Bayesian Analysis

Sequential Bayesian Analysis of Multivariate Count Data

Tevfik Aktekin, Nick Polson, and Refik Soyer

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We develop a new class of dynamic multivariate Poisson count models that allow for fast online updating. We refer to this class as multivariate Poisson-scaled beta (MPSB) models. The MPSB model allows for serial dependence in count data as well as dependence with a random common environment across time series. Notable features of our model are analytic forms for state propagation, predictive likelihood densities, and sequential updating via sufficient statistics for the static model parameters. Our approach leads to a fully adapted particle learning algorithm and a new class of predictive likelihoods and marginal distributions which we refer to as the (dynamic) multivariate confluent hyper-geometric negative binomial distribution (MCHG-NB) and the dynamic multivariate negative binomial (DMNB) distribution, respectively. To illustrate our methodology, we use a simulation study and empirical data on weekly consumer non-durable goods demand.

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Bayesian Anal., Volume 13, Number 2 (2018), 385-409.

First available in Project Euclid: 23 March 2017

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state space count time series multivariate poisson scaled beta prior particle learning

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Aktekin, Tevfik; Polson, Nick; Soyer, Refik. Sequential Bayesian Analysis of Multivariate Count Data. Bayesian Anal. 13 (2018), no. 2, 385--409. doi:10.1214/17-BA1054.

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  • Abramowitz, M. and Stegun, I. (1968). Handbook of Mathematical Functions Number 55. Applied Mathematical Series.
  • Aktekin, T. and Soyer, R. (2011). “Call center arrival modeling: A Bayesian state-space approach.” Naval Research Logistics, 58(1): 28–42.
  • Aktekin, T. and Soyer, R. (2012). “Bayesian analysis of queues with impatient customers: Applications to call centers.” Naval Research Logistics, 59(2): 441–456.
  • Aktekin, T., Soyer, R., and Xu, F. (2013). “Assessment of Mortgage Default Risk via Bayesian State Space Models.” Annals of Applied Statistics, 7(3): 1450–1473.
  • Aktekin, T., Polson, N., and Soyer, R. (2017). “Supplementary Appendices for “Sequential Bayesian Analysis of Multivariate Count Data”.” Bayesian Analysis.
  • Al-Osh, M. A. and Alzaid, A. A. (1987). “First-order integer valued autoregressive (INAR(1)) Process.” Journal of Time Series Analysis, 8(3): 261–275.
  • Arbous, A. G. and Kerrich, J. (1951). “Accident statistics and the concept of accident proneness.” Biometrics, 7: 340–432.
  • Carter, C. K. and Kohn, R. (1994). “On Gibbs sampling for state space models.” Biometrika, 81(3): 541–553.
  • Carvalho, C., Johannes, M. S., Lopes, H. F., and Polson, N. (2010a). “Particle learning and smoothing.” Statistical Science, 25(1): 88–106.
  • Carvalho, C. M., Lopes, H. F., Polson, N. G., and Taddy, M. A. (2010b). “Particle learning for general mixtures.” Bayesian Analysis, 5(4): 709–740.
  • Cox, D. R. (1981). “Statistical Analysis of Time Series: Some Recent Developments.” Scandinavian Journal of Statistics, 8: 93–115.
  • Davis, R., Holan, S., Lund, R., and Ravishanker, N. (2015). Handbook of Discrete-Valued Time Series. Chapman and Hall/CRC.
  • Ding, M., He, L., Dunson, D., and Carin, L. (2012). “Nonparametric Bayesian segmentation of a multivariate inhomogeneous space-time Poisson process.” Bayesian Analysis, 7(4): 813.
  • Durbin, J. and Koopman, S. (2000). “Time Series Analysis of Non-Gaussian Observations based on state space models from both classical and Bayesian perspectives.” Journal of the Royal Statistical Society, Series B, 62(1): 3–56.
  • Freeland, R. K. and McCabe, B. P. M. (2004). “Analysis of Low Count Time Series Data by Poisson Autocorrelation.” Journal of Time Series Analysis, 25(5): 701–722.
  • Fruhwirth-Schnatter, S. (1994). “Data Augmentation and Dynamic Linear Models.” Journal of Time Series Analysis, 15(2): 183–202.
  • Fruhwirth-Schnatter, S. and Wagner, H. (2006). “Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling.” Biometrika, 93(4): 827–841.
  • Gamerman, D., Dos-Santos, T. R., and Franco, G. C. (2013). “A non-Gaussian family of state-space models with exact marginal likelihood.” Journal of Time Series Analysis, 34(6): 625–645.
  • Gordon, N. J., Salmon, D. J., and Smith, A. F. (1993). “Novel approach to nonlinear/non-Gaussian Bayesian state estimation.” IEE Proceedings F (Radar and Signal Processing), 140(2): 107–113.
  • Gordy, M. B. (1998). “Computationally convenient distributional assumptions for common-value auctions.” Computational Economics, 12(1): 61–78.
  • Gramacy, R. B. and Polson, N. G. (2011). “Particle learning of Gaussian process models for sequential design and optimization.” Journal of Computational and Graphical Statistics, 20(1).
  • Hankin, R. K. S. (2006). “Special functions in R: Introducing the gsl package.” R News, 6.
  • Harvey, A. C. and Fernandes, C. (1989). “Time Series Models for Count or Qualitative Observations.” Journal of Business and Economic Statistics, 7(4): 407–417.
  • Johannes, M. S., Polson, N. G., and Stroud, J. R. (2009). “Optimal filtering of jump diffusions: Extracting latent states from asset prices.” Review of Financial Studies, 22(7): 2759–2799.
  • Jorgensen, B., Lundbye-Christensen, S., Song, P., and Sun, L. (1999). “A State Space Model for Multivariate Longitudinal Count Data.” Biometrika, 86: 169–181.
  • Kim, B. (2013). “Essays in Dynamic Bayesian Models in Marketing.” Ph.D. thesis, The George Washington University.
  • Lindley, D. V. and Singpurwalla, N. D. (1986). “Multivariate Distributions for the Life Lengths of Components of a System Sharing a Common Environment.” Journal of Applied Probability, 23: 418–431.
  • Lopes, H. F. and Polson, N. G. (2016). “Particle Learning for Fat-Tailed Distributions.” Econometric Reviews, 1–26.
  • Ord, K., Fernandes, C., and Harvey, A. C. (1993). “Time series models for multivariate series of count data.” In Developments in Time Series Analysis: In honour of Maurice B. Priestley, T. Subba Rao (ed.), 295–309.
  • Pedeli, X. and Karlis, D. (2011). “A bivariate INAR(1) model with application.” Statistical Modelling, 11: 325–349.
  • Pedeli, X. and Karlis, D. (2012). “On composite likelihood estimation of a multivariate INAR(1) model.” Journal of Time Series Analysis, 33: 903–915.
  • Polson, N. G., Stroud, J. R., and Müller, P. (2008). “Practical filtering with sequential parameter learning.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(2): 413–428.
  • Ravishanker, N., Serhiyenko, V., and Willig, M. R. (2014). “Hierarchical dynamic models for multivariate times series of counts.” Statistics and Its Interface, 7: 559–570.
  • Ravishanker, N., Venkatesan, R., and Hu, S. (2015). “Dynamic models for time series of counts with a marketing application.” In Handbook of discrete-valued time series, R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker (eds.), 423–445.
  • Serhiyenko, V., Ravishanker, N., and Venkatesan, R. (2015). “Approximate Bayesian estimation for multivariate count time series models.” In Ordered Data Analysis, Modeling and Health Research Methods, P. K. Choudhary et al. (eds.), 155–167.
  • Smith, R. and Miller, J. E. (1986). “A Non-Gaussian State Space Model and Application to Prediction of Records.” Journal of the Royal Statistical Society, Series B, 48(1): 79–88.
  • Storvik, G. (2002). “Particle filters for state-space models with the presence of unknown static parameters.” IEEE Transactions on Signal Processing, 50(2): 281–289.
  • Taddy, M. A. (2010). “Autoregressive mixture models for dynamic spatial Poisson processes: Application to tracking intensity of violent crime.” Journal of the American Statistical Association, 105(492): 1403–1417.
  • Taddy, M. A. and Kottas, A. (2012). “Mixture modeling for marked Poisson processes.” Bayesian Analysis, 7(2): 335–362.
  • Weinberg, J., Brown, L. D., and Stroud, J. R. (2007). “Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data.” Journal of the American Statistical Association, 102(480): 1185–1198.
  • Zeger, S. L. (1988). “A regression model for time series of counts.” Biometrika, 75(4): 621–629.

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