Description

Sequential Monte Carlo (SMC) Methods fall into two groups, those aimed at sequential data assimilation (commonly referred to as particle filters) and those designed for generating samples from stationary distributions(Andrieu et al., 2004). In general SMC methods aim to approximate the distribution of interest by a properly weighted set of particles generated by importance sampling. Particle filters (Liu and Chen, 1998; Pitt and Shephard, 1999) overcome the two main limitations of Kalman filter based techniques, that is the approximation by second order moments and assumption of normal distributions. Particle filters allow any probability distribution to be used to represent the error on the observed data series or, if required, the state corrections and parameter evolution. Also by generating a sample of particles from the full distribution they capture the full moments of the distribution. This additional theoretical completeness comes at the cost of computational efficiency due to the large amount of sampling involved in generating and evolving the particles. With certain sampling strategies the computational scheme may require monitoring to confirm that particle sample does not become highly correlated. There are several published applications of particle filters in environmental settings (Kitagawa et al., 2001; Moradkhani et al., 2005). SMC techniques for the generation of samples from stationary distributions(Cappe et al., 2004) are less well known in environmental modelling, though they over an interesting alternative to Markov Chain Monte Carlo samplers (Andrieu et al., 2004). These techniques are generally based upon sequential importance sampling from the distribution of interest, or from a series of artificial distribution that move slowly from an initial (easily sampled distribution) to the complex distribution of interest.

Software

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Advantages

• Filter algorithms that allow a free choice of probability distribution for the observational errors;

• Sampling within the filtering algorithm that is dependant upon the whole distribution of interest, not just it’s first two moments;

• Computationally simple and robust

Disadvantages

• Computationally expensive in terms of model evaluations;

• Computational scheme may need careful monitoring;

• Additional thought has to given to the selection of error distributions used.

References and Further Reading

Andrieu, C., Doucet, A. and Robert, C.P., 2004. Computational advances for and from Bayesian Analysis. Statistical science, 19(1). Cappe, O., Guillin, A., Marin, J.M. and Robert, C.P., 2004. Population Monte Carlo. Journal of Computational and Graphical Statistics, 13(4): 907-929.

Kitagawa, G., Takanami, T. and Matsumoto, N., 2001. Signal extraction problems in seismology. International Statistical Review, 69(1): 129-152.

Liu, J.S. and Chen, R., 1998. Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association, 93(443): 1032-1044.

Moradkhani, H., Hsu, K.-L., Gupta, H. and Sorooshian, S., 2005. Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter. Water Resources Research, 41: doi:10.1029/2004WR003604.

Pitt, M.K. and Shephard, N., 1999. Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association, 94(446): 590-599.