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A simple algorithm for computing the probabilities of count models based on pure birth processes

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Abstract

Recently, non-monotonic rate sequences of pure birth processes have been the focus of much attention in the analysis of count data due to their ability to provide a combination of over-, under-, and equidispersed distributions without the need to reuse covariates (traditional methods). They also permit the modeling of excess counts, a frequent issue arising when using count models based on monotonic rate sequences such as the Poisson, gamma, Weibull, Conway-Maxwell-Poisson (CMP), Faddy (1997), etc. Matrix-exponential approaches have always been used for computing the probabilities for count models based on pure birth processes, although none have been proposed for them as a specific algorithm. It is intractable to calculate these pure birth probabilities numerically in an analytic form because severe numerical cancellations may occur. However, we circumvent this difficulty by exploiting a Taylor series expansion, and then a new analytic form is derived. We developed a simple algorithm for efficiently implementing the new formula and conducted numerical experiments to study the efficiency and accuracy of the developed algorithm. The results indicate that this new approach is faster and more accurate than the matrix-exponential methods.

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References

  • Ball F (1995) A note on variation in birth processes. Math Sci 20:50–55

    MathSciNet  Google Scholar 

  • Banks WD, Martin G (2013) Optimal primitive sets with restricted primes. Integers 13:A69

    MathSciNet  Google Scholar 

  • Barreto-Souza W, Simas AB (2016) General mixed Poisson regression models with varying dispersion. Stat Comput 26(6):1263–1280

    Article  MathSciNet  Google Scholar 

  • Bartlett MS (1978) An introduction to stochastic processes. Cambridge University Press, Cambridge

    Google Scholar 

  • Bourguignon M, de Medeiros RMR (2022) A simple and useful regression model for fitting count data. TEST 31(3):790–827

    Article  MathSciNet  Google Scholar 

  • Chanialidis C, Evers L, Neocleous T et al (2018) Efficient Bayesian inference for COM-Poisson regression models. Stat Comput 28:595–608

    Article  MathSciNet  Google Scholar 

  • Cheney EW, Kincaid DR (2004) Numerical mathematics and computing, 5th edn. Brooks/Cole Publishing Company, Belmont

    Google Scholar 

  • Conway RW, Maxwell WL (1962) A queuing model with state dependent service rates. J Ind Eng 12(2):132–136

    Google Scholar 

  • Cox DR, Miller H (1965) The Theory of Stochastic Processes. Chapman and Hall, London

    Google Scholar 

  • Crawford FW, Ho LST, Suchard MA (2018) Computational methods for birth-death processes. WIREs Comput Stat 10(2):1–22

    Article  MathSciNet  Google Scholar 

  • Daniels HE (1982) The saddlepoint approximation for a general birth process. J Appl Probab 19(1):20–28

    Article  MathSciNet  Google Scholar 

  • Eddelbuettel D, Francois R, Allaire J, et al (2023) Rcpp: seamless R and C++ integration. https://CRAN.R-project.org/package=Rcpp, R package version 1.0.10

  • Faddy MJ (1994) On variation in Poisson processes. Math Sci 19(1):47–51

    MathSciNet  Google Scholar 

  • Faddy MJ (1997) Extended Poisson process modelling and analysis of count data. Biom J 39(4):431–440

    Article  Google Scholar 

  • Faddy MJ, Smith DM (2008) Extended Poisson process modelling of dilution series data. J Roy Stat Soc: Ser C (Appl Stat) 57(4):461–471

    MathSciNet  Google Scholar 

  • Faddy MJ, Smith DM (2011) Analysis of count data with covariate dependence in both mean and variance. J Appl Stat 38(12):2683–2694

    Article  MathSciNet  Google Scholar 

  • Faddy MJ, Smith DM (2012) Extended Poisson process modelling and analysis of grouped binary data. Biom J 54(3):426–435

    Article  MathSciNet  Google Scholar 

  • Forthmann B, Gühne D, Doebler P (2020) Revisiting dispersion in count data item response theory models: The Conway-Maxwell-Poisson counts model. Br J Math Stat Psychol 73:32–50

    Article  Google Scholar 

  • Goulet V, Dutang C, Maechler M, et al (2022) expm: matrix exponential, log, ‘etc’. https://CRAN.R-project.org/package=expm, R package version 0.999-7

  • Guikema SD, Goffelt JP (2008) A flexible count data regression model for risk analysis. Risk Anal Int J 28(1):213–223

    Article  Google Scholar 

  • Higham NJ (2005) The scaling and squaring method for the matrix exponential revisited. SIAM J Matrix Anal Appl 26(4):1179–1193

    Article  MathSciNet  Google Scholar 

  • Higham NJ (2008) Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, Philadelphia

    Book  Google Scholar 

  • Higham NJ (2009) The scaling and squaring method for the matrix exponential revisited. SIAM Rev 51(4):747–764

    Article  MathSciNet  Google Scholar 

  • Huang J, Zhu F (2021) A new first-order integer-valued autoregressive model with Bell innovations. Entropy 23(6):713

    Article  MathSciNet  Google Scholar 

  • Jensen A (1953) Markoff chains as an aid in the study of Markoff processes. Scand Actuar J 1953(sup1):87–91

    Article  MathSciNet  Google Scholar 

  • Jung RC, Ronning G, Tremayne AR (2005) Estimation in conditional first order autoregression with discrete support. Stat Pap 46(2):195–224

    Article  MathSciNet  Google Scholar 

  • Kharrat T, Boshnakov GN (2022) Countr: flexible univariate count models based on renewal processes. https://CRAN.R-project.org/package=Countr, R package version 3.5.6

  • Kharrat T, Boshnakov GN, McHale I et al (2019) Flexible regression models for count data based on renewal processes. J Stat Softw 90(13):1–35

    Article  Google Scholar 

  • Maechler M, Maechler MM, MPFR S, et al (2024) Package ‘Rmpfr’. https://CRAN.R-project.org/package=Rmpfr, R package version 0.9-5

  • MATLAB (2017) version 9.2.0.538062 (R2017a). The MathWorks, Inc., Natick, Massachusetts, United States

  • McShane B, Adrian M, Bradlow ET et al (2008) Count models based on Weibull interarrival times. J Bus Econ Stat 26(3):369–378

    Article  MathSciNet  Google Scholar 

  • Minka T, Shmueli G, Kadane J, et al (2003) Computing with the COM-poisson distribution. Technical report CMU Statistics Department 776

  • Moler C, Van Loan C (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev 45(3):3–49

    Article  MathSciNet  Google Scholar 

  • Morales-Otero M, Núñez-Antón V (2021) Comparing Bayesian spatial conditional overdispersion and the Besag-York-Mollié models: application to infant mortality rates. Mathematics 9(3):282

    Article  Google Scholar 

  • Peluso A, Vinciotti V, Yu K (2019) Discrete Weibull generalized additive model: an application to count fertility data. J Roy Stat Soc Ser C (Appl Stat) 68(3):565–583

    Article  MathSciNet  Google Scholar 

  • Philipson P, Huang A (2023) A fast look-up method for Bayesian mean-parameterised Conway-Maxwell-Poisson regression models. Stat Comput 33(81):1–16

    MathSciNet  Google Scholar 

  • Podlich HM, Faddy MJ, Smyth GK (2004) Semi-parametric extended Poisson process models for count data. Stat Comput 14:311–321

    Article  MathSciNet  Google Scholar 

  • R Core Team (2022) R: a language and environment for statistical computing. R foundation for statistical computing, Vienna, Austria, https://www.R-project.org/

  • Renshaw E (2011) Stochastic population processes: analysis, approximations, simulations. Oxford University Press, Oxford

    Book  Google Scholar 

  • Rovenţa I, Temereancă LE (2019) A note on the positivity of the even degree complete homogeneous symmetric polynomials. Mediterr J Math 16(1):1–16

    Article  MathSciNet  Google Scholar 

  • Saez-Castillo AJ, Conde-Sanchez A (2013) A hyper-Poisson regression model for overdispersed and underdispersed count data. J Comput Stat Data Anal 61:148–157

    Article  MathSciNet  Google Scholar 

  • Sellers KF, Shmueli G (2009) A regression model for count data with observation-level dispersion. In: Paper presented at the 24th international workshop on statistical modelling, Cornell University, New York, Accessed 20–24 July 2009

  • Sellers KF, Shmueli G (2010) A flexible regression model for count data. Ann Appl Stat 4(2):943–961

    Article  MathSciNet  Google Scholar 

  • Sellers KF, Lotze T, Raim AM (2019) COMPoissonReg: Conway-Maxwell Poisson (COM-Poisson) regression. https://CRAN.R-project.org/package=COMPoissonReg, R package version 0.7.0

  • Skulpakdee W, Hunkrajok M (2022a) 3: a sue-Poisson inarch(1) model. In: Paper presented at the 6th international conference on compute and data analysis, Shanghai, China. Accessed 25-27 February 2022

  • Skulpakdee W, Hunkrajok M (2022) Unusual-event processes for count data. SORT Stat Oper Res Trans 46(1):39–66

    MathSciNet  Google Scholar 

  • Smith DM, Faddy MJ (2016) Mean and variance modelling of under- and overdispersed count data. J Stat Softw 69(6):1–23

    Article  Google Scholar 

  • Smith DM, Faddy MJ (2018) CountsEPPM: Mean and variance modeling of count data. https://CRAN.R-project.org/package=CountsEPPM, R package version 3.0

  • Smith DM, Faddy MJ (2019a) BinaryEPPM: mean and variance modeling of binary data. https://CRAN.R-project.org/package=BinaryEPPM, R package version 2.3

  • Smith DM, Faddy MJ (2019) Mean and variance modeling of under-dispersed and over-dispersed grouped binary data. J Stat Softw 90(8):1–20

    Article  Google Scholar 

  • Smyth GK, Podlich HM (2002) An improved saddlepoint approximation based on the negative binomial distribution for the general birth process. Comput Stat 17(1):17–28

    Article  MathSciNet  Google Scholar 

  • Watkins DS (2010) Fundamentals of matrix computations, 3rd edn. Wiley, New Jersey

    Google Scholar 

  • Weiß CH (2018) An introduction to discrete-valued time series, 1st edn. Wiley, New Jersey

    Book  Google Scholar 

  • Weiß CH, Zhu F, Hoshiyar A (2022) Softplus INGARCH models. Stat Sin 32(2):099–1120

    MathSciNet  Google Scholar 

  • Winkelmann R (1995) Duration dependence and dispersion in count-data models. J Bus Econ Stat 13(4):467–474

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the associate editor and anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. Independent Researcher, Bangkok, Thailand

    Mongkol Hunkrajok

  2. Graduate School of Applied Statistics, National Institute of Development Administration, Bangkok, Thailand

    Wanrudee Skulpakdee

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  1. Mongkol Hunkrajok
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  2. Wanrudee Skulpakdee
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Correspondence to Wanrudee Skulpakdee.

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Hunkrajok, M., Skulpakdee, W. A simple algorithm for computing the probabilities of count models based on pure birth processes. Comput Stat (2024). https://doi.org/10.1007/s00180-024-01491-4

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