Advertisement

Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events

  • Paula R. BouzasEmail author
  • Nuria Ruiz-Fuentes
  • Carmen Montes-Gijón
  • Juan Eloy Ruiz-Castro
Regular Article
First Online:
  • 19 Downloads

Abstract

Compound Cox processes (CCP) are flexible marked point processes due to the stochastic nature of their intensity. This paper states closed-form expressions of their counting and time statistics in terms of the intensity and of the mean processes. They are forecast by means of principal components prediction models applied to the mean process in order to reach attainable results. A proposition proves that only weak restrictions are needed to estimate the probability of a new occurrence. Additionally, the phase type process is introduced, which important feature is that its marginal distributions are phase type with random parameters. Since any non-negative variable can be approximated by a phase-type distribution, the new stochastic process is proposed to model the intensity process of any point process. The CCP with this type of intensity provides an especially general model. Several simulations and the corresponding study of the estimation errors illustrate the results and their accuracy. Finally, an application to real data is performed; extreme temperatures in the South of Spain are modeled by a CPP and forecast.

Keywords

Compound Cox process Estimation Principal components prediction Phase type process 

Mathematics Subject Classification

60G25 60G51 60G55 62M20 62M99 90C15 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This work was supported by Ministerio de Economía y Competitividad (project MTM2013-47929-P) and Consejería de Innovación de la Junta de Andalucía (Grants FQM-307 and FQM-246), all in Spain.

References

  1. Aguilera AM, Ocaña FA, Valderrama MJ (1997) An approximated principal component prediction model for continuous-time stochastic processes. Appl Stoch Models Data Anal 13:61–72MathSciNetCrossRefzbMATHGoogle Scholar
  2. Asha G, Nair UN (2010) Reliability properties of mean time to failure in age replacement models. Int J Reliab Qual Saf Eng 17:15–26CrossRefGoogle Scholar
  3. Barta P, Miller M, Qiu A (2005) A stochastic model for studying the laminar structure of cortex from mri. IEEE Trans Med Imaging 24:728–742CrossRefGoogle Scholar
  4. Bieniek M, Goroncy A (2017) Sharp lower bounds on expectations of GOS based on DGFR distributions. Stat Pap.  https://doi.org/10.1007/s00362-017-0972-y
  5. Bouzas PR, Aguilera AM, Valderrama MJ (2002) Forecasting a class of doubly stochastic Poisson processes. Stat Pap 43:507–523MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bouzas PR, Valderrama MJ, Aguilera AM, Ruiz-Fuentes N (2006) Modelling the mean of a doubly stochastic poisson process by functional data analysis. Comput Stat Data Anal 50:2655–2667MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bouzas PR, Ruiz-Fuentes N, Ocaña FM (2007) Functional approach to the random mean of a compound Cox process. Comput Stat 22:467–479MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bouzas PR, Ruiz-Fuentes N, Matilla A, Aguilera AM, Valderrama MJ (2010a) A cox model for radioactive counting measure: inference on the intensity process. Chemom Intell Lab Syst 103:116–121CrossRefGoogle Scholar
  9. Bouzas PR, Ruiz-Fuentes N, Ruiz-Castro JE (2010) Forecasting a compound cox process by means of PCP. In: Lechevallier Y, Saporta G (eds) Proceedings of COMPSTAT’2010. Physica-Verlag, Heidelberg, pp 839–846Google Scholar
  10. Bouzas PR, Aguilera AM, Ruiz-Fuentes N (2012) Functional estimation of the random rate of a Cox process. Methodol Comput Appl Probab 14:57–69MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chen F, Hall P (2013) Inference for a non-stationary self-exciting point process with an application in ultra-high frequency financial data modeling. J Appl Probab 50:1006–1024MathSciNetCrossRefzbMATHGoogle Scholar
  12. Chertok AV, Korolev VY, Korchagin AY (2016) Modeling high-frequency non-homogeneous order flows by compound Cox processes. J Math Sci 214:44–68MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dousse O, Baccelli F, Thiran P (2005) Impact of interferences on connectivityin ad hoc networks. IEEE/ACM Trans Netw 13:425–436CrossRefGoogle Scholar
  14. Economou A (2003) On the control of a compound inmigration process through total catastrophes. Eur J Oper Res 147:522–529MathSciNetCrossRefzbMATHGoogle Scholar
  15. Genaro AD, Simonis A (2015) Estimating doubly stochastic Poisson process with affine intensities by Kalman filter. Stat Pap 56:723–748MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gospodinov D, Rotondi R (2001) Exploratory analysis of marked Poisson processes applied to Balkan earthquake sequences. J Balkan Geophys Soc 4:61–68Google Scholar
  17. Greenberg D, Houweling A, Kerr J (2008) Population imaging of ongoing neuronal activity in the visual cortex of awake rats. Nat Neurosci 11:749–751CrossRefGoogle Scholar
  18. He Q (2014) Fundamentals of matrix-analytic methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  19. Lefebvre M, Belsalma F (2015) Modeling and forecasting river flows by means of filtered Poisson processes. Appl Math Model 39:230–243MathSciNetCrossRefzbMATHGoogle Scholar
  20. Lin XS, Pavlova KP (2006) The compound Poisson risk model with a threshold dividend strategy. Insur Math Econ 38:57–80MathSciNetCrossRefzbMATHGoogle Scholar
  21. Neuts MF (1975) Probability distributions of phase type. Liber. Amicorum Prof. Emeritus. H. Florin. Department of Mathematics, University of Louvain, BelgiumGoogle Scholar
  22. Neuts MF (1981) Matrix geometric solutions in stochastic models: an algorithmic approach. Dover, New YorkzbMATHGoogle Scholar
  23. Ogata Y (1998) Space–time point-process models for earthquake occurrences. Ann Inst Stat Math 50:379–402CrossRefzbMATHGoogle Scholar
  24. Park C, Padgett W (2005) Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Anal 11:511–527MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ruiz-Castro JE (2016a) Complex multi-state systems modelled through marked markovian arrival processes. Eur J Oper Res 252:852–865MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ruiz-Castro JE (2016b) Markov counting and reward processes for analyzing the performance of a complex system subject to random inspections. Reliab Eng Syst Saf 145:155–168CrossRefGoogle Scholar
  27. Russell JR, Engle RF (2010) Analysis of high-frequency data. In: Handbook of financial econometrics: tools and techniques. Elsevier, Amsterdam, pp 383–426Google Scholar
  28. Sepehrifar M, Yarahmadian S (2017) Decreasing renewal dichotomous Markov noise shock model with hypothesis testing applications. Stat Pap 58:1115–1124MathSciNetCrossRefzbMATHGoogle Scholar
  29. Si S (2001) Random irreversible phenomena: entropy in subordination. Chaos Solitons Fractals 12:2873–2876CrossRefzbMATHGoogle Scholar
  30. Singh S, Tripathi YM (2018) Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring. Stat Pap 12:2873–2876MathSciNetzbMATHGoogle Scholar
  31. Snyder DL, Miller MI (1991) Random point processes in time and space, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  32. Tank F, Eryilmaz S (2015) The distributions of sum, minima and maxima of generalized geometric random variables. Stat Pap 56:1191–1203MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Paula R. Bouzas
    • 1
    Email author
  • Nuria Ruiz-Fuentes
    • 2
  • Carmen Montes-Gijón
    • 1
  • Juan Eloy Ruiz-Castro
    • 1
  1. 1.Department of Statistics and Operations ResearchUniversity of GranadaGranadaSpain
  2. 2.Department of Statistics and Operations ResearchUniversity of JaénJaénSpain

Personalised recommendations