAI, X.; ZHANG, L. Generalized poisson hurdle model for count data and its application in ear disease. Entropy, Basel, v. 23, n. 9, p. 1-16, 2021. DOI: https://doi.org/10.3390/e23091206. DOI: https://doi.org/10.3390/e23091206
ZUUR, A.; IENO, E.; WALKER, N.; SAVELIEV, A.; SMITH, G. Mixed effects models and extensions in ecology with R. New York: Springer, 2009. DOI: https://doi.org/10.1007/978-0-387-87458-6
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