Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Aguilera AM, Ocaña FA, Valderrama MJ (1997) An approximated principal component prediction model for continuous-time stochastic processes. Appl Stoch Models Data Anal 13:61–72

    Article  MathSciNet  Google Scholar 

  • Asha G, Nair UN (2010) Reliability properties of mean time to failure in age replacement models. Int J Reliab Qual Saf Eng 17:15–26

    Article  Google Scholar 

  • Barta P, Miller M, Qiu A (2005) A stochastic model for studying the laminar structure of cortex from mri. IEEE Trans Med Imaging 24:728–742

    Article  Google Scholar 

  • 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

  • Bouzas PR, Aguilera AM, Valderrama MJ (2002) Forecasting a class of doubly stochastic Poisson processes. Stat Pap 43:507–523

    Article  MathSciNet  Google Scholar 

  • 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 50:2655–2667

    Article  MathSciNet  Google Scholar 

  • Bouzas PR, Ruiz-Fuentes N, Ocaña FM (2007) Functional approach to the random mean of a compound Cox process. Comput Stat 22:467–479

    Article  MathSciNet  Google Scholar 

  • 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 103:116–121

    Article  Google Scholar 

  • 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 839–846

    Google Scholar 

  • Bouzas PR, Aguilera AM, Ruiz-Fuentes N (2012) Functional estimation of the random rate of a Cox process. Methodol Comput Appl Probab 14:57–69

    Article  MathSciNet  Google Scholar 

  • 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 50:1006–1024

    Article  MathSciNet  Google Scholar 

  • Chertok AV, Korolev VY, Korchagin AY (2016) Modeling high-frequency non-homogeneous order flows by compound Cox processes. J Math Sci 214:44–68

    Article  MathSciNet  Google Scholar 

  • Dousse O, Baccelli F, Thiran P (2005) Impact of interferences on connectivityin ad hoc networks. IEEE/ACM Trans Netw 13:425–436

    Article  Google Scholar 

  • Economou A (2003) On the control of a compound inmigration process through total catastrophes. Eur J Oper Res 147:522–529

    Article  Google Scholar 

  • Genaro AD, Simonis A (2015) Estimating doubly stochastic Poisson process with affine intensities by Kalman filter. Stat Pap 56:723–748

    Article  MathSciNet  Google Scholar 

  • Gospodinov D, Rotondi R (2001) Exploratory analysis of marked Poisson processes applied to Balkan earthquake sequences. J Balkan Geophys Soc 4:61–68

    Google Scholar 

  • Greenberg D, Houweling A, Kerr J (2008) Population imaging of ongoing neuronal activity in the visual cortex of awake rats. Nat Neurosci 11:749–751

    Article  Google Scholar 

  • He Q (2014) Fundamentals of matrix-analytic methods. Springer, New York

    Book  Google Scholar 

  • Lefebvre M, Belsalma F (2015) Modeling and forecasting river flows by means of filtered Poisson processes. Appl Math Model 39:230–243

    Article  MathSciNet  Google Scholar 

  • Lin XS, Pavlova KP (2006) The compound Poisson risk model with a threshold dividend strategy. Insur Math Econ 38:57–80

    Article  MathSciNet  Google Scholar 

  • Neuts MF (1975) Probability distributions of phase type. Liber. Amicorum Prof. Emeritus. H. Florin. Department of Mathematics, University of Louvain, Belgium

  • Neuts MF (1981) Matrix geometric solutions in stochastic models: an algorithmic approach. Dover, New York

    MATH  Google Scholar 

  • Ogata Y (1998) Space–time point-process models for earthquake occurrences. Ann Inst Stat Math 50:379–402

    Article  Google Scholar 

  • Park C, Padgett W (2005) Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Anal 11:511–527

    Article  MathSciNet  Google Scholar 

  • Ruiz-Castro JE (2016a) Complex multi-state systems modelled through marked markovian arrival processes. Eur J Oper Res 252:852–865

    Article  MathSciNet  Google Scholar 

  • 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 145:155–168

    Article  Google Scholar 

  • Russell JR, Engle RF (2010) Analysis of high-frequency data. In: Handbook of financial econometrics: tools and techniques. Elsevier, Amsterdam, pp 383–426

  • Sepehrifar M, Yarahmadian S (2017) Decreasing renewal dichotomous Markov noise shock model with hypothesis testing applications. Stat Pap 58:1115–1124

    Article  MathSciNet  Google Scholar 

  • Si S (2001) Random irreversible phenomena: entropy in subordination. Chaos Solitons Fractals 12:2873–2876

    Article  Google Scholar 

  • Singh S, Tripathi YM (2018) Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring. Stat Pap 12:2873–2876

    MathSciNet  MATH  Google Scholar 

  • Snyder DL, Miller MI (1991) Random point processes in time and space, 2nd edn. Springer, New York

    Book  Google Scholar 

  • Tank F, Eryilmaz S (2015) The distributions of sum, minima and maxima of generalized geometric random variables. Stat Pap 56:1191–1203

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Granada, Granada, Spain

    Paula R. Bouzas, Carmen Montes-Gijón & Juan Eloy Ruiz-Castro

  2. Department of Statistics and Operations Research, University of Jaén, Jaén, Spain

    Nuria Ruiz-Fuentes

Authors
  1. Paula R. Bouzas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nuria Ruiz-Fuentes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carmen Montes-Gijón
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Juan Eloy Ruiz-Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paula R. Bouzas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouzas, P.R., Ruiz-Fuentes, N., Montes-Gijón, C. et al. Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events. Stat Papers 62, 235–265 (2021). https://doi.org/10.1007/s00362-019-01092-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-019-01092-0

Keywords

Mathematics Subject Classification

Search

Navigation