Simulating of poisson point process using conditional intensity function (Hazard function)

  • Authors

    • Behrouz Fathi-Vajargah
    • Hassan Khoshkar-Foshtomi University of guilan
    2014-04-24
    https://doi.org/10.14419/ijasp.v2i1.2165

  • In this article we first study linear point process then based on introducing conditional intensity function (Hazard function), we present an algorithm for simulating Monte Carlo point process and try to test and study its behaviors on some individuals of time points.

     

    Keywords: Conditional Intensity Function, Monte Carlo Simulation, Point Process, Time Points.

  • References

    1. Badar, M. G., and Priest, A. M., “Statistical aspects of fiber and bundle strength in hybrid composites”, Progress in Science and Engineering Composites, Hayashi, T., Kawata, K. and Umekawa, S. (Eds.), ICCM- IV, Tokyo, (1982) 1129-1136.
    2. Bennett, S., “Logistic regression model for survival data”, Applied Statistics, 32, (1983) 165-171.
    3. Bjerkedal, T., “Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli”, American Journal of Hygiene, 72, (1960) 130-148.
    4. Bremaud, P. and Massoulie, L., “Hawkes branching point processes without ancestors”, J. Appl. Probab, 38, (2001) 122–135.
    5. Brown, T. M. and Nair, M., “A simple proof of the multivariate random time change theorem for point processes”, J. Appl. Probab, 25, (1988) 210–214.
    6. Carlsson, H. and Nerman, O., “An alternative proof of Lorden’s renewal inequality”, Adv. Appl. Probab, 18, (1986) 1015–1016.
    7. Chung, K. L., “Crudely stationary point processes”, Amer. Math. Monthly, 79, (1972) 867–877.
    8. Copson, E. C., An introduction to the theory of functions of a complex variable, Oxford University Press, Oxford, (1935).
    9. Daley, D. J. and Vere-Jones, D., An introduction to the theory of point processes, Volume I: Elementary Theory and Methods. Springer, New York, 2nd edition, (2003).
    10. Daley, D. J. and Vere-Jones, D., An introduction to the theory of point processes, Volume II: General Theory and Structure. Springer, New York, 2nd edition, (2008).
    11. Daley, D. J., “On a class of renewal functions”, Proc. Cambridge Philos. Soc. 61, (1965) 519–526.
    12. Daley, D.J. and Vere-Jones, D., “A summary of the theory of point processes”, In Lewis, (1972) 299–383.
    13. Davies, R.B., “Testing the hypothesis that a point process is Poisson”, Adv. Appl. Probab, 9, (1977) 724–746.
    14. Desmond, A. F., “Stochastic models of failure in random environments”, Canadian Journal of Statistics, 13, (1985) 171-183.
    15. Diggle, P.J., Gratton, R.J., “Monte Carlo methods of inference for implicit statistical models (with discussion)”, J. Royal Statist. Soc. Ser. B, 46, (1984) 193–227.
    16. Doob, J.L., “Renewal theory from the point of view of the theory of probability”, Trans. Amer. Math. Soc. 63, (1948) 422–438.
    17. Glaser, R. A., “Bathtub and related failure rate characterization”, Journal of the American Statistical Association, 75, (1980) 667-672.
    18. Gupta, R. C., Kannan, N. and RayChoudhuri, A., “Analysis of log-normal survival data”, Mathematical Biosciences, 139, (1997) 103-115.
    19. Hawkes, A. G., “Point spectra of some mutually exciting point processes”, Journal of the Royal Statistical Society, Series B, 33, (1971) 438–443.
    20. Hawkes, A. G., “Spectra of some self-exciting and mutually exciting point processes”, Journal of Biometrika, 58(1), (1971) 83–90.
    21. Hawkes, A. G., “Spectra of some mutually exciting point processes with associated variables”, In P. A. W. Lewis, editor, Stochastic Point Processes, Wiley, New York, (1972) 261–271.
    22. Hawkes, A. G. and Oakes, D., “A cluster representation of a self-exciting process”, Journal of Applied Probability, 11, (1974) 493–503.
    23. Isham, V. and Westcott, M., “A self-correcting point process”, Stoch. Proc. Appl., 8, (1979) 335–347.
    24. Ludwig, J.A., Reynolds, J.F., Statistical ecology: A primer on methods and computing, John Wiley & Sons, New York, (1988).
    25. Mann, N. R., Schafer, R. E., and Singpurwalla, N. D., Methods for statistical analysis of reliability and life data, John Wiley & Sons, New York, (1974).
    26. Møller, J. and Rasmussen, J. G., “Perfect simulation of hawkes processes”, Adv. in Appl. Probab., 37(3), (2005) 629–646.
    27. Møller, J. and Rasmussen, J. G., “Approximate simulation of hawkes processes”, Methodol. Comput. Appl. Probab., 8, (2006) 53–65.
    28. Møller, J. and Waagepetersen, R. P., Statistical inference and simulation for spatial point processes. Chapman & Hall, Boca Raton, Florida, (2004).
    29. Ogata, Y., “On Lewis’ simulation method for point processes”, IEEE Trans- actions on Information Theory, IT-27(1), (1981) 23–31.
    30. Ogata, Y., “Statistical models for earthquake occurrences and residual analysis for point processes”, Journal of the American Statistical Association, 83(401), (1988) 9–27.
    31. Ogata, Y., “Space-time point-process models for earthquake occurrences”, Annals of the Institute of Statistical Mathematics, 50(2), (1998) 379–402.
    32. Parzen, E., Stochastic processes, Holden-Day, San Francisco, (1962).
    33. Picinbono B., “Time intervals and counting in point processes”, IEEE Trans. Inform. Theory, 50, (2004) 1336-1340.
    34. Pinelis, I., “Monotonicity properties of the relative error of a Pade approximation for Mills' ratio”, Journal of Inequalities in Pure and Applied Mathematics, 3, 20, (2002).
    35. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, U.K. (1991).
    36. Rasmussen, J. G., “Temporal point processes the conditional intensity function”, January 24, (2011).
    37. Ross, Sh. M., Introduction to probability models, Academic Press, San Diego, 8nd edition, (2003).
    38. Rubenstein, R. Y., Simulation and the Monte Carlo method, Wiley, New York, (1981).
    39. Schmeiser, B. W., “Random variate generation a survey”, Proc, Winter Simulation Conf., Orlando, FL, (1980) 79-104.
    40. Xie, F.C. and Wei, B.C., “Diagnostics analysis for log-Birnbaum-Saunders re- gression models”, Computational Statistics & Data Analysis, 51, (2007) 4692-4706.

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