Optimal projection filters with information geometry. (English) Zbl 07789565
Summary: We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247-252, 1998, Bernoulli 5(3):495-534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1-33, 2016) where a direct \(L^2\) metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176-213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.
MSC:
| 62M20 | Inference from stochastic processes and prediction |
| 93E11 | Filtering in stochastic control theory |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
| 62B10 | Statistical aspects of information-theoretic topics |
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 58A20 | Jets in global analysis |
Keywords:
stochastic partial differential equations; stochastic differential equations; SPDEs projection on a submanifold; Stratonovich projection; Itô-vector projection; Itô-jet projection; nonlinear filtering; projection filters; Stratonovich projection filters; optimal projection filtersReferences:
| [1] | Aggarwal, NL; Kampé de Fériet, J.; Picard, CF, Sur l’information de fisher, Théories de l’Information, 111-117 (1974), Berlin: Springer, Berlin · Zbl 0299.94016 · doi:10.1007/BFb0065772 |
| [2] | Ahmed, NU, Linear and Nonlinear Filtering for Scientists and Engineers (1998), Singapore: World Scientific, Singapore · Zbl 0918.93001 · doi:10.1142/3911 |
| [3] | Amari, S., Differential Geometrical Methods in Statistics (1985), Berlin: Springer, Berlin · Zbl 0559.62001 · doi:10.1007/978-1-4612-5056-2 |
| [4] | Armstrong, J., Brigo, D.: Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples (2013). arXiv preprint. arXiv:1303.6236 |
| [5] | Armstrong, J.; Brigo, D.; Nielsen, F.; Critchley, F.; Dodson, K., Extrinsic projection of Itô SDEs on submanifolds with applications to non-linear filtering, Computational Information Geometry for Image and Signal Processing (2016), Berlin: Springer, Berlin |
| [6] | Armstrong, J.; Brigo, D., Nonlinear filtering via stochastic PDE projection on mixture manifolds in \(L^2\) direct metric, Math. Control Signals Syst., 28, 1, 1-33 (2016) · Zbl 1338.93365 · doi:10.1007/s00498-015-0154-1 |
| [7] | Armstrong, J., Brigo, D.: Ito stochastic differential equations as 2-jets. In: Geometric Science of Information, Proceedings of the 2017 Conference, Paris, France, pp. 543-551. Springer, Berlin (2017) · Zbl 1428.60074 |
| [8] | Armstrong, J., Brigo, D.: Intrinsic stochastic differential equations as jets. Proc. R. Soc. A: Math. Phys. Eng. Sci. 474 (2018) · Zbl 1402.34064 |
| [9] | Armstrong, J., Brigo, D., Rossi Ferrucci, E.: Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry (2023, to appear). Earlier arXiv preprint (2018) in arXiv:1810.03923 |
| [10] | Armstrong, J.; Brigo, D.; Rossi Ferrucci, E., Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections, Proc. Lond. Math. Soc., 119, 1, 176-213 (2019) · Zbl 1421.58008 · doi:10.1112/plms.12226 |
| [11] | Azimi-Sadjadi, B.; Krishnaprasad, PS, Approximate nonlinear filtering and its application in navigation, Automatica, 41, 6, 945-956 (2005) · Zbl 1091.93039 · doi:10.1016/j.automatica.2004.12.013 |
| [12] | Bain, A.; Crisan, D., Fundamentals of Stochastic Filtering (2009), Berlin: Springer, Berlin · Zbl 1176.62091 · doi:10.1007/978-0-387-76896-0 |
| [13] | Barndorff-Nielsen, O., Information and Exponential Families (1978), New York: Wiley, New York · Zbl 0387.62011 |
| [14] | Berefelt, F., Hamberg, J., Robinson, J.W.C.: Geometric aspects of nonlinear filtering. Technical report, Swedish Defence Research Agency (2003) |
| [15] | Brigo, D., On the nice behaviour of the Gaussian projection filter with small observation noise, Syst. Control Lett., 26, 5, 363-370 (1995) · Zbl 0877.93128 · doi:10.1016/0167-6911(95)00037-2 |
| [16] | Brigo, D.: Filtering by projection on the manifold of exponential densities. Free University of Amsterdam, PhD dissertation (1996) |
| [17] | Brigo, D., New results on the Gaussian projection filter with small observation noise, Syst. Control Lett., 28, 5, 273-279 (1996) · Zbl 0875.93556 · doi:10.1016/0167-6911(96)00038-2 |
| [18] | Brigo, D.; Hanzon, B., On some filtering problems arising in mathematical finance, Insur.: Math. Econ., 22, 1, 53-64 (1998) · Zbl 1093.91514 |
| [19] | Brigo, D.; Hanzon, B.; LeGland, F., A differential geometric approach to nonlinear filtering: the projection filter, IEEE Trans. Autom. Control, 43, 2, 247-252 (1998) · Zbl 0904.93038 · doi:10.1109/9.661075 |
| [20] | Brigo, D.; Hanzon, B.; LeGland, F., Approximate nonlinear filtering by projection on exponential manifolds of densities, Bernoulli, 5, 3, 495-534 (1999) · Zbl 0930.93074 · doi:10.2307/3318714 |
| [21] | Brigo, D., Pistone, G.: Dimensionality reduction for measure-valued evolution equations in statistical manifolds. In: Proceedings of the conference on Computational Information Geometry for Image and Signal Processing. Springer, Berlin (2016) |
| [22] | Brigo, D., Pistone, G.: Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods (2016). arXiv preprint. arXiv:1603.04348 |
| [23] | Bröcker, J., Parlitz, U.: Noise reduction and filtering of chaotic time series. In: Proceedings of NOLTA 2000, Dresden (2000) · Zbl 1080.37598 |
| [24] | Cipra, B.: Engineers look to Kalman Filtering for Guidance. SIAM News 26 (1993) |
| [25] | Cohen, S., Fausti, E.: Stability and approximation of projection filters. Seminar presented at Imperial College London, Department of Mathematics, 18 April 2023 |
| [26] | Cohen, S., Fausti, E.: Exponential contractions and robustness for approximate Wonham filters (2023). arXiv preprint arXiv:2305.02256 |
| [27] | Elworthy, D.: Geometric aspects of diffusions on manifolds. In: École d’Été de Probabilités de Saint-Flour XV-XVII, 1985-1987, pp. 277-425. Springer, Berlin (1988) · Zbl 0658.58040 |
| [28] | Elworthy, D., Jan, Y.L., Li, X.-M.: The Geometry of Filtering. Frontiers in Mathematics. Birkhäuser, Basel (2010) · Zbl 1219.58017 |
| [29] | Emery, M., Stochastic Calculus in Manifolds (1989), Heidelberg: Springer, Heidelberg · Zbl 0697.60060 · doi:10.1007/978-3-642-75051-9 |
| [30] | Gao, Q.; Zhang, G.; Petersen, IR, An exponential quantum projection filter for open quantum systems, Automatica, 99, 59-68 (2019) · Zbl 1407.81112 · doi:10.1016/j.automatica.2018.10.014 |
| [31] | Hanzon, B., Hut, R.: New results on the projection filter. In: Proceedings of the European Control Conference, Grenoble, France, pp. 623-628. ECC91 (1991) |
| [32] | Hanzon, B.; Dodson, CTJ, A differential-geometric approach to approximate nonlinear filtering, Geometrization of Statistical Theory, 219-223 (1987), Lancaster: ULMD Publications, University of Lancaster, Lancaster |
| [33] | Harel, Y.; Meir, R.; Opper, M.; Cortes, C.; Lawrence, N.; Lee, D.; Sugiyama, M.; Garnett, R., A tractable approximation to optimal point process filtering: application to neural encoding, Advances in Neural Information Processing Systems (2015), New York: Curran Associates, Inc., New York |
| [34] | Hazewinkel, M.; Marcus, SI; Sussmann, HJ, Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem, Syst. Control Lett., 3, 6, 331-340 (1983) · Zbl 0539.93077 · doi:10.1016/0167-6911(83)90074-9 |
| [35] | Hsu, EP, Stochastic Analysis on Manifolds (2002), Providence: American Mathematical Society, Providence · Zbl 0994.58019 · doi:10.1090/gsm/038 |
| [36] | Jazwinski, AH, Stochastic Processes and Filtering Theory (1970), New York: Academic Press, New York · Zbl 0203.50101 |
| [37] | Jones, ES; Soatto, S., Visual-inertial navigation, mapping and localization: a scalable real-time causal approach, Int. J. Robot. Res., 30, 4, 407-430 (2011) · doi:10.1177/0278364910388963 |
| [38] | Kushner, HJ, Approximations to optimal nonlinear filters, IEEE Trans. Autom. Control, 12, 5, 546-556 (1967) · doi:10.1109/TAC.1967.1098671 |
| [39] | Kutschireiter, A., Rast, L., Drugowitsch, J.: Projection filtering with observed state increments with applications in continuous-time circular filtering. IEEE Trans. Signal Process. 70, 686-700 (2022) · Zbl 07911290 |
| [40] | Lermusiaux, PFJ, Uncertainty estimation and prediction for interdisciplinary ocean dynamics, J. Comput. Phys., 217, 1, 176-199 (2006) · Zbl 1146.86002 · doi:10.1016/j.jcp.2006.02.010 |
| [41] | Liptser, RS; Shiryayev, AN, Statistics of Random Processes I, General Theory (1978), Berlin: Springer, Berlin · Zbl 0369.60001 |
| [42] | Ma, Y.; Zhao, Y.; Chen, L.; Shuai, C., Hazard position estimation of sudden accidents based on projection filter, Syst. Eng. Theory Pract., 35, 3, 651 (2015) |
| [43] | Maybeck, PS, Stochastic Models, Estimation, and Control (1982), Cambridge: Academic Press, Cambridge · Zbl 0546.93063 |
| [44] | Newton, NJ, An infinite-dimensional statistical manifold modelled on Hilbert space, J. Funct. Anal., 263, 6, 1661-1681 (2012) · Zbl 1257.46044 · doi:10.1016/j.jfa.2012.06.007 |
| [45] | Newton, N.J.: Information geometric nonlinear filtering. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18(2), 1550014 (2015) · Zbl 1317.93253 |
| [46] | Pardoux, É.: Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. In: Ecole d’Eté de Probabilités de Saint-Flour XIX 1989, pp. 68-163. Springer, Berlin (1991) · Zbl 0732.60050 |
| [47] | Pistone, G.; Sempi, C., An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Stat., 23, 5, 1543-1561, 10 (1995) · Zbl 0848.62003 · doi:10.1214/aos/1176324311 |
| [48] | Rao, C.R.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 81-91 (1945) · Zbl 0063.06420 |
| [49] | Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 2: Ito Calculus. Cambridge University Press, Cambridge (1987) · Zbl 0627.60001 |
| [50] | Surace, S.; Pfister, J-P, Online maximum likelihood estimation of the parameters of partially observed diffusion processes, IEEE Trans. Autom. Control, 64, 09 (2017) |
| [51] | Tronarp, F., Särkkä, S.: Updates in Bayesian filtering by continuous projections on a manifold of densities. In: ICASSP 2019—2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, pp. 5032-5036 (2019) |
| [52] | van Handel, R., Mabuchi, H.: Quantum projection filter for a highly nonlinear model in cavity QED. J. Opt. B: Quantum Semiclass. Opt. 7, S226-S236 (2005) |
| [53] | Vellekoop, MH; Clark, JMC, A nonlinear filtering approach to changepoint detection problems: Direct and differential-geometric methods, SIAM Rev., 48, 2, 329-356 (2006) · Zbl 1092.60016 · doi:10.1137/050647438 |
| [54] | Zhang, XM; Wang, RW; Wu, HB; Xu, B., Automated measurements of fiber diameters in melt-blown nonwovens, J. Ind. Text., 43, 4, 593-605 (2014) · doi:10.1177/1528083712471696 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
