| Title:
|
Spatio-temporal modelling of a Cox point process sampled by a curve, filtering and Inference (English) |
| Author:
|
Frcalová, Blažena |
| Author:
|
Beneš, Viktor |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
6 |
| Year:
|
2009 |
| Pages:
|
912-930 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with Cox point processes in time and space with Lévy based driving intensity. Using the generating functional, formulas for theoretical characteristics are available. Because of potential applications in biology a Cox process sampled by a curve is discussed in detail. The filtering of the driving intensity based on observed point process events is developed in space and time for a parametric model with a background driving compound Poisson field delimited by special test sets. A hierarchical Bayesian model with point process densities yields the posterior. Markov chain Monte Carlo "Metropolis within Gibbs" algorithm enables simultaneous filtering and parameter estimation. Posterior predictive distributions are used for model selection and a numerical example is presented. The new approach to filtering is related to the residual analysis of spatio-temporal point processes. (English) |
| Keyword:
|
Cox point process |
| Keyword:
|
filtering |
| Keyword:
|
spatio-temporal process |
| MSC:
|
60D05 |
| MSC:
|
60G55 |
| MSC:
|
62M30 |
| idZBL:
|
Zbl 1192.60073 |
| idMR:
|
MR2650073 |
| . |
| Date available:
|
2010-06-02T19:24:33Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140029 |
| . |
| Reference:
|
[1] A. Baddeley, R. Turner, J. Møller, and M. Hazelton: Residual analysis for spatial point processes (with discussion).J. Royal Stat. Soc. B 67 (2005), 617–666. MR 2210685 |
| Reference:
|
[2] O. Barndorff-Nielsen and N. Shephard: Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial economics.J. Royal Stat. Soc. B 63 (2001), 167–241. MR 1841412 |
| Reference:
|
[3] O. Barndorff-Nielsen and J. Schmiegel: Lévy based tempo-spatial modelling; with applications to turbulence.Usp. Mat. Nauk 159 (2004), 63–90. MR 2068843 |
| Reference:
|
[4] V. Beneš and B. Frcalová: Modelling and simulation of a neurophysiological experiment by spatio-temporal point processes.Image Anal. Stereol. 1 (2008), 27, 47–52. |
| Reference:
|
[5] P. Brémaud: Point Process and Queues.Springer, New York 1981. MR 0636252 |
| Reference:
|
[6] A. Brix and J. Møller: Space-time multi type log Gaussian Cox processes with a view to modelling weed data.Scand. J. Statist. 28 (2002), 471–488. MR 1858412 |
| Reference:
|
[7] A. Brix and P. Diggle: Spatio-temporal prediction for log-Gaussian Cox processes.J. Royal Statist. Soc. B 63 (2001), 823–841. MR 1872069 |
| Reference:
|
[8] M. A. Clyde and R. L. Wolpert: Nonparametric function estimation using overcomplete dictionaries.Bayesian Statistics 8 (2007), 1–24. MR 2433190 |
| Reference:
|
[9] R. Cont and P. Tankov: Financial Modelling with Jump Processes.Chapman and Hall/CRC, Boca Raton 2004. MR 2042661 |
| Reference:
|
[10] D. Daley and D. Vere-Jones: An Introduction to the Theory of Point Processes I, II.Springer, New York 2003, 2008. MR 0950166 |
| Reference:
|
[11] A. Ergun, R. Barbieri, U. T. Eden, M. A. Wilson, and E. N. Brown: Construction of point process adaptive filter algorithms for neural systems using sequential Monte Carlo methods.IEEE Trans. Biomed. Engrg. 54 (2007), 3, 307–326. |
| Reference:
|
[12] P. M. Fishman and D. Snyder: The statistical analysis of space-time point processes.IEEE Trans. Inform. Theory 22 (1976), 257–274. MR 0418216 |
| Reference:
|
[13] G. Hellmund, M. Prokešová, and E. Vedel Jensen: Lévy based Cox point processes.Adv. Appl. Probab. 40 (2008), 3, 603–629. MR 2454025 |
| Reference:
|
[14] M. Jacobsen: Point Processes Theory and Applications.Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston 2006. MR 2189574 |
| Reference:
|
[15] A. F. Karr: Point Processes and Their Statistical Inference.Marcel Dekker, New York 1985. MR 1113698 |
| Reference:
|
[16] P. Lánský and J. Vaillant: Stochastic model of the overdispersion in the place cell discharge.BioSystems 58 (2000), 27–32. |
| Reference:
|
[17] R. Lechnerová, K. Helisová, and V. Beneš: Cox point processes driven by Ornstein–Uhlenbeck type processes.Method. Comp. Appl. Probab. 10 (2008), 3, 315–336. |
| Reference:
|
[18] J. Møller and R. Waagepetersen: Statistics and Simulations of Spatial Point Processes.World Sci., Singapore 2003. |
| Reference:
|
[19] J. Møller and C. Diaz-Avalos: Structured spatio-temporal shot-noise Cox point process models, with a view to modelling forest fires.Scand. J. Statist. (2009), to appear. MR 2675937 |
| Reference:
|
[20] Y. Ogata: Space-time point process models for eartquake occurences.Ann. Inst. Statist. Math. 50 (1998), 379–402. |
| Reference:
|
[21] J. Pedersen: The Lévy–Ito Decomposition of Independently Scattered Random Measure.Res. Report 2, MaPhySto, University of Aarhus 2003. |
| Reference:
|
[22] R. D. Peng, F. P. Schoenberg, and J. Woods: A space-time conditional intensity model for evaluating a wildfire hazard risk.J. Amer. Statist. Assoc. 100 (2005), 469, 26–35. MR 2166067 |
| Reference:
|
[23] B. S. Rajput and J. Rosinski: Spectral representations of infinitely divisible processes.Probab. Theory Related Fields 82 (1989), 451–487. MR 1001524 |
| Reference:
|
[24] G. Roberts, O. Papaspiliopoulos, and P. Dellaportas: Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes.J. Royal Statist. Soc. B 66 (2004), 369–393. MR 2062382 |
| Reference:
|
[25] D. L. Snyder: Filtering and detection for doubly stochastic Poisson processes.IEEE Trans. Inform. Theory 18 (1972), 91–102. Zbl 0227.62055, MR 0408953 |
| Reference:
|
[26] F. P. Schoenberg, D. R. Brillinger, and P. M. Guttorp: Point processes, spatial-temporal.In: Encycl. of Environmetrics (A. El. Shaarawi and W. Piegorsch, eds.), Vol. 3, Wiley, New York 2003, pp. 1573–1577. |
| Reference:
|
[27] J. Zhuang: Second-order residual analysis of spatiotemporal point processes and applications in model evaluation.J. Royal Statist. Soc. B 68 (2006), 635–653. Zbl 1110.62128, MR 2301012 |
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