A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system

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2014, 11(3): 573-597. doi: 10.3934/mbe.2014.11.573

A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system

Laura Martín-Fernández ^1, , Gianni Gilioli ^2, , Ettore Lanzarone ^3, , Joaquín Míguez ^4, , Sara Pasquali ^3, , Fabrizio Ruggeri ^3, and Diego P. Ruiz ^1,

1.	Departamento de Física Aplicada, Universidad de Granada, Avda. Fuentenueva s/n, 18071 Granada, Spain, Spain
2.	Department of Molecular and Translational Medicine, University of Brescia, Viale Europa 11, 25125 Brescia
3.	CNR-IMATI, Via Bassini 15, 20133 Milano, Italy
4.	Departamento de Teoría de la Señal y Comunicaciones, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

Received September 2012 Revised November 2013 Published January 2014

Abstract
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Functional response estimation and population tracking in predator-prey systems are critical problems in ecology. In this paper we consider a stochastic predator-prey system with a Lotka-Volterra functional response and propose a particle filtering method for: (a) estimating the behavioral parameter representing the rate of effective search per predator in the functional response and (b) forecasting the population biomass using field data. In particular, the proposed technique combines a sequential Monte Carlo sampling scheme for tracking the time-varying biomass with the analytical integration of the unknown behavioral parameter. In order to assess the performance of the method, we show results for both synthetic and observed data collected in an acarine predator-prey system, namely the pest mite Tetranychus urticae and the predatory mite Phytoseiulus persimilis.

Keywords: state-space model, Prey-predator system, parameter estimation, Rao-Blackwellized particle filter., population tracking.

Mathematics Subject Classification: Primary: 62F15, 65C35, 92D25; Secondary: 65C3.

Citation: Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573

References:

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[5]	A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Springer, 2008.
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[7]	G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions, Ecological Modelling, 170 (2003), 155-171. doi: 10.1016/S0304-3800(03)00223-0.
[8]	O. Cappé, S. J. Godsill and E. Moulines, An overview of existing methods and recent advances in sequential Monte Carlo, Proceedings of the IEEE, 95 (2007), 899-924.
[9]	J. Carpenter, P. Clifford and P. Fearnhead, Improved particle filter for nonlinear problems, IEE Proceedings - Radar, Sonar and Navigation, 146 (1999), 2-7. doi: 10.1049/ip-rsn:19990255.
[10]	S. R. Carpenter, K. L. Cottingham and C. A. Stow, Fitting predator-prey models to time series with observation errors, Ecology, 75 (1994), 1254-1264. doi: 10.2307/1937451.
[11]	R. Chen and J. S. Liu, Mixture Kalman filters, Journal of the Royal Statistics Society B, 62 (2000), 493-508. doi: 10.1111/1467-9868.00246.
[12]	N. Chopin, P. E. Jacob and O. Papaspiliopoulos, SMC2: An efficient algorithm for sequential analysis of state space models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), (2012). doi: 10.1111/j.1467-9868.2012.01046.x.
[13]	J. A. Comiskey, F. Dallmeier and A. Alonso, Framework for Assessment and Monitoring of Biodiversity, Academic Press, New York, 1999.
[14]	R. Douc, O. Cappe and E. Moulines, Comparison of resampling schemes for particle filtering, in ISPA 2005: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005, 64-69. doi: 10.1109/ISPA.2005.195385.
[15]	A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice, Springer, New York (USA), 2001.
[16]	A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208.
[17]	M. Dowd, A sequential Monte Carlo approach to marine ecological prediction, Environmetrics, 17 (2006), 435-455. doi: 10.1002/env.780.
[18]	M. Dowd, Estimating parameters for a stochastic dynamic marine ecological system, Environmetrics, 22 (2011), 501-515. doi: 10.1002/env.1083.
[19]	M. Dowd and R. Joy, Estimating behavioral parameters in animal movement models using a state-augmented particle filter, Ecology, 92 (2011), 568-575. doi: 10.1890/10-0611.1.
[20]	G. B. Durham and A. R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, J. Bus. Econ. Stat., 20 (2002), 297-316. doi: 10.1198/073500102288618397.
[21]	O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions, Econometrica, 69 (2001), 959-993. doi: 10.1111/1468-0262.00226.
[22]	B. Eraker, MCMC analysis of diffusion models with application to finance, J. Bus. Econ. Stat., 19 (2001), 177-191. doi: 10.1198/073500101316970403.
[23]	G. Gilioli, S. Pasquali and F. Ruggeri, Bayesian inference for functional response in a stochastic predator-prey system, Bulletin of Mathematical Biology, 70 (2008), 358-381. doi: 10.1007/s11538-007-9256-3.
[24]	G. Gilioli, S. Pasquali and F. Ruggeri, Nonlinear functional response parameter estimation in a stochastic predator-prey model, Mathematical Biosciences and Engineering, 9 (2012), 75-96. doi: 10.3934/mbe.2012.9.75.
[25]	G. Gilioli and V. Vacante, Aspetti della dinamica di popolazione del sistema tetranychus urticae- phytoseiulus persimilis in pieno campo: implicazioni per le strategie di lotta biologica, in Atti del Convegno "La difesa delle colture in agricoltura biologica" - Notiziario sulla protezione delle piante, vol. 13, 2001.
[26]	A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximations, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.
[27]	A. Golightly and D. J. Wilkinson, Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo, Interface Focus, 1 (2011), 807-820. doi: 10.1098/rsfs.2011.0047.
[28]	N. Gordon, D. Salmond and A. F. M. Smith, Novel approach to nonlinear and non-Gaussian Bayesian state estimation, IEE Proceedings-F, 140 (1993), 107-113. doi: 10.1049/ip-f-2.1993.0015.
[29]	C. Jost and S. P. Ellner, Testing for predator dependence in predator-prey dynamics: a non-parametric approach, Proc. Roy. Soc. Lond. B, 267 (2000), 1611-1620. doi: 10.1098/rspb.2000.1186.
[30]	R. E. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552.
[31]	J. Knape and P. de Valpine, Fitting complex population models by combining particle filters with Markov chain Monte Carlo, Ecology, 93 (2012), 256-263. doi: 10.1890/11-0797.1.
[32]	J. Liu and M. West, Combined parameter and state estimation in simulation-based filtering, in Sequential Monte Carlo Methods in Practice (eds. A. Doucet, N. de Freitas and N. Gordon), chap. 10, Springer, 2001, 197-223.
[33]	J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems, Journal of the American Statistical Association, 93 (1998), 1032-1044. doi: 10.1080/01621459.1998.10473765.
[34]	J. Míguez, D. Crisan and P. M. Djurić, On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization, Statistics and Computing.
[35]	P. D. Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, 2004. doi: 10.1007/978-1-4684-9393-1.
[36]	M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches, Ecology, 77 (1996), 337-349. doi: 10.2307/2265613.
[37]	A. R. Pedersen, A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scand. J. Stat., 22 (1995), 55-71.
[38]	M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, Journal of the American statistical association, 94 (1999), 590-599. doi: 10.1080/01621459.1999.10474153.
[39]	B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, Arnold, London, 1999.
[40]	C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, 2004.
[41]	2.0.CO;2] K. Shea, H. P. Possingham, W. W. Murdoch and R. Roush, Active adaptive management in insect pest and weed control: intervention with a plan for learning, Ecological Applications, 12 (2002), 927-936.
[42]	H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: a survey, Int. Stat. Rev., 72 (2004), 337-354. doi: 10.1111/j.1751-5823.2004.tb00241.x.
[43]	O. Stramer and J. Yan, On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation, J. Comput. Graph. Stat., 16 (2007). doi: 10.1198/106186007X237306.
[44]	J. Y. Xia, R. Rabbinge and W. van der Werf, Multistage functional responses in a leadybeetle-aphid system: scaling up from the laboratory to the field, Environmental Entomology, 32 (2003), 151-162. doi: 10.1603/0046-225X-32.1.151.

show all references

References:

[1]	Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica, 70 (2002), 223-262. doi: 10.1111/1468-0262.00274.
[2]	B. D. O. Anderson and J. B. Moore, Optimal Filtering, Englewood Cliffs, 1979. doi: 10.1109/TSMC.1982.4308806.
[3]	C. Andrieu, A. Doucet and R. Holenstein, Particle Markov chain Monte Carlo methods, Journal of the Royal Statistical Society Series B-Statistical Methodology, 72 (2010), 269-342. doi: 10.1111/j.1467-9868.2009.00736.x.
[4]	C. Andrieu, A. Doucet, S. S. Singh and V. B. Tadić, Particle methods for change detection, system identification and control, Proceedings of the IEEE, 92 (2004), 423-438. doi: 10.1109/JPROC.2003.823142.
[5]	A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Springer, 2008.
[6]	A. Beskos, O. Papaspiliopoulos, G. O. Roberts and P. Fearnhead, Exact an computationally efficient likelihood-based estimation for discretely observed diffusion processes, J. Roy. Stat. Soc. Ser. B, 68 (2006), 333-382. doi: 10.1111/j.1467-9868.2006.00552.x.
[7]	G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions, Ecological Modelling, 170 (2003), 155-171. doi: 10.1016/S0304-3800(03)00223-0.
[8]	O. Cappé, S. J. Godsill and E. Moulines, An overview of existing methods and recent advances in sequential Monte Carlo, Proceedings of the IEEE, 95 (2007), 899-924.
[9]	J. Carpenter, P. Clifford and P. Fearnhead, Improved particle filter for nonlinear problems, IEE Proceedings - Radar, Sonar and Navigation, 146 (1999), 2-7. doi: 10.1049/ip-rsn:19990255.
[10]	S. R. Carpenter, K. L. Cottingham and C. A. Stow, Fitting predator-prey models to time series with observation errors, Ecology, 75 (1994), 1254-1264. doi: 10.2307/1937451.
[11]	R. Chen and J. S. Liu, Mixture Kalman filters, Journal of the Royal Statistics Society B, 62 (2000), 493-508. doi: 10.1111/1467-9868.00246.
[12]	N. Chopin, P. E. Jacob and O. Papaspiliopoulos, SMC2: An efficient algorithm for sequential analysis of state space models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), (2012). doi: 10.1111/j.1467-9868.2012.01046.x.
[13]	J. A. Comiskey, F. Dallmeier and A. Alonso, Framework for Assessment and Monitoring of Biodiversity, Academic Press, New York, 1999.
[14]	R. Douc, O. Cappe and E. Moulines, Comparison of resampling schemes for particle filtering, in ISPA 2005: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005, 64-69. doi: 10.1109/ISPA.2005.195385.
[15]	A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice, Springer, New York (USA), 2001.
[16]	A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208.
[17]	M. Dowd, A sequential Monte Carlo approach to marine ecological prediction, Environmetrics, 17 (2006), 435-455. doi: 10.1002/env.780.
[18]	M. Dowd, Estimating parameters for a stochastic dynamic marine ecological system, Environmetrics, 22 (2011), 501-515. doi: 10.1002/env.1083.
[19]	M. Dowd and R. Joy, Estimating behavioral parameters in animal movement models using a state-augmented particle filter, Ecology, 92 (2011), 568-575. doi: 10.1890/10-0611.1.
[20]	G. B. Durham and A. R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, J. Bus. Econ. Stat., 20 (2002), 297-316. doi: 10.1198/073500102288618397.
[21]	O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions, Econometrica, 69 (2001), 959-993. doi: 10.1111/1468-0262.00226.
[22]	B. Eraker, MCMC analysis of diffusion models with application to finance, J. Bus. Econ. Stat., 19 (2001), 177-191. doi: 10.1198/073500101316970403.
[23]	G. Gilioli, S. Pasquali and F. Ruggeri, Bayesian inference for functional response in a stochastic predator-prey system, Bulletin of Mathematical Biology, 70 (2008), 358-381. doi: 10.1007/s11538-007-9256-3.
[24]	G. Gilioli, S. Pasquali and F. Ruggeri, Nonlinear functional response parameter estimation in a stochastic predator-prey model, Mathematical Biosciences and Engineering, 9 (2012), 75-96. doi: 10.3934/mbe.2012.9.75.
[25]	G. Gilioli and V. Vacante, Aspetti della dinamica di popolazione del sistema tetranychus urticae- phytoseiulus persimilis in pieno campo: implicazioni per le strategie di lotta biologica, in Atti del Convegno "La difesa delle colture in agricoltura biologica" - Notiziario sulla protezione delle piante, vol. 13, 2001.
[26]	A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximations, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.
[27]	A. Golightly and D. J. Wilkinson, Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo, Interface Focus, 1 (2011), 807-820. doi: 10.1098/rsfs.2011.0047.
[28]	N. Gordon, D. Salmond and A. F. M. Smith, Novel approach to nonlinear and non-Gaussian Bayesian state estimation, IEE Proceedings-F, 140 (1993), 107-113. doi: 10.1049/ip-f-2.1993.0015.
[29]	C. Jost and S. P. Ellner, Testing for predator dependence in predator-prey dynamics: a non-parametric approach, Proc. Roy. Soc. Lond. B, 267 (2000), 1611-1620. doi: 10.1098/rspb.2000.1186.
[30]	R. E. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552.
[31]	J. Knape and P. de Valpine, Fitting complex population models by combining particle filters with Markov chain Monte Carlo, Ecology, 93 (2012), 256-263. doi: 10.1890/11-0797.1.
[32]	J. Liu and M. West, Combined parameter and state estimation in simulation-based filtering, in Sequential Monte Carlo Methods in Practice (eds. A. Doucet, N. de Freitas and N. Gordon), chap. 10, Springer, 2001, 197-223.
[33]	J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems, Journal of the American Statistical Association, 93 (1998), 1032-1044. doi: 10.1080/01621459.1998.10473765.
[34]	J. Míguez, D. Crisan and P. M. Djurić, On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization, Statistics and Computing.
[35]	P. D. Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, 2004. doi: 10.1007/978-1-4684-9393-1.
[36]	M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches, Ecology, 77 (1996), 337-349. doi: 10.2307/2265613.
[37]	A. R. Pedersen, A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scand. J. Stat., 22 (1995), 55-71.
[38]	M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, Journal of the American statistical association, 94 (1999), 590-599. doi: 10.1080/01621459.1999.10474153.
[39]	B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, Arnold, London, 1999.
[40]	C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, 2004.
[41]	2.0.CO;2] K. Shea, H. P. Possingham, W. W. Murdoch and R. Roush, Active adaptive management in insect pest and weed control: intervention with a plan for learning, Ecological Applications, 12 (2002), 927-936.
[42]	H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: a survey, Int. Stat. Rev., 72 (2004), 337-354. doi: 10.1111/j.1751-5823.2004.tb00241.x.
[43]	O. Stramer and J. Yan, On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation, J. Comput. Graph. Stat., 16 (2007). doi: 10.1198/106186007X237306.
[44]	J. Y. Xia, R. Rabbinge and W. van der Werf, Multistage functional responses in a leadybeetle-aphid system: scaling up from the laboratory to the field, Environmental Entomology, 32 (2003), 151-162. doi: 10.1603/0046-225X-32.1.151.

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