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


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.

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  1. 1.

    It should be noted that the model proposed by Blight (1989) only sporadically meets an ARMA autocorrelation structure.

  2. 2.

    Out of 1713 stations, 603 are assigned to an AR model and 691 to an MA model, for both bvARMA and NDARMA. In contrast, 319 stations were assigned to bvAR(1) and NDMA(1) models whereas 100 stations to bvMA(1) and NDAR(1) models.


  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 23–37

    Google Scholar 

  2. Berchtold A (2002) High-order extensions of the double chain Markov model. Stoch models 18:193–227

    Article  Google Scholar 

  3. Blight PA (1989) Time series formed from the superposition of discrete renewal processes. J Appl Probab 26:189–195

    Article  Google 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 23:1367–1378

    Article  Google Scholar 

  5. Cui Y, Lund R (2009) A new look at time series of counts. Biometrika 96:781–792

    Article  Google Scholar 

  6. Deckmyn A, Minka TP, Brownrigg R, Becker RA, Wilks AR (2017) maps: draw geographical maps. R package version 320

  7. Delleur JW, Chang TJ, Kavvas ML (1989) Simulation models of sequences of dry and wet days. J Irrig Drain Eng 115:344–357

    Article  Google 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 4:42

    Article  CAS  Google Scholar 

  9. Grunwald G, Hyndman RJ, Tedesco L, Tweedie RL (2000) Non-gaussian conditional linear AR(1) models. Aust Nz J Statist 42:479–495

    Article  Google Scholar 

  10. Jacobs PA, Lewis PAW (1983) Stationary discrete autoregressive-moving average time series generated by mixtures. J Time Ser Anal 4:19–36

    Article  Google Scholar 

  11. Khoo WC, Ong SH, Biswas A (2017) Modeling time series of counts with a new class of INAR(1) model. Stat Pap 58:393–416

    Article  Google 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 22:1441–1453

    Article  Google Scholar 

  13. Lacombe G, McCartney M (2014) Uncovering consistencies in indian rainfall trends observed over the last half century. Clim Change 2:287–299

    Article  Google Scholar 

  14. Maldonado AD, Aguilera PA, Salmerón A (2016) Continuous Bayesian networks for probabilistic environmental risk mapping. Stoch Environ Res Risk Assess 30:1441–1455

    Article  Google Scholar 

  15. Pavlopoulos H, Karlis D (2008) INAR(1) modeling of overdispersed count series with an environmental application. Environmetrics 19:369–393

    Article  Google Scholar 

  16. Pegram GGS (1980) An autoregressive model for multilag markov chains. J Appl Probab 17:350–362

    Article  Google 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 7:46466

    Article  CAS  Google Scholar 

  18. R Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  19. Scotto MG, Weiß CH, Silva ME, Pereira I (2014) Bivariate binomial autoregressive models. J Multivar Anal 125:233–251

    Article  Google Scholar 

  20. Steutel FW, van Harn K (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899

    Article  Google Scholar 

  21. Thyregod P, Carstensen J, Madsen H, Arnbjerg-Nielsen K (1999) Integer valued autoregressive models for tipping bucket rainfall measurements. Environmetrics 10:395–411

    Article  Google 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 142:04016008

    Article  Google Scholar 

  23. Weiß CH (2013) Serial dependence of NDARMA processes. Comput Stat Data Anal 68:213–238

    Article  Google Scholar 

  24. Weiß CH, Göb R (2008) Measuring serial dependence in categorical time series. AStA Adv Stat Anal 92:71–89

    Article  Google 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 65

  26. Wickham H (2009) ggplot2: elegant graphics for data analysis. Chapman & Hall/CRC monographs on statistics and applied probability. Springer, New York

    Google 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 Raton

    Google Scholar 

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

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Corresponding author

Correspondence to Christian H. Weiß.

Additional information

This research was supported by the German Academic Exchange Service (DAAD) and the Fundação para a Ciência e a Tecnologia (FCT), under the program “Ações Integradas Luso-Alemãs” and the Grants 57212119 and A-38/16. This work was partially supported by FCT, with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020–within IEETA/UA project UID/CEC/00127/2013 (Instituto de Engenharia Electrónica e Informática de Aveiro, Aveiro) and CIDMA/UA project UID/MAT/04106/2013 (Centro de I&D em Matemática e Aplicações, Aveiro). S. Gouveia acknowledges the postdoctoral grant by FCT (ref. SFRH/BPD/87037/2012).



Proof of Theorem 2

For \(k\ge 1\), by conditioning,

$$\begin{aligned} \gamma&(k)\ =\ Cov[ X_{t}, X_{t-k} ] \\&=\ E\big [ Cov[ X_{t}, X_{t-k} | X_{t-1}, \dots , \epsilon _{t}, \epsilon _{t-1}, \dots ] \big ] + Cov\big [ E[ X_{t} | X_{t-1}, \dots , \epsilon _{t}, \epsilon _{t-1}, \dots ], X_{t-k}\big ]\\&\quad =\ 0 + Cov\Big [ \sum _{i=1}^{{p }} \phi _{i} X_{t-i} + \sum _{j=0}^{{q }} \phi _{-j} \epsilon _{t-j}, X_{t-k}\Big ]\\& =\ \sum _{i=1}^{{p }} \phi _{i} Cov[X_{t-i}, X_{t-k}] + \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, X_{t-k}]\\& =\ \sum _{i=1}^{{p }} \phi _{i} \gamma ( | k-i| ) + \sum _{j=k}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, X_{t-k}]. \end{aligned}$$

Let us now look at the cross-covariances \(a_{k}:= Cov[X_{t}, \epsilon _{t-k}]\) with \(k\in {\mathbb {Z}}\), which satisfies \(a_{k}=0\) for \(k<0\). At lag 0, we obtain by conditioning (in analogy to above)

$$\begin{aligned} a_{0}\ =\ Cov[X_{t}, \epsilon _{t}] \ =\ \sum _{i=1}^{{p }} \phi _{i} Cov[X_{t-i}, \epsilon _{t}]\ +\ \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, \epsilon _{t}] \ =\ \phi _{0} \sigma _{\epsilon }^{2}. \end{aligned}$$

For lags \(k\ge 1\),

$$\begin{aligned} a_{k} \ =\ \sum _{i=1}^{{p }} \phi _{i} Cov[ X_{t-i}, \epsilon _{t-k}]\ +\ \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, \epsilon _{t-k}] \ =\ \sum _{i=1}^{{p }} \phi _{i} a_{k-i}\ +\ \phi _{-k}\sigma _{\epsilon }^{2}. \end{aligned}$$

If we now define \(b_{k}:=a_{k}/\sigma _{\epsilon }^{2}\) for \(k=0,1,\dots , {q }\), we get the recursion stated in Theorem 2.

Derivation of conditional variance

Using (8) for the conditional factorial moments, we obtain

$$\begin{aligned} \begin{array}{l} V[X_{t} | X_{t-1}, \dots , \epsilon _{t-1}, \dots ]\ =\ E[(X_t)_{(2)}\ |\ \ldots ] + E[ X_{t}\ |\ \ldots ] - E[ X_{t}\ |\ \ldots ]^{2}\\ \ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]\ +\ \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}\, X_{t-i}^{2} +\ \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\, \epsilon _{t-j}^{2}\\ \quad + \phi _0\cdot \mu \ +\ \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _i\, X_{t-i}\ +\ \mathop {\sum }\nolimits _{j=1}^{{q }}\ \phi _{-j}\, \epsilon _{t-j}\\ \quad - \left( \phi _0\cdot \mu \ +\ \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _i\, X_{t-i}\ +\ \mathop {\sum }\nolimits _{j=1}^{{q }}\ \phi _{-j}\, \epsilon _{t-j}\right) ^{2}\\ \ =\ \phi _{0} \sigma _{\epsilon }^{2} + \phi _{0} ( 1- \phi _{0}) \mu ^{2} + \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}\, X_{t-i}^{2} - \mathop {\sum }\nolimits _{i=1}^{{p }} \mathop {\sum }\nolimits _{k=1}^{{p }} \phi _{i} \phi _{k} X_{t-i}X_{t-k}\\ \quad + (1-2\phi _{0}\mu ) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i} X_{t-i} + \left( 1- \tfrac{1}{n}\right) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\, \epsilon _{t-j}^{2}\\ \quad - \mathop {\sum }\nolimits _{j=1}^{{q }} \mathop {\sum }\nolimits _{l=1}^{{q }} \phi _{-j}\phi _{-l} \epsilon _{t-j}\epsilon _{t-l}+ (1-2\phi _{0}\mu ) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\epsilon _{t-j} - 2\mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}X_{t-i} \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}. \end{array} \end{aligned}$$

So we obtain

$$\begin{aligned} \begin{array}{l} V[X_{t} | X_{t-1}, \dots , \epsilon _{t-1}, \dots ]\\ \ =\ \phi _{0} \sigma _{\epsilon }^{2} + \phi _{0} ( 1- \phi _{0}) \mu ^{2} + (1-\tfrac{1}{n})\, \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _{i} X_{t-i}^{2} - \mathop {\sum }\nolimits _{i=1}^{{p }} \mathop {\sum }\nolimits _{k=1}^{{p }} \phi _{i} \phi _{k} X_{t-i}X_{t-k}\\ \quad + (1-\tfrac{1}{n})\, \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}^{2}- \mathop {\sum }\nolimits _{j=1}^{{q }} \mathop {\sum }\nolimits _{l=1}^{{q }} \phi _{-j}\phi _{-l} \epsilon _{t-j}\epsilon _{t-l}\\ \quad + (1-2\phi _{0}\mu ) \left( \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i} X_{t-i} + \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\epsilon _{t-j}\right) - 2\mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}X_{t-i} \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}. \end{array} \end{aligned}$$

Note that the first-order autoregressive case (\({p }=1\)) gives

$$\begin{aligned} \begin{array}{l} V[X_t\ |\ X_{t-1}]\ =\ E[(X_t)_{(2)}\ |\ X_{t-1}]\ +\ E[X_t\ |\ X_{t-1}]\ -\ E^2[X_t\ |\ X_{t-1}]\\ \ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]+(1-\phi _0)\,\frac{n-1}{n}\,X_{t-1}^2 \\ \quad +\ \phi _0\cdot \mu +(1-\phi _0)\,X_{t-1} \ -\ \big (\phi _0\cdot \mu +(1-\phi _0)\,X_{t-1}\big )^2\\ \ =\ \phi _0\cdot \sigma _{\epsilon }^2 + \phi _0(1-\phi _0)\,\mu ^2 \\ \quad +\ (1-\phi _0)\,(\phi _0-\frac{1}{n})\,X_{t-1}^2 \ +\ (1-\phi _0)\,(1-2\,\phi _0\, \mu )\,X_{t-1}. \end{array} \end{aligned}$$

Again from (8) with \(r=2\), we obtain the unconditional 2nd factorial moment

$$\begin{aligned} E[(X_t)_{(2)}]\ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]\ +\ \tfrac{n_{(2)}}{n^2}\,\sum _{i=1}^{{p }}\ \phi _i\, E[X_{t-i}^2]\ +\ \tfrac{n_{(2)}}{n^2}\,\sum _{j=1}^{{q }}\ \phi _{-j}\, E[\epsilon _{t-j}^2]. \end{aligned}$$

Using the stationarity, it follows that

$$\begin{aligned} \left( 1 - (1-\tfrac{1}{n})\, \sum _{i=1}^{{p }}\phi _{i}\right) \, E[X_t^2]\ =\ (1-\phi _0)\mu \ +\ \phi _0\cdot E[\epsilon _t^2]\ +\ (1-\tfrac{1}{n})\,\sum _{j=1}^{{q }}\ \phi _{-j}\, E[\epsilon _{t-j}^2]. \end{aligned}$$

So the unconditional variance \(\sigma ^{2}=V[X_t]\) of the stationary process is given by

$$\begin{aligned} \left( 1 - (1-\tfrac{1}{n})\, \sum _{i=1}^{{p }}\phi _{i}\right) \,\sigma ^{2} \ =\ \phi _{0} \sigma _{\epsilon }^{2} + (1-\tfrac{1}{n})\, \sum _{j=1}^{{q }} \phi _{-j} \sigma _{\epsilon }^{2} +(1-\phi _{0})\mu \left( 1-\frac{\mu }{n}\right) . \end{aligned}$$

So Eq. (9) for the BID follows.

Cohen’s \(\kappa \) for bvAR(1) and bvMA(1) models

The computation of Cohen’s \(\kappa (h)\) (2) requires knowledge about the bivariate joint distributions with lag h. The bvAR(1) process constitutes a finite Markov chain with the 1-step-ahead transition probabilities \(p_{k|l}\) given in Eq. 11; let \(\mathbf{P }=(p_{k|l})_{k,l=0,\ldots ,n}\) denote the corresponding transition matrix. Then the h-step-ahead transition probabilities \(P(X_t=k\ |\ X_{t-h}=l)\) are the entries of the matrix \(\mathbf{P }^h\), and the stationary marginal distribution is given by the solution of the invariance equation \(\mathbf{P }\, \varvec{\pi }\ =\ \varvec{\pi }\), i. e., \(P(X_t=j)=\pi _j\).

For the bvMA(1) model, the stationary marginal distribution equals

$$\begin{aligned} P(X=x)\ =\ \phi _{0}\, P(\epsilon =x)\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=x), \end{aligned}$$

see Example 2. Since \(X_t\) and \(X_{t-h}\) are independent for lags \(h>1\), we then have \(\kappa (h)=0\); so it suffices to compute the bivariate probabilities \(P(X_t=k, X_{t-1}=l)\) for time lag 1. Using that

$$\begin{aligned} P(X_t=x\ |\ \epsilon _t=u,\epsilon _{t-1}=v)\ =\ \phi _{0}\, \delta _{x,u}\ +\ \phi _{-1}\, P({bv _{n}(v)}=x), \end{aligned}$$

and denoting \(p_{\epsilon ,u}:=P(\epsilon _t=u)\), we obtain

$$\begin{aligned} \begin{array}{l} P(X_t=k, X_{t-1}=l)\ =\ \mathop {\sum }\nolimits _{a,b,c} P(X_t=k, X_{t-1}=l,\ \epsilon _t=a,\epsilon _{t-1}=b,\epsilon _{t-2}=c)\\ \quad =\ \mathop {\sum }\nolimits _{a,b,c} P(X_t=k\ |\ \epsilon _t=a,\epsilon _{t-1}=b)\, P(X_{t-1}=l\ |\ \epsilon _{t-1}=b,\epsilon _{t-2}=c)\, p_{\epsilon ,a}p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{a,b,c} \big (\phi _{0}\, \delta _{k,a}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(c)}=l)\big )\, p_{\epsilon ,a}p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{b,c} \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(c)}=l)\big )\, p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{b} \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=l)\big )\, p_{\epsilon ,b}\\ \quad =\ \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(l)}=k)\big )\, \phi _{0}\, p_{\epsilon ,l}\\ \qquad +\ \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=k)\big )\, \phi _{-1}\, P({bv _{n}(\epsilon )}=l)\\ \quad =\ \phi _{0}^2\, p_{\epsilon ,k} p_{\epsilon ,l}\ +\ \phi _{0}\phi _{-1}\, p_{\epsilon ,l}\, P({bv _{n}(l)}=k)\\ \qquad +\ \phi _{0}\phi _{-1}\, p_{\epsilon ,k}\, P({bv _{n}(\epsilon )}=l)\ +\ \phi _{-1}^2\, P({bv _{n}(\epsilon )}=k)\, P({bv _{n}(\epsilon )}=l).\\ \end{array} \end{aligned}$$

Simulation results from Sect. 4.2

See Figs. 9, 10, 11, 12, 13, 14, 15, 16 and 17.

Fig. 9

Violin plots of estimated minus true value of \(\phi _{-1}\), \(\phi _{0}\), \(\phi _{1}\), \(\pi \) and d, for fitting bvARMA(1, 1) model to DGP: bvARMA(1, 1) with \(n=7\), \(\phi _{-1}=0.1\), \(\phi _{0}=0.7\), \(\phi _{1}=0.2\), \(\pi =0.5\), \(d=1.8\)

Fig. 10

DGP as in Fig. 9, but fitting misspecified bvMA(1) model

Fig. 11

DGP as in Fig. 9, but fitting misspecified bvAR(1) model

Fig. 12

Violin plots of estimated minus true value of \(\phi _{-1}\), \(\phi _{0}\), \(\phi _{1}\), \(\pi \) and d, for fitting bvMA(1) model to DGP: bvMA(1) with \(n=7\), \(\phi _{-1}=0.1\), \(\phi _{0}=0.9\), \(\phi _{1}=0.0\), \(\pi =0.5\), \(d=1.8\)

Fig. 13

DGP as in Fig. 12, but fitting misspecified bvARMA(1, 1) model

Fig. 14

DGP as in Fig. 12, but fitting misspecified bvAR(1) model

Fig. 15

Violin plots of estimated minus true value of \(\phi _{-1}\), \(\phi _{0}\), \(\phi _{1}\), \(\pi \) and d, for fitting bvAR(1) model to DGP: bvAR(1) with \(n=7\), \(\phi _{-1}=0.0\), \(\phi _{0}=0.8\), \(\phi _{1}=0.2\), \(\pi =0.5\), \(d=1.8\)

Fig. 16

DGP as in Fig. 15, but fitting misspecified bvARMA(1, 1) model

Fig. 17

DGP as in Fig. 15, but fitting misspecified bvMA(1) model

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Gouveia, S., Möller, T.A., Weiß, C.H. et al. A full ARMA model for counts with bounded support and its application to rainy-days time series. Stoch Environ Res Risk Assess 32, 2495–2514 (2018). https://doi.org/10.1007/s00477-018-1584-3

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