## The Annals of Applied Statistics

### Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts

#### Abstract

This paper examines a bivariate count time series with some curious statistical features: Saffir–Simpson Category 3 and stronger annual hurricane counts in the North Atlantic and eastern Pacific Ocean Basins. As land and ocean temperatures on our planet warm, an intense climatological debate has arisen over whether hurricanes are becoming more numerous, or whether the strengths of the individual storms are increasing. Recent literature concludes that an increase in hurricane counts occurred in the Atlantic Basin circa 1994. This increase persisted through 2012; moreover, the 1994–2012 period was one of relative inactivity in the eastern Pacific Basin. When Atlantic activity eased in 2013, heavy activity in the eastern Pacific Basin commenced. When examined statistically, a Poisson white noise model for the annual severe hurricane counts is difficult to resoundingly reject. Yet, decadal cycles (longer term dependence) in the hurricane counts is plausible. This paper takes a statistical look at the issue, developing a stationary multivariate count time series model with Poisson marginal distributions and a flexible autocovariance structure. Our auto- and cross-correlations can be negative and have long-range dependence; features that most previous count models cannot achieve in tandem. Our model is new in the literature and is based on categorizing and super-positioning multivariate Gaussian time series. We derive the autocovariance function of the model and propose a method to estimate model parameters. In the end, we conclude that severe hurricane counts are indeed negatively correlated across the two ocean basins. Some evidence for long-range dependence is also presented; however, with only a 49-year record, this issue cannot be definitively judged without additional data.

#### Article information

Source
Ann. Appl. Stat., Volume 12, Number 1 (2018), 408-431.

Dates
Received: June 2017
Revised: September 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1520564478

Digital Object Identifier
doi:10.1214/17-AOAS1098

Mathematical Reviews number (MathSciNet)
MR3773399

Zentralblatt MATH identifier
06894712

#### Citation

Livsey, James; Lund, Robert; Kechagias, Stefanos; Pipiras, Vladas. Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts. Ann. Appl. Stat. 12 (2018), no. 1, 408--431. doi:10.1214/17-AOAS1098. https://projecteuclid.org/euclid.aoas/1520564478

#### References

• Alzaid, A. A. and Al-Osh, M. (1990). An integer-valued $p$th-order autoregressive structure (INAR($p$)) process. J. Appl. Probab. 27 314–324.
• Barndorff-Nielsen, O. E., Lunde, A., Shephard, N. and Veraart, A. E. D. (2014). Integer-valued trawl processes: A class of stationary infinitely divisible processes. Scand. J. Stat. 41 693–724.
• Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes. Springer, Heidelberg.
• Blight, P. A. (1989). Time series formed from the superposition of discrete renewal processes. J. Appl. Probab. 26 189–195.
• Brockwell, P. J. and Davis, R. A. (2006). Time Series: Theory and Methods. Springer, New York. Reprint of the second (1991) edition.
• Chu, P. S. and Zhao, Z. (2004). Bayesian change-point analysis of tropical cyclone activity: The Central North Pacific case. J. Climate 17 4893–4901.
• Cui, Y. and Lund, R. (2009). A new look at time series of counts. Biometrika 96 781–792.
• Czado, C., Gneiting, T. and Held, L. (2009). Predictive model assessment for count data. Biometrics 65 1254–1261.
• Davis, R. A. and Dunsmuir, W. T. M. (2016). State space models for count time series. In Handbook of Discrete-Valued Time Series (R. A. Davis, S. Holan, R. B. Lund and N. Ravishanker, eds.) CRC Press, Boca Raton, FL.
• Davis, R. A., Holan, S. H., Lund, R. B. and Ravishanker, N., eds. (2016). Handbook of Discrete-Valued Time Series. CRC Press, Boca Raton.
• Doukhan, P., Oppenheim, G. and Taqqu, M. S. (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston, MA.
• Dunsmuir, W. T. M. (2016). Generalized linear autoregressive moving average models. In Handbook of Discrete-Valued Time Series (R. A. Davis, S. Holan, R. B. Lund and R. N. Ravishanker, eds.) CRC Press, Boca Raton, FL.
• Elsner, J. and Jagger, T. H. (2006). Prediction models for annual US hurricane counts. J. Climate 19 2935–2952.
• Elsner, J., Jagger, T. and Niu, X. F. (2000). Changes in the rates of North Atlantic major hurricane activity during the 20th century. Geophys. Res. Lett. 27 1743–1746.
• Elsner, J. B. and Kocher, B. (2000). Global tropical cyclone activity: A link to the North Atlantic Oscillation. Geophys. Res. Lett. 27 129–132.
• Elsner, J., Kossin, J. P. and Jagger, T. H. (2008). The increasing intensity of the strongest tropical cyclones. Nature 455 92–95.
• Enciso-Mora, V., Neal, P. and Subba Rao, T. (2009). Efficient order selection for integer-valued ARMA processes. J. Time Series Anal. 30 1–18.
• Fokianos, K. and Kedem, B. (2003). Regression theory for categorical time series. Statist. Sci. 18 357–376.
• Giraitis, L., Koul, H. L. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. Imperial College Press, London.
• Goldenberg, S. B., Landsea, C. W. and Mestas-Nunez, A. M. (2001). The recent increase in Atlantic hurricane activity: Causes and implications. Science 293 474–479.
• Gray, W. M. (1984). Atlantic seasonal hurricane frequency. Part I: El Nino and 30 mb quasi-biennial oscillation influences. Mon. Weather Rev. 112 1649–1668.
• Hasselmann, K. (1976). Stochastic climate models part I. Theory. Tellus 28 473–485.
• Helgason, H., Pipiras, V. and Abry, P. (2011). Fast and exact synthesis of stationary multivariate Gaussian time series using circulant embedding. Signal Process. 91 1123–1133.
• Jia, Y. and Lund, R. B. (2016). Superpositioned stationary count time series. J. Appl. Probab.. To appear.
• Joe, H. (1996). Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Probab. 33 664–677.
• Kachour, M. and Yao, J. F. (2009). First-order rounded integer-valued autoregressive ($\mathrm{RINAR}(1)$) process. J. Time Series Anal. 30 417–448.
• Karlis, D. (2016). Models for multivariate count time series. In Handbook of Discrete-Valued Time Series (R. A. Davis, S. Holan, R. B. Lund and N. Ravishanker, eds.) 407–424. CRC Press, Boca Raton, FL.
• Karlis, D. and Meligkotsidou, L. (2007). Finite mixtures of multivariate Poisson distributions with application. J. Statist. Plann. Inference 137 1942–1960.
• Kechagias, S. and Pipiras, V. (2015). Definitions and representations of multivariate long-range dependent time series. J. Time Series Anal. 36 1–25.
• Kechagias, S. and Pipiras, V. (2017). Identification, estimation and applications of a bivariate long-range dependent time series model with general phase. Preprint.
• Kerss, A., Leonenko, N. and Sikorskii, A. (2014). Fractional Skellam processes with applications to finance. Fract. Calc. Appl. Anal. 17 532–551.
• Kolassa, S. (2016). Evaluating predictive count data distributions in retail sales forecasting. Int. J. Forecast. 32 788–803.
• Lund, R. B., Holan, S. H. and Livsey, J. (2016). Long memory discrete-valued time series. In Handbook of Discrete-Valued Time Series (R. A. Davis, S. Holan, R. B. Lund and N. Ravishanker, eds.) 447–458. CRC Press, Boca Raton, FL.
• Lund, R. B. and Livsey, J. (2016). Renewal-based count time series. In Handbook of Discrete-Valued Time Series (R. A. Davis, S. Holan, R. B. Lund and N. Ravishanker, eds.) 101. CRC Press, Boca Raton.
• Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer, Berlin.
• MacDonald, I. L. and Zucchini, W. (2016). Hidden Markov models for discrete-valued time series. In Handbook of Discrete-Valued Time Series (R. Davis, S. Holan, R. B. Lund and N. Ravishanker, eds.) CRC Press, Boca Raton, FL.
• McDonnell, K. A. and Holbrook, N. J. (2004). A Poisson regression model of tropical cyclogenesis for the Australian-southwest Pacific Ocean region. Weather Forecast. 19 440–455.
• McKenzie, E. (2003). Discrete variate time series. In Stochastic Processes: Modelling and Simulation. Handbook of Statistics 21 573–606. North-Holland, Amsterdam.
• Mooley, D. A. (1981). Applicability of the Poisson probability model to severe cyclonic storms striking the cost around the bay of bengal. Sankyha 43 187–197.
• Mooney, C. C. (2007). Storm World. Hurricanes, Politics, and the Battle over Global Warming. Harcourt, New York.
• Mudelsee, M. (2013). Climate Time Series Analysis. Springer, Berlin.
• Neal, P. and Subba Rao, T. (2007). MCMC for integer-valued ARMA processes. J. Time Series Anal. 28 92–110.
• Palma, W. (2007). Long-Memory Time Series. Wiley, New Jersey.
• Parisi, F. and Lund, R. B. (2000). Seasonality and return periods of landfalling Atlantic basin hurricanes. Aust. N. Z. J. Stat. 42 271–282.
• Park, K. and Willinger, W. (2000). Self-Similar Network Traffic and Performance Evaluation. Wiley Online Library.
• Percival, D. B., Overland, J. E. and Mofjeld, H. O. (2001). Interpretation of North Pacific variability as a short-and long-memory process. J. Climate 14 4545–4559.
• Pipiras, V. and Taqqu, M. S. (2017). Long-Range Dependence and Self-Similarity 45. Cambridge Univ. Press, Cambridge.
• Quoreshi, A. M. M. S. (2014). A long-memory integer-valued time series model, INARFIMA, for financial application. Quant. Finance 14 2225–2235.
• Robbins, M. W., Lund, R. B., Gallagher, C. M. and Lu, Q. (2011). Changepoints in the North Atlantic tropical cyclone record. J. Amer. Statist. Assoc. 106 89–99.
• Robinson, P. M. (2003). Long-memory time series. In Time Series with Long Memory. Adv. Texts Econometrics 4–32. Oxford Univ. Press, Oxford.
• Sela, R. J. (2010). Three essays in econometrics: Multivariate long memory time series and applying regression trees to longitudinal data. Ph.D. thesis, New York Univ.
• Skellam, J. G. (1946). The frequency distribution of the difference between two Poisson variates belonging to different populations. J. R. Stat. Soc. 109 296.
• Solow, A. (1989). Statistical modeling of storm counts. J. Climate 2 131–136.
• Steutel, F. W. and Van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Ann. Probab. 7 893–899.
• Thompson, M. L. and Guttorp, P. (1986). A probability model for severe cyclonic storms striking the cost around the bay of bengal. Mon. Weather Rev. 114 2267–2271.
• Van Vleck, J. H. and Middleton, D. (1966). The spectrum of clipped noise. Proc. IEEE 54 2–19.
• Varotsos, C. and Efstathiou, M. N. (2013). Is there any long-term memory effect in the tropical cyclones? Theor. Appl. Climatol. 114 643–650.
• Villarini, G., Vecchi, G. A. and Smith, J. A. (2010). Modeling the dependence of tropical storm counts in the North Atlantic basin on climate indices. Mon. Weather Rev. 138 2681–2705.
• Xiao, S., Kottas, A. and Sansó, B. (2015). Modeling for seasonal marked point processes: An analysis of evolving hurricane occurrences. Ann. Appl. Stat. 9 353–382.
• Yuan, N., Fu, Z. and Liu, S. (2014). Extracting climate memory using fractional integrated statistical model: A new perspective on climate prediction. Sci. Rep. 4.