Bayesian Analysis
- Bayesian Anal.
- Volume 7, Number 2 (2012), 335-362.
Mixture Modeling for Marked Poisson Processes
Matthew A. Taddy and Athanasios Kottas
Full-text: Open access
Abstract
We propose a general inference framework for marked Poisson processes observed over time or space. Our modeling approach exploits the connection of nonhomogeneous Poisson process intensity with a density function. Nonparametric Dirichlet process mixtures for this density, combined with nonparametric or semiparametric modeling for the mark distribution, yield flexible prior models for the marked Poisson process. In particular, we focus on fully nonparametric model formulations that build the mark density and intensity function from a joint nonparametric mixture, and provide guidelines for straightforward application of these techniques. A key feature of such models is that they can yield flexible inference about the conditional distribution for multivariate marks without requiring specification of a complicated dependence scheme. We address issues relating to choice of the Dirichlet process mixture kernels, and develop methods for prior specification and posterior simulation for full inference about functionals of the marked Poisson process. Moreover, we discuss a method for model checking that can be used to assess and compare goodness of fit of different model specifications under the proposed framework. The methodology is illustrated with simulated and real data sets.
Article information
Source
Bayesian Anal. Volume 7, Number 2 (2012), 335-362.
Dates
First available in Project Euclid: 16 June 2012
Permanent link to this document
https://projecteuclid.org/euclid.ba/1339878891
Digital Object Identifier
doi:10.1214/12-BA711
Mathematical Reviews number (MathSciNet)
MR2934954
Zentralblatt MATH identifier
1330.62200
Keywords
Bayesian nonparametrics Beta mixtures Dirichlet process Marked point process Multivariate normal mixtures Non-homogeneous Poisson process Nonparametric regression
Citation
Taddy, Matthew A.; Kottas, Athanasios. Mixture Modeling for Marked Poisson Processes. Bayesian Anal. 7 (2012), no. 2, 335--362. doi:10.1214/12-BA711. https://projecteuclid.org/euclid.ba/1339878891
References
- Andrews, D. F. and Herzberg, A. M. (1985). Data, A Collection of Problems from Many Fields for the Student and Research Worker. Springer-Verlag.Zentralblatt MATH: 0567.62002
- Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics 2, 1152 – 1174.Mathematical Reviews (MathSciNet): MR365969
Zentralblatt MATH: 0335.60034
Digital Object Identifier: doi:10.1214/aos/1176342871
Project Euclid: euclid.aos/1176342871 - Baddeley, A., R. Turner, J. Møller, and M. Hazelton (2005). Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society, Series B 67, 617–666.Mathematical Reviews (MathSciNet): MR2210685
Zentralblatt MATH: 1112.62302
Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00519.x - Best, N. G., K. Ickstadt, and R. L. Wolpert (2000). Spatial Poisson regression for health and exposure data measured at disparate resolutions. Journal of the American Statistical Association 95, 1076–1088.
- Brix, A. and P. J. Diggle (2001). Spatiotemporal prediction for log-Gaussian Cox processes. Journal of the Royal Statistical Society, Series B 63, 823–841.Mathematical Reviews (MathSciNet): MR1872069
Zentralblatt MATH: 0996.62076
Digital Object Identifier: doi:10.1111/1467-9868.00315 - Brix, A. and J. Møller (2001). Space-time multi type log Gaussian Cox processes with a view to modeling weeds. Scandinavian Journal of Statistics 28, 471–488.
- Brown, E. M., R. Barbieri, V. Ventura, R. E. Kass, and L. M. Frank (2001). The time-rescaling theorem and its application to neural spike train data analysis. Neural Computation 14, 325–346.
- Brunner, L. J. and A. Y. Lo (1989). Bayes methods for a symmetric uni-modal density and its mode. Annals of Statistics 17, 1550–1566.Mathematical Reviews (MathSciNet): MR1026299
Zentralblatt MATH: 0697.62003
Digital Object Identifier: doi:10.1214/aos/1176347381
Project Euclid: euclid.aos/1176347381 - Bush, C., J. Lee, and S. N. MacEachern (2010). Minimally informative prior distributions for nonparametric Bayesian analysis. Journal of the Royal Statistical Society: Series B 72, 253–268.Mathematical Reviews (MathSciNet): MR2830767
Digital Object Identifier: doi:10.1111/j.1467-9868.2009.00735.x - Cressie, N. A. C. (1993). Statistics for Spatial Data (Revised edition). Wiley.
- Daley, D. J. and D. Vere-Jones (2003). An Introduction to the Theory of Point Processes (2nd ed.). Springer.Mathematical Reviews (MathSciNet): MR950166
- Diaconis, P. and D. Ylvisaker (1985). Quantifying prior opinion. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith (Eds.), Bayesian Statistics, Volume 2. North-Holland.
- Diggle, P. J. (1990). A point process modelling approach to raised incidence of a rare phenomenon in the vicinity of a prespecified point. Journal of the Royal Statistical Society, Series A 153, 349–362.
- Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns (Second Edition). London, Arnold.Mathematical Reviews (MathSciNet): MR743593
- Escobar, M. and M. West (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90, 577–588.
- Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics 1, 209–230.Mathematical Reviews (MathSciNet): MR350949
Zentralblatt MATH: 0255.62037
Digital Object Identifier: doi:10.1214/aos/1176342360
Project Euclid: euclid.aos/1176342360 - Grunwald, G. K., A. E. Raftery, and P. Guttorp (1993). Time series of continuous proportions. Journal of the Royal Statistical Society, Series B 55, 103–116.
- Gutiérrez-Peña, E. and L. E. Nieto-Barajas (2003). Bayesian nonparametric inference for mixed Poisson processes. In J. Bernardo, M. Bayarri, J. Berger, A. Dawid, D. Heckerman, A. Smith, and M. West (Eds.), Bayesian Statistics 7, pp. 163–179. Oxford University Press.Mathematical Reviews (MathSciNet): MR2003172
- Guttorp, P. (1995). Stochastic Modelling of Scientific Data. Chapman & Hall.
- Heikkinen, J. and E. Arjas (1998). Non-parametric Bayesian estimation of a spatial Poisson intensity. Scandinavian Journal of Statistics 25, 435–450.
- Heikkinen, J. and E. Arjas (1999). Modeling a Poisson forest in variable elevations: a nonparametric Bayesian approach. Biometrics 55, 738–745.
- Hjort, N. L. (1990). Nonparametric Bayes Estimators based on beta processes in models for life history data. Annals of Statistics 18, 1259–1294.Mathematical Reviews (MathSciNet): MR1062708
Zentralblatt MATH: 0711.62033
Digital Object Identifier: doi:10.1214/aos/1176347749
Project Euclid: euclid.aos/1176347749 - Ickstadt, K. and R. L. Wolpert (1999). Spatial regression for marked point processes. In J. M. Bernardo, J. O. Berger, P. Dawid, and A. F. M. Smith (Eds.), Bayesian Statistics 6, pp. 323–341. Oxford University Press.
- Ishwaran, H. and L. F. James (2004). Computational methods for multiplicative intensity models using weighted gamma processes: Proportional hazards, marked point processes, and panel count data. Journal of the American Statistical Association 99, 175–190.Mathematical Reviews (MathSciNet): MR2054297
Zentralblatt MATH: 1089.62520
Digital Object Identifier: doi:10.1198/016214504000000179 - Ishwaran, H. and M. Zarepour (2002). Exact and approximate sum representations for the Dirichlet process. The Canadian Journal of Statistics 30, 269–283.
- Ji, C., D. Merl, T. B. Kepler, and M. West (2009). Spatial mixture modelling for unobserved point processes: Examples in immunofluorescence histology. Bayesian Analysis 4, 297 – 316.
- Kingman, J. F. C. (1993). Poisson Processes. Clarendon Press.Mathematical Reviews (MathSciNet): MR1207584
- Kottas, A. and A. E. Gelfand (2001). Bayesian semiparametric median regression modeling. Journal of the American Statistical Association 96, 1458–1468.Mathematical Reviews (MathSciNet): MR1946590
Digital Object Identifier: doi:10.1198/016214501753382363 - Kottas, A. and B. Sansó (2007). Bayesian mixture modeling for spatial Poisson process intensities, with applications to extreme value analysis. Journal of Statistical Planning and Inference 137, 3151–3163.Mathematical Reviews (MathSciNet): MR2365118
Zentralblatt MATH: 1114.62100
Digital Object Identifier: doi:10.1016/j.jspi.2006.05.022 - Kottas, A. and S. Behseta (2010). Bayesian nonparametric modeling for comparison of single-neuron firing intensities. Biometrics 66, 277–286.Mathematical Reviews (MathSciNet): MR2756715
Digital Object Identifier: doi:10.1111/j.1541-0420.2009.01230.x - Kottas, A., S. Behseta, D. E. Moorman, V. Poynor, and C. R. Olson (2012). Bayesian nonparametric analysis of neuronal intensity rates. Journal of Neuroscience Methods, 203, 241–253.
- Kuo, L. and S. K. Ghosh (1997). Bayesian nonparametric inference for nonhomogeneous Poisson processes. Technical report, Department of Statistics, University of Connecticut.
- Kuo, L. and T. Y. Yang (1996). Bayesian computation for nonhomogeneous Poisson processes in software reliability. Journal of the American Statistical Association 91, 763–773.
- Liang, S., B. P. Carlin, and A. E. Gelfand (2009). Analysis of Minnesota colon and rectum cancer point patterns with spatial and nonspatial covariate information. The Annals of Applied Statistics 3, 943–962.Mathematical Reviews (MathSciNet): MR2750381
Zentralblatt MATH: 1196.62143
Digital Object Identifier: doi:10.1214/09-AOAS240
Project Euclid: euclid.aoas/1254773273 - Lo, A. Y. (1992). Bayesian inference for Poisson process models with censored data. Journal of Nonparametric Statistics 2, 71–80.Mathematical Reviews (MathSciNet): MR1256374
Zentralblatt MATH: 05143373
Digital Object Identifier: doi:10.1080/10485259208832544 - Lo, A. Y. and C.-S. Weng (1989). On a class of Bayesian nonparametric estimates: II. Hazard rate estimates. Annals of the Institute for Statistical Mathematics 41, 227–245.Mathematical Reviews (MathSciNet): MR1006487
Zentralblatt MATH: 0716.62043
Digital Object Identifier: doi:10.1007/BF00049393 - MacEachern, S. N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Communications in Statistics: Simulation and Computation 23, 727–741.Mathematical Reviews (MathSciNet): MR1293996
Zentralblatt MATH: 0825.62053
Digital Object Identifier: doi:10.1080/03610919408813196 - MacEachern, S. N., M. Clyde, and J. S. Liu (1999). Sequential importance sampling for nonparametric Bayes models: The next generation. The Canadian Journal of Statistics 27, 251–267.
- Møller, J., A. R. Syversveen, and R. P. Waagepetersen (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics 25, 451–452.
- Møller, J. and R. P. Waagepetersen (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC.Mathematical Reviews (MathSciNet): MR2004226
- Müller, P., A. Erkanli, and M. West (1996). Bayesian curve fitting using multivariate normal mixtures. Biometrika 83, 67–79.Mathematical Reviews (MathSciNet): MR1399156
Zentralblatt MATH: 0865.62029
Digital Object Identifier: doi:10.1093/biomet/83.1.67 - Neal, R. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics 9, 249–265.Mathematical Reviews (MathSciNet): MR1823804
- Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In T. S. Ferguson, L. S. Shapley, and J. B. MacQueen (Eds.), Statistics, Probability and Game Theory, IMS Lecture Notes – Monograph Series, 30, pp. 245–267.
- Rathburn, S. L. and N. Cressie (1994). A space-time survival point process for a longleaf pine forest in southern Georgia. Journal of the American Statistical Association 89, 1164–1174.
- Rodriguez, A., D. B. Dunson, and A. E. Gelfand (2009). Nonparametric functional data analysis through Bayesian density estimation. Biometrika 96, 149–162.Mathematical Reviews (MathSciNet): MR2482141
Zentralblatt MATH: 1163.62027
Digital Object Identifier: doi:10.1093/biomet/asn054 - Sethuraman, J. (1994). A constructive definition of the Dirichlet prior. Statistica Sinica 4, 639–650.
- Taddy, M. (2010). Autoregressive mixture models for dynamic spatial Poisson processes: Application to tracking intensity of violent crime Journal of the American Statistical Association 105, 1403–1417.Mathematical Reviews (MathSciNet): MR2796559
Digital Object Identifier: doi:10.1198/jasa.2010.ap09655 - Taddy, M. and A. Kottas (2010). A Bayesian nonparametric approach to inference for quantile regression. Journal of Business and Economic Statistics 28, 357–369.Mathematical Reviews (MathSciNet): MR2723605
- Wolpert, R. L. and K. Ickstadt (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85, 251–267.
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