Space-time Integer-valued ARMA modelling for time series of counts. (English) Zbl 07784501
Summary: This paper introduces a new class of space-time integer-valued ARMA models referred to as STINARMA. This class arises as the natural space-time extension of the INARMA models and, simultaneously, as the integer-valued counterpart of the conventional STARMA models. In this work, the moving average subclass STINMA\((q_{m_{1}}, \ldots, m_q)\) is studied in detail. Particular attention is given to the derivation of first- and second-order moments, including space-time autocorrelations. Due to its large potential use in real-data applications, the Poisson STINMA\((1_1)\) process is analyzed in further detail. Estimation methods are also addressed and their performance is demonstrated through a simulation study and by analysing the daily number of hospital admissions observed over time in three Portuguese locations.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62H11 | Directional data; spatial statistics |
| 62H12 | Estimation in multivariate analysis |
| 62P12 | Applications of statistics to environmental and related topics |
| 68T09 | Computational aspects of data analysis and big data |
Keywords:
space-time series of counts; STINARMA models; autoregressive moving-average processes; binomial thinning operator; Poisson distributionSoftware:
R.utils; RootSolve; RcppArmadillo; ggsn; optimParallel; HelpersMG; doParallel; ggplot2; ggforce; R; foreach; patchwork; dplyr; Matrix; RcppReferences:
| [1] | ALDOR-NOIMAN, S., BROWN, L. D., FOX, E. B. and STINE, R. A. (2016). Spatio-Temporal Low Count Processes with Application to Violent Crime Events. Statistica Sinica 26 1587-1610. Digital Object Identifier: 10.5705/ss.2014.217t Google Scholar: Lookup Link MathSciNet: MR3586230 · Zbl 1356.62131 · doi:10.5705/ss.2014.217t |
| [2] | ALEKSANDROV, B. and WEIß, C. H. (2020). Parameter Estimation and Diagnostic Tests for INMA(1) Processes. TEST 29 196-232. MathSciNet: MR4063388 · Zbl 1460.62142 |
| [3] | SANTOS BAQUERO, O. (2019). ggsn: North Symbols and Scale Bars for Maps Created with ‘ggplot2’ or ‘ggmap’ R package version 0.5.0. |
| [4] | BATES, D. and MAECHLER, M. (2021). Matrix: Sparse and Dense Matrix Classes and Methods R package version 1.4-0. |
| [5] | BENGTSSON, H. (2021). R.utils: Various Programming Utilities R package version 2.11.0. |
| [6] | BOROVKOVA, S., LOPUHAA, H. and NURANI, B. (2002). Generalized STAR Model with Experimental Weights. In Proceedings of the 17th International Workshop on Statistical Modelling 139-147. |
| [7] | BRÄNNÄS, K. and HALL, A. (2001). Estimation in Integer-Valued Moving Average Models. Applied Stochastic Models in Business and Industry 17 277-291. MathSciNet: MR1864091 · Zbl 0979.62066 |
| [8] | BRÄNNÄS, K. and QUORESHI, S. (2010). Integer-Valued Moving Average Modelling of the Number of Transactions in Stocks. Applied Financial Economics 20 1429-1440. |
| [9] | BUTEIKIS, A. and LEIPUS, R. (2020). An Integer-Valued Autoregressive Process for Seasonality. Journal of Statistical Computation and Simulation 90 391-411. MathSciNet: MR4044916 · Zbl 07194291 |
| [10] | MICROSOFT CORPORATION and WESTON, S. (2020). doParallel: Foreach Parallel Adaptor for the ‘parallel’ Package R package version 1.0.16. |
| [11] | DAVIS, R. A., HOLAN, S. H., LUND, R. and RAVISHANKER, N. (2016). Handbook of Discrete-valued Time Series. CRC Press. MathSciNet: MR3642975 · Zbl 1331.62003 |
| [12] | DAVIS, R. A., FOKIANOS, K., HOLAN, S. H., JOE, H., LIVSEY, J., LUND, R., PIPIRAS, V. and RAVISHANKER, N. (2021). Count Time Series: A Methodological Review. Journal of the American Statistical Association 116 1533-1547. MathSciNet: MR4309291 · Zbl 1510.62356 |
| [13] | DION, J. P., GAUTHIER, G. and LATOUR, A. (1995). Branching Processes with Immigration and Integer-Valued Time Series. Serdica Mathematical Journal 21 123-136. MathSciNet: MR1338812 · Zbl 0819.62067 |
| [14] | EDDELBUETTEL, D. and BALAMUTA, J. J. (2018). Extending R with C++: A Brief Introduction to Rcpp. The American Statistician 72 28-36. Digital Object Identifier: 10.1080/00031305.2017.1375990 Google Scholar: Lookup Link MathSciNet: MR3790565 · Zbl 07663915 · doi:10.1080/00031305.2017.1375990 |
| [15] | EDDELBUETTEL, D. and SANDERSON, C. (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra. Computational Statistics and Data Analysis 71 1054-1063. MathSciNet: MR3132026 · Zbl 1471.62055 |
| [16] | Efron, B. and Tibshirani, R. J. (1994). An Introduction to the Bootstrap. CRC Press. MathSciNet: MR1270903 · Zbl 0835.62038 |
| [17] | FRANKE, J. and SUBBA RAO, T. (1993). Multivariate First-Order Integer-Valued Autoregressions Technical Report, University of Kaiserslautern. |
| [18] | GERBER, F. and FURRER, R. (2019). optimParallel: An R Package Providing a Parallel Version of the L-BFGS-B Optimization Method. The R Journal 11 352-358. Digital Object Identifier: 10.32614/RJ-2019-030 Google Scholar: Lookup Link · doi:10.32614/RJ-2019-030 |
| [19] | GHODSI, A. (2015). Conditional Maximum Likelihood Estimation of the First-Order Spatial Integer-Valued Autoregressive (SINAR \((1, 1))\) Model. Journal of the Iranian Statistical Society 14 15-36. Digital Object Identifier: 10.7508/jirss.2015.02.002 Google Scholar: Lookup Link MathSciNet: MR3518270 · Zbl 1412.62125 · doi:10.7508/jirss.2015.02.002 |
| [20] | GHODSI, A., SHITAN, M. and BAKOUCH, H. S. (2012). A First-Order Spatial Integer-Valued Autoregressive SINAR (1, 1) Model. Communications in Statistics-Theory and Methods 41 2773-2787. MathSciNet: MR2946646 · Zbl 1270.62128 |
| [21] | GIRONDOT, M. (2021). HelpersMG: Tools for Environmental Analyses, Ecotoxicology and Various R Functions R package version 4.8. |
| [22] | HUDA, N., MUKHAIYAR, U. and PASARIBU, U. (2021). The Approximation of GSTAR Model for Discrete Cases through INAR Model. In Journal of Physics: Conference Series 1722 012100. IOP Publishing. |
| [23] | JOWAHEER, V., SUNECHER, Y. and MAMODE KHAN, N. (2016). A Non-Stationary BINAR(1) Process with Negative Binomial Innovations for Modeling the Number of Goals in the First and Second half: The Case of Arsenal Fotball Club. Communications in Statistics: Case Studies, Data Analysis and Applications 2 21-33. MathSciNet: MR3523164 |
| [24] | LATOUR, A. (1997). The Multivariate GINAR (p) Process. Advances in Applied Probability 29 228-248. MathSciNet: MR1432938 · Zbl 0871.62073 |
| [25] | MAHAMUNULU, D. (1967). A Note on Regression in the Multivariate Poisson Distribution. Journal of the American Statistical Association 62 251-258. MathSciNet: MR0216613 · Zbl 0147.37802 |
| [26] | MARTIN, R. L. and OEPPEN, J. (1975). The Identification of Regional Forecasting Models using Space: Time Correlation Functions. Transactions of the Institute of British Geographers 95-118. |
| [27] | MARTINS, A., SCOTTO, M. G., DEUS, R., MONTEIRO, A. and GOUVEIA, S. (2021). Association between Respiratory Hospital Admissions and Air quality in Portugal: A Count Time Series Approach. PLOS One 16 e0253455. |
| [28] | MCKENZIE, E. (1988). Some ARMA Models for Dependent Sequences of Poisson Counts. Advances in Applied Probability 20 822-835. MathSciNet: MR0968000 · Zbl 0664.62089 |
| [29] | PEBESMA, E. (2018). Simple Features for R: Standardized Support for Spatial Vector Data. The R Journal 10 439-446. Digital Object Identifier: 10.32614/RJ-2018-009 Google Scholar: Lookup Link · doi:10.32614/RJ-2018-009 |
| [30] | PEDELI, X. and KARLIS, D. (2011). A Bivariate INAR(1) Process with Application. Statistical modelling 11 325-349. MathSciNet: MR2906704 · Zbl 1420.62393 |
| [31] | PEDELI, X. and KARLIS, D. (2013). Some Properties of Multivariate INAR(1) Processes. Computational Statistics & Data Analysis 67 213-225. MathSciNet: MR3079598 · Zbl 1471.62158 |
| [32] | PEDERSEN, T. L. (2019). ggforce: Accelerating ‘ggplot2’ R package version 0.3.1. |
| [33] | PEDERSEN, T. L. (2020). patchwork: The Composer of Plots R package version 1.1.1. |
| [34] | PFEIFER, P. E. (1979). Spatial-Dynamic Modeling, PhD thesis, Georgia Institute of Technology. |
| [35] | PFEIFER, P. E. and DEUTSCH, S. J. (1980a). A Three-Stage Iterative Procedure for Space-Time Modeling. Technometrics 22 35-47. · Zbl 0429.62066 |
| [36] | PFEIFER, P. E. and DEUTSCH, S. J. (1980b). Identification and Interpretation of First Order Space-Time ARMA Models. Technometrics 22 397-408. · Zbl 0459.62078 |
| [37] | QUORESHI, A. S. (2006). Bivariate Time Series Modeling of Financial Count Data. Communications in Statistics-Theory and Methods 35 1343-1358. MathSciNet: MR2328481 · Zbl 1105.62107 |
| [38] | QUORESHI, A. S. (2008). A Vector Integer-Valued Moving Average Model for High Frequency Financial Count Data. Economics Letters 101 258-261. |
| [39] | SANTOS, C., PEREIRA, I. and SCOTTO, M. G. (2021). On the Theory of Periodic Multivariate INAR Processes. Statistical Papers 62 1291-1348. MathSciNet: MR4262195 · Zbl 1477.62252 |
| [40] | SCOTTO, M. G., WEIß, C. H. and GOUVEIA, S. (2015). Thinning-Based Models in the Analysis of Integer-Valued Time Series: A Review. Statistical Modelling 15 590-618. MathSciNet: MR3441230 · Zbl 07259004 |
| [41] | SCOTTO, M. G., WEIß, C. H., SILVA, M. E. and PEREIRA, I. (2014). Bivariate Binomial Autoregressive Models. Journal of Multivariate Analysis 125 233-251. MathSciNet: MR3163841 · Zbl 1283.62188 |
| [42] | SILVA, I., SILVA, M. E. and TORRES, C. (2020). Inference for Bivariate Integer-Valued Moving Average Models based on Binomial Thinning Operation. Journal of Applied Statistics 47 2546-2564. MathSciNet: MR4149569 · Zbl 1521.62479 |
| [43] | SOETAERT, K. (2009). rootSolve: Nonlinear Root Finding, Equilibrium and Steady-State Analysis of Ordinary Differential Equations R package 1.6. |
| [44] | STEUTEL, F. W. and VAN HARN, K. (1979). Discrete Analogues of Self-Decomposability and Stability. The Annals of Probability 893-899. MathSciNet: MR0542141 · Zbl 0418.60020 |
| [45] | SUBBA RAO, T. and COSTA ANTUNES, A. M. (2004). Spatio-Temporal Modelling of Temperature Time Series: A Comparative Study. In Time series analysis and applications to geophysical systems 123-150. Springer. MathSciNet: MR2111482 · Zbl 1061.62186 |
| [46] | SUNECHER, Y. (2021). Application of the BINARMA(1, 1) Model with NB Innovations on Accident Data. In 2021 IEEE Asia-Pacific Conference on Computer Science and Data Engineering (CSDE) 1-4. IEEE. |
| [47] | SUNECHER, Y., MAMODE KHAN, N. and JOWAHEER, V. (2017). Estimating the Parameters of a BINMA Poisson Model for a Non-Stationary Bivariate Time Series. Communications in Statistics-Simulation and Computation 46 6803-6827. MathSciNet: MR3764940 · Zbl 1384.65016 |
| [48] | SUNECHER, Y., MAMODE KHAN, N. and JOWAHEER, V. (2019). The Non-Stationary BINARMA(1, 1) Process with Poisson Innovations: An Application on Accident Data. International Journal of Mathematical and Computational Sciences 13 193-196. |
| [49] | SUNECHER, Y., MAMODE KHAN, N. and JOWAHEER, V. (2020). BINMA(1) Model with COM-Poisson Innovations: Estimation and Application. Communications in Statistics-Simulation and Computation 49 1631-1652. MathSciNet: MR4120951 · Zbl 07553270 |
| [50] | TABANDEH, A. and GHODSI, A. (2022). First-Order Spatial Random Coefficient Non-Negative Integer-Valued Autoregressive (SRCINAR(1,1)) model. Communications in Statistics - Simulation and Computation 0 1-13. Digital Object Identifier: 10.1080/03610918.2022.2083164 Google Scholar: Lookup Link · doi:10.1080/03610918.2022.2083164 |
| [51] | R CORE TEAM (2021). R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing, Vienna, Austria. |
| [52] | WEIß, C. H. (2008). Serial Dependence and Regression of Poisson INARMA Models. Journal of Statistical Planning and Inference 138 2975-2990. MathSciNet: MR2442225 · Zbl 1140.62069 |
| [53] | WEIß, C. H. (2012). Fully Observed INAR (1) Processes. Journal of Applied Statistics 39 581-598. MathSciNet: MR2880436 · Zbl 1514.62175 |
| [54] | WEIß, C. H. (2021). Stationary Count Time Series Models. Wiley Interdisciplinary Reviews: Computational Statistics 13 e1502. MathSciNet: MR4186769 · Zbl 1461.62163 |
| [55] | WEIß, C. H., FELD, M. H.-J., MAMODE KHAN, N. and SUNECHER, Y. (2019). INARMA Modeling of Count Time Series. Stats 2 284-320. |
| [56] | WICKHAM, H. (2016). ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. · Zbl 1397.62006 |
| [57] | WICKHAM, H., FRANÇOIS, R., HENRY, L. and MÜLLER, K. (2020). dplyr: A Grammar of Data Manipulation R package version 1.0.2. |
| [58] | ZUCCHINI, W. and MACDONALD, I. L. (2009). Hidden Markov Models for Time Series: an Introduction using R. Chapman and Hall/CRC. MathSciNet: MR2523850 · Zbl 1180.62130 |
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