Advertisement

Stochastic Environmental Research and Risk Assessment

, Volume 32, Issue 9, pp 2495–2514 | Cite as

A full ARMA model for counts with bounded support and its application to rainy-days time series

  • Sónia Gouveia
  • Tobias A. Möller
  • Christian H. Weiß
  • Manuel G. Scotto
Original Paper

Abstract

Motivated by a large dataset containing time series of weekly number of rainy days collected over two thousand locations across Europe and Russia for the period 2000–2010, we propose a new class of ARMA-like model for time series of bounded counts, which can also handle extra-binomial variation. We abbreviate this model as bvARMA, as it is based upon a novel operation referred to as binomial variation. After having discussed important stochastic properties and proposed a model-fitting approach relying on maximum likelihood estimation, we apply the bvARMA model family to the rainy-days time series. Results show that both bvAR and bvMA models are adequate and exhibit a similar performance. Furthermore, bvARMA results outperform those obtained by fitting ordinary discrete ARMA (NDARMA) models of the same order.

Keywords

Binomial variation Count time series ARMA structure Rainy-days time series 

Notes

Acknowledgements

The authors thank the Associate Editor and the referees for carefully reading the article and for their comments, which greatly improved the article.

Supplementary material

References

  1. Agnew MD, Goodess CM, Hemming D, Giannakopoulos C, Bindi M, Dibari C, El-Askary H, El-Hattab M, El-Raey M, Ferrise R, Harzallah A, Hatzak M, Kostopoulou E, Lionello P, Abed SS, Sánchez-Arcilla A, Senouci M, Sommer R, Zoheir Taleb M, Tanzarella A (2013) Stakeholders. In: Navarra A, Tubiana L (eds) Regional assessment of climate change in the mediterranean. Springer International Publishing, Berlin, pp 23–37CrossRefGoogle Scholar
  2. Berchtold A (2002) High-order extensions of the double chain Markov model. Stoch models 18:193–227CrossRefGoogle Scholar
  3. Blight PA (1989) Time series formed from the superposition of discrete renewal processes. J Appl Probab 26:189–195CrossRefGoogle Scholar
  4. Chang TJ, Kavvas ML, Delleur JW (1984) Modeling of sequences of wet and dry days by binary discrete autoregressive moving average processes. J Clim Appl Meteor 23:1367–1378CrossRefGoogle Scholar
  5. Cui Y, Lund R (2009) A new look at time series of counts. Biometrika 96:781–792CrossRefGoogle Scholar
  6. Deckmyn A, Minka TP, Brownrigg R, Becker RA, Wilks AR (2017) maps: draw geographical maps. R package version 320Google Scholar
  7. Delleur JW, Chang TJ, Kavvas ML (1989) Simulation models of sequences of dry and wet days. J Irrig Drain Eng 115:344–357CrossRefGoogle Scholar
  8. Ehelepola NBD, Ariyaratne K, Buddhadasa WMNP, Ratnayake S, Wickramasinghe M (2015) A study of the correlation between dengue and weather in Kandy City, Sri Lanka (2003–2012) and lessons learned. Infect Dis Poverty 4:42CrossRefGoogle Scholar
  9. Grunwald G, Hyndman RJ, Tedesco L, Tweedie RL (2000) Non-gaussian conditional linear AR(1) models. Aust Nz J Statist 42:479–495CrossRefGoogle Scholar
  10. Jacobs PA, Lewis PAW (1983) Stationary discrete autoregressive-moving average time series generated by mixtures. J Time Ser Anal 4:19–36CrossRefGoogle Scholar
  11. Khoo WC, Ong SH, Biswas A (2017) Modeling time series of counts with a new class of INAR(1) model. Stat Pap 58:393–416CrossRefGoogle Scholar
  12. Klein Tank AMG, Wijngaard JB, Können GP, Böhm R, Demarée G, Gocheva A, Mileta M, Pashiardis S, Hejkrlik L, Kern-Hansen C, Heino R, Bessemoulin P, Müller-Westermeier G, Tzanakou M, Szala S, Pálsdóttir T, Fitzgerald D, Rubin S, Capaldo M, Maugeri M, Leitass A, Bukantis A, Aberfeld R, van Engelen AFV, Forland E, Mietus M, Coelho F, Mares C, Razuvaev V, Nieplova E, Cegnar T, Antonio López J, Dahlström B, Moberg A, Kirchhofer W, Ceylan A, Pachaliuk O, Alexander LV, Petrovic P (2002) Daily dataset of 20th-century surface air temperature and precipitation series for the european climate assessment. Int J Climatol 22:1441–1453CrossRefGoogle Scholar
  13. Lacombe G, McCartney M (2014) Uncovering consistencies in indian rainfall trends observed over the last half century. Clim Change 2:287–299CrossRefGoogle Scholar
  14. Maldonado AD, Aguilera PA, Salmerón A (2016) Continuous Bayesian networks for probabilistic environmental risk mapping. Stoch Environ Res Risk Assess 30:1441–1455CrossRefGoogle Scholar
  15. Pavlopoulos H, Karlis D (2008) INAR(1) modeling of overdispersed count series with an environmental application. Environmetrics 19:369–393CrossRefGoogle Scholar
  16. Pegram GGS (1980) An autoregressive model for multilag markov chains. J Appl Probab 17:350–362CrossRefGoogle Scholar
  17. Pohl B, Macron C, Monerie PA (2017) Fewer rainy days and more extreme rainfall by the end of the century in Southern Africa. Sci Rep 7:46466CrossRefGoogle Scholar
  18. R Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  19. Scotto MG, Weiß CH, Silva ME, Pereira I (2014) Bivariate binomial autoregressive models. J Multivar Anal 125:233–251CrossRefGoogle Scholar
  20. Steutel FW, van Harn K (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899CrossRefGoogle Scholar
  21. Thyregod P, Carstensen J, Madsen H, Arnbjerg-Nielsen K (1999) Integer valued autoregressive models for tipping bucket rainfall measurements. Environmetrics 10:395–411CrossRefGoogle Scholar
  22. Tian D, Martinez CJ, Asefa T (2016) Improving short-term urban water demand forecasts with reforecast analog ensembles. J Water Resour Plan Manage 142:04016008CrossRefGoogle Scholar
  23. Weiß CH (2013) Serial dependence of NDARMA processes. Comput Stat Data Anal 68:213–238CrossRefGoogle Scholar
  24. Weiß CH, Göb R (2008) Measuring serial dependence in categorical time series. AStA Adv Stat Anal 92:71–89CrossRefGoogle Scholar
  25. WHO (2014) Climatic factors and the occurrence of dengue fever, dysentery and leptospirosis in Sri Lanka 1996–2010: a retrospective study. Technical report 65Google Scholar
  26. Wickham H (2009) ggplot2: elegant graphics for data analysis. Chapman & Hall/CRC monographs on statistics and applied probability. Springer, New YorkCrossRefGoogle Scholar
  27. Zucchini W, MacDonald IL (2009) Hidden Markov models for time series: an introduction using R. Chapman & Hall/CRC monographs on statistics & applied probability. CRC Press, Boca RatonCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de Engenharia Eletrónica e Informática de Aveiro (IEETA) and Centro de I&D em Matemática e Aplicações (CIDMA)Universidade de AveiroAveiroPortugal
  2. 2.Department of Mathematics and StatisticsHelmut Schmidt UniversityHamburgGermany
  3. 3.Departamento de Matemática and CEMAT, ISTUniversidade de LisboaLisboaPortugal

Personalised recommendations