A method for high-dimensional smoothing. (English) Zbl 1415.62070
Summary: We consider the problem of the computation of smoothed additive functionals, which are some integrals with respect to the joint smoothing distribution. It is a key issue in inference for general state-space models as these quantities appear naturally for maximum likelihood parameter inference. The computation of smoothed additive functionals is very challenging as exact computations are not possible for non-linear non-Gaussian state-space models. It becomes even more difficult when the hidden state lies in a high dimensional space because traditional numerical methods suffer from the curse of dimensionality. We propose a new algorithm to efficiently calculate the smoothed additive functionals in an online manner for a specific family of high-dimensional state-space models in discrete time, which is named the Space-Time Forward Smoothing (STFS) algorithm. The cost of this algorithm is at least \(O(N^2 d^2 T)\), which is polynomial in \(d\). \(T\) and \(N\) denote the number of time steps and the number of particles respectively, while \(d\) is the dimension of the hidden state space. Its superior performance over other existing methods is illustrated by various simulation studies. Moreover, STFS algorithm is successfully applied to perform maximum likelihood estimation for static model parameters both in an online and an offline manner.
MSC:
| 62M20 | Inference from stochastic processes and prediction |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 65C05 | Monte Carlo methods |
| 65C40 | Numerical analysis or methods applied to Markov chains |
References:
| [1] | Andersson Naesseth, C.; Lindsten, F.; Schön, T., Nested Sequential Monte Carlo Methods, (32nd international conference on machine learning, Lille, France, 6-11 July, 2015, Vol. 37 (2015), Journal of Machine Learning Research (Online)), 1292-1301 |
| [2] | Andrieu, C.; Doucet, A.; Holenstein, R., Particle Markov chain Monte Carlo methods, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 72, 3, 269-342 (2010) · Zbl 1411.65020 |
| [3] | Asai, M.; McAleer, M.; Yu, J., Multivariate stochastic volatility: a review, Econometric Reviews, 25, 2-3, 145-175 (2006) · Zbl 1107.62108 |
| [4] | Bertsekas, D. P.; Shreve, S. E., Stochastic optimal control: The discrete time case, vol. 23 (1978), Academic Press New York · Zbl 0471.93002 |
| [5] | Beskos, A.; Crisan, D.; Jasra, A.; Kamatani, K.; Zhou, Y., A stable particle filter for a class of high-dimensional state-space models, Advances in Applied Probability, 49, 1, 24-48 (2017) · Zbl 1426.62274 |
| [6] | Bickel, P.; Li, B.; Bengtsson, T., Sharp failure rates for the bootstrap particle filter in high dimensions, (Pushing the limits of contemporary statistics: contributions in honor of Jayanta K. Ghosh (2008), Institute of Mathematical Statistics), 318-329 · Zbl 1159.62004 |
| [7] | Briers, M.; Doucet, A.; Maskell, S., Smoothing algorithms for state-space models, Annals of the Institute of Statistical Mathematics, 62, 1, 61-89 (2010) · Zbl 1422.62297 |
| [8] | Cappé, O.; Moulines, E.; Rydén, T., Inference in Hidden Markov Models (2005), Springer · Zbl 1080.62065 |
| [9] | Chib, S.; Nardari, F.; Shephard, N., Analysis of high dimensional multivariate stochastic volatility models, Journal of Econometrics, 134, 2, 341-371 (2006) · Zbl 1418.62377 |
| [10] | Chopin, N.; Jacob, P. E.; Papaspiliopoulos, O., \( \text{SMC}^2\): an efficient algorithm for sequential analysis of state space models, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 75, 3, 397-426 (2013) · Zbl 1411.62242 |
| [11] | Del Moral, P., Feynman-kac formulae: Genealogical and interacting particle systems with applications (2004), New York: Springer · Zbl 1130.60003 |
| [12] | Del Moral, P., Doucet, A., & Singh, S. (2010). Forward smoothing using sequential Monte Carlo. arXiv preprint arXiv:1012.5390; Del Moral, P., Doucet, A., & Singh, S. (2010). Forward smoothing using sequential Monte Carlo. arXiv preprint arXiv:1012.5390 |
| [13] | Doucet, A. (1998). On sequential simulation-based methods for Bayesian filtering.; Doucet, A. (1998). On sequential simulation-based methods for Bayesian filtering. |
| [14] | Doucet, A.; Godsill, S.; Andrieu, C., On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10, 3, 197-208 (2000) |
| [15] | Doucet, A.; Johansen, A. M., A tutorial on particle filtering and smoothing: Fifteen years later, Handbook of Nonlinear Filtering, 12, 656-704 (2009) · Zbl 1513.60043 |
| [16] | Durbin, J.; Koopman, S. J., Time series analysis by state space methods, vol. 38 (2012), OUP Oxford · Zbl 1270.62120 |
| [17] | Elliott, R. J.; Aggoun, L.; Moore, J. B., Hidden Markov models: estimation and control, vol. 29 (2008), Springer Science & Business Media |
| [18] | Finke, A.; Singh, S. S., approximate smoothing and parameter estimation in high-dimensional state-space models, IEEE Transactions on Signal Processing, 65, 22, 5982-5994 (2017) · Zbl 1414.94197 |
| [19] | Godsill, S. J.; Doucet, A.; West, M., Monte Carlo smoothing for nonlinear time series, Journal of the American Statistical Association, 99, 465, 156-168 (2004) · Zbl 1089.62517 |
| [20] | Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42, 2, 281-300 (1987) |
| [21] | Ishihara, T.; Omori, Y., Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors, Computational Statistics & Data Analysis, 56, 11, 3674-3689 (2012) · Zbl 1255.62066 |
| [22] | Kantas, N.; Doucet, A.; Singh, S. S.; Maciejowski, J. M., An overview of sequential Monte Carlo methods for parameter estimation in general state-space models, IFAC Proceedings Volumes, 42, 10, 774-785 (2009) |
| [23] | Kantas, N.; Doucet, A.; Singh, S. S.; Maciejowski, J.; Chopin, N., On particle methods for parameter estimation in state-space models, Statistical Science, 30, 3, 328-351 (2015) · Zbl 1332.62096 |
| [24] | Kitagawa, G., Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, 5, 1, 1-25 (1996) |
| [25] | Kitagawa, G.; Sato, S., Monte Carlo smoothing and self-organising state-space model, (Sequential Monte Carlo methods in practice (2001), Springer), 177-195 · Zbl 1056.93581 |
| [26] | Klaas, M.; De Freitas, N.; Doucet, A., Toward practical \(N^2\) Monte Carlo: the marginal particle filter, Uncertainty in Artificial Intelligence, 308-315 (2005) |
| [27] | Olsson, J.; Cappé, O.; Douc, R.; Moulines, E., Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models, Bernoulli, 14, 1, 155-179 (2008) · Zbl 1155.62055 |
| [28] | Pitt, M. K.; Shephard, N., Filtering via simulation: Auxiliary particle filters, Journal of the American Statistical Association, 94, 446, 590-599 (1999) · Zbl 1072.62639 |
| [29] | Platanioti, K.; McCoy, E.; Stephens, D., (A review of stochastic volatility: univariate and multivariate models. Technical report (2005), working paper) |
| [30] | Poyiadjis, G.; Doucet, A.; Singh, S. S., Particle approximations of the score and observed information matrix in state space models with application to parameter estimation, Biometrika, 98, 1, 65-80 (2011) · Zbl 1214.62093 |
| [31] | Rebeschini, P.; Van Handel, R., Can local particle filters beat the curse of dimensionality?, Annals of Applied Probability, 25, 5, 2809-2866 (2015) · Zbl 1325.60058 |
| [32] | Snyder, C.; Bengtsson, T.; Bickel, P.; Anderson, J., Obstacles to high-dimensional particle filtering, Monthly Weather Review, 136, 12, 4629-4640 (2008) |
| [33] | Spantini, A., Bigoni, D., & Marzouk, Y. (2017). Inference via low-dimensional couplings. arXiv preprint arXiv:1703.06131; Spantini, A., Bigoni, D., & Marzouk, Y. (2017). Inference via low-dimensional couplings. arXiv preprint arXiv:1703.06131 |
| [34] | Storvik, G., Particle filters for state-space models with the presence of unknown static parameters, IEEE Transactions on Signal Processing, 50, 2, 281-289 (2002) |
| [35] | Vergé, C.; Dubarry, C.; Del Moral, P.; Moulines, E., On parallel implementation of sequential Monte Carlo methods: the island particle model, Statistics and Computing, 25, 2, 243-260 (2015) · Zbl 1331.65023 |
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