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

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

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.
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
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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. Google Scholar

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S. R. Carpenter, K. L. Cottingham and C. A. Stow, Fitting predator-prey models to time series with observation errors,, Ecology, 75 (1994), 1254. doi: 10.2307/1937451. Google Scholar

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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. Google Scholar

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A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering,, Statistics and Computing, 10 (2000), 197. Google Scholar

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B. Eraker, MCMC analysis of diffusion models with application to finance,, J. Bus. Econ. Stat., 19 (2001), 177. doi: 10.1198/073500101316970403. Google Scholar

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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. doi: 10.1007/s11538-007-9256-3. Google Scholar

[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. doi: 10.3934/mbe.2012.9.75. Google Scholar

[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, (2001). Google Scholar

[26]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximations,, Biometrics, 61 (2005), 781. doi: 10.1111/j.1541-0420.2005.00345.x. Google Scholar

[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. doi: 10.1098/rsfs.2011.0047. Google Scholar

[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. doi: 10.1049/ip-f-2.1993.0015. Google Scholar

[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. doi: 10.1098/rspb.2000.1186. Google Scholar

[30]

R. E. Kalman, A new approach to linear filtering and prediction problems,, Journal of Basic Engineering, 82 (1960), 35. doi: 10.1115/1.3662552. Google Scholar

[31]

J. Knape and P. de Valpine, Fitting complex population models by combining particle filters with Markov chain Monte Carlo,, Ecology, 93 (2012), 256. doi: 10.1890/11-0797.1. Google Scholar

[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, (2001), 197. Google Scholar

[33]

J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems,, Journal of the American Statistical Association, 93 (1998), 1032. doi: 10.1080/01621459.1998.10473765. Google Scholar

[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., (). Google Scholar

[35]

P. D. Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications,, Springer, (2004). doi: 10.1007/978-1-4684-9393-1. Google Scholar

[36]

M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches,, Ecology, 77 (1996), 337. doi: 10.2307/2265613. Google Scholar

[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. Google Scholar

[38]

M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters,, Journal of the American statistical association, 94 (1999), 590. doi: 10.1080/01621459.1999.10474153. Google Scholar

[39]

B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes,, Arnold, (1999). Google Scholar

[40]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods,, Springer, (2004). Google Scholar

[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. Google Scholar

[42]

H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: a survey,, Int. Stat. Rev., 72 (2004), 337. doi: 10.1111/j.1751-5823.2004.tb00241.x. Google Scholar

[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. Google Scholar

[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. doi: 10.1603/0046-225X-32.1.151. Google Scholar

show all references

References:
[1]

Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach,, Econometrica, 70 (2002), 223. doi: 10.1111/1468-0262.00274. Google Scholar

[2]

B. D. O. Anderson and J. B. Moore, Optimal Filtering,, Englewood Cliffs, (1979). doi: 10.1109/TSMC.1982.4308806. Google Scholar

[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. doi: 10.1111/j.1467-9868.2009.00736.x. Google Scholar

[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. doi: 10.1109/JPROC.2003.823142. Google Scholar

[5]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering,, Springer, (2008). Google Scholar

[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. doi: 10.1111/j.1467-9868.2006.00552.x. Google Scholar

[7]

G. Buffoni and G. Gilioli, A lumped parameter model for acarine predator-prey population interactions,, Ecological Modelling, 170 (2003), 155. doi: 10.1016/S0304-3800(03)00223-0. Google Scholar

[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. Google Scholar

[9]

J. Carpenter, P. Clifford and P. Fearnhead, Improved particle filter for nonlinear problems,, IEE Proceedings - Radar, 146 (1999), 2. doi: 10.1049/ip-rsn:19990255. Google Scholar

[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. doi: 10.2307/1937451. Google Scholar

[11]

R. Chen and J. S. Liu, Mixture Kalman filters,, Journal of the Royal Statistics Society B, 62 (2000), 493. doi: 10.1111/1467-9868.00246. Google Scholar

[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. Google Scholar

[13]

J. A. Comiskey, F. Dallmeier and A. Alonso, Framework for Assessment and Monitoring of Biodiversity,, Academic Press, (1999). Google Scholar

[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. doi: 10.1109/ISPA.2005.195385. Google Scholar

[15]

A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice,, Springer, (2001). Google Scholar

[16]

A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering,, Statistics and Computing, 10 (2000), 197. Google Scholar

[17]

M. Dowd, A sequential Monte Carlo approach to marine ecological prediction,, Environmetrics, 17 (2006), 435. doi: 10.1002/env.780. Google Scholar

[18]

M. Dowd, Estimating parameters for a stochastic dynamic marine ecological system,, Environmetrics, 22 (2011), 501. doi: 10.1002/env.1083. Google Scholar

[19]

M. Dowd and R. Joy, Estimating behavioral parameters in animal movement models using a state-augmented particle filter,, Ecology, 92 (2011), 568. doi: 10.1890/10-0611.1. Google Scholar

[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. doi: 10.1198/073500102288618397. Google Scholar

[21]

O. Elerian, S. Chib and N. Shephard, Likelihood inference for discretely observed nonlinear diffusions,, Econometrica, 69 (2001), 959. doi: 10.1111/1468-0262.00226. Google Scholar

[22]

B. Eraker, MCMC analysis of diffusion models with application to finance,, J. Bus. Econ. Stat., 19 (2001), 177. doi: 10.1198/073500101316970403. Google Scholar

[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. doi: 10.1007/s11538-007-9256-3. Google Scholar

[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. doi: 10.3934/mbe.2012.9.75. Google Scholar

[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, (2001). Google Scholar

[26]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximations,, Biometrics, 61 (2005), 781. doi: 10.1111/j.1541-0420.2005.00345.x. Google Scholar

[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. doi: 10.1098/rsfs.2011.0047. Google Scholar

[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. doi: 10.1049/ip-f-2.1993.0015. Google Scholar

[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. doi: 10.1098/rspb.2000.1186. Google Scholar

[30]

R. E. Kalman, A new approach to linear filtering and prediction problems,, Journal of Basic Engineering, 82 (1960), 35. doi: 10.1115/1.3662552. Google Scholar

[31]

J. Knape and P. de Valpine, Fitting complex population models by combining particle filters with Markov chain Monte Carlo,, Ecology, 93 (2012), 256. doi: 10.1890/11-0797.1. Google Scholar

[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, (2001), 197. Google Scholar

[33]

J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems,, Journal of the American Statistical Association, 93 (1998), 1032. doi: 10.1080/01621459.1998.10473765. Google Scholar

[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., (). Google Scholar

[35]

P. D. Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications,, Springer, (2004). doi: 10.1007/978-1-4684-9393-1. Google Scholar

[36]

M. A. Pascual and K. Kareiva, Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches,, Ecology, 77 (1996), 337. doi: 10.2307/2265613. Google Scholar

[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. Google Scholar

[38]

M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters,, Journal of the American statistical association, 94 (1999), 590. doi: 10.1080/01621459.1999.10474153. Google Scholar

[39]

B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes,, Arnold, (1999). Google Scholar

[40]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods,, Springer, (2004). Google Scholar

[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. Google Scholar

[42]

H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: a survey,, Int. Stat. Rev., 72 (2004), 337. doi: 10.1111/j.1751-5823.2004.tb00241.x. Google Scholar

[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. Google Scholar

[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. doi: 10.1603/0046-225X-32.1.151. Google Scholar

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