September  2021, 3(3): 647-675. doi: 10.3934/fods.2021025

Stability of non-linear filter for deterministic dynamics

1. 

Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India-400076

2. 

International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India-560089

* Corresponding author: Amit Apte

Received  March 2021 Revised  August 2021 Published  September 2021 Early access  September 2021

Fund Project: The work of ASR was partially supported by Infosys Foundation Excellence Program of ICTS. AA acknowledges support from US Office of Naval Research under grant N00014-18-1-2204. Authors acknowledge the support of the Department of Atomic Energy, Government of India, under projects no.12-R & D-TFR-5.10-1100, and no.RTI4001

This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.

Citation: Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025
References:
[1]

B. D. O. Anderson and J. B. Moore, New results in linear system stability, SIAM J. Control, 7 (1969), 398-414.  doi: 10.1137/0307029.  Google Scholar

[2]

A. Apte, M. Hairer, A. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50–64. doi: 10.1016/j.physd.2006.06.009.  Google Scholar

[3]

M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974546.pt1.  Google Scholar

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R. Atar, Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain, Ann. Probab., 26 (1998), 1552-1574.  doi: 10.1214/aop/1022855873.  Google Scholar

[5]

R. Atar and O. Zeitouni, Exponential stability for nonlinear filtering, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 697-725.  doi: 10.1016/S0246-0203(97)80110-0.  Google Scholar

[6]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, 60, Springer, New York, 2009. doi: 10.1007/978-0-387-76896-0.  Google Scholar

[7]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics, 148, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/gsm/148.  Google Scholar

[8]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[9]

A. N. Bishop and P. Del Moral, On the stability of Kalman–Bucy diffusion processes, SIAM J. Control Optim., 55 (2017), 4015-4047.  doi: 10.1137/16M1102707.  Google Scholar

[10]

M. BocquetK. S. GurumoorthyA. ApteA. CarrassiC. Grudzien and C. K. R. T. Jones, Degenerate Kalman filter error covariances and their convergence onto the unstable subspace, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 304-333.  doi: 10.1137/16M1068712.  Google Scholar

[11]

A. Budhiraja, Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 919-941.  doi: 10.1016/S0246-0203(03)00022-0.  Google Scholar

[12]

A. Budhiraja and D. Ocone, Exponential stability in discrete-time filtering for non-ergodic signals, Stochastic Process. Appl., 82 (1999), 245-257.  doi: 10.1016/S0304-4149(99)00032-0.  Google Scholar

[13]

A. Budhiraja and D. Ocone, Exponential stability of discrete-time filters for bounded observation noise, Systems Control Lett., 30 (1997), 185-193.  doi: 10.1016/S0167-6911(97)00012-1.  Google Scholar

[14]

A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, WIREs Clim. Change, 9 (2018). doi: 10.1002/wcc.535.  Google Scholar

[15]

F. Cérou, Long Time Asymptotics for Some Dynamical Noise Free Non-Linear Filtering Problems, New Cases, Research Report, RR-2541, INRIA, 1995. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.57.1616&rep=rep1&type=pdf. Google Scholar

[16]

F. Cérou, Long time behavior for some dynamical noise free nonlinear filtering problems, SIAM J. Control Optim., 38 (2000), 1086-1101.  doi: 10.1137/S0363012995290124.  Google Scholar

[17]

P. Chigansky, Stability of nonlinear filters: A survey, Mini-Course Lecture Notes, Petropolis, Brazil, 2006. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.169.1819&rep=rep1&type=pdf. Google Scholar

[18]

P. Chigansky, R. Liptser and R. Van Handel, Intrinsic methods in filter stability, in Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 319-351.  Google Scholar

[19]

J. M. C. ClarkD. L. Ocone and C. Coumarbatch, Relative entropy and error bounds for filtering of Markov processes, Math. Control Signals Systems, 12 (1999), 346-360.  doi: 10.1007/PL00009856.  Google Scholar

[20]

A. D'Aristotile, P. Diaconis and D. Freedman, On merging of probabilities, Sankhyā Ser. A, 50 (1988), 363–380.  Google Scholar

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.  Google Scholar

[22]

H. E. Gollwitzer, A note on a functional inequality, Proc. Amer. Math. Soc., 23 (1969), 642-647.  doi: 10.1090/S0002-9939-1969-0247016-9.  Google Scholar

[23]

K. S. GurumoorthyC. GrudzienA. ApteA. Carrassi and C. K. R. T. Jones, Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution, SIAM J. Control Optim., 55 (2017), 741-759.  doi: 10.1137/15M1025839.  Google Scholar

[24]

G. Kallianpur, Stochastic Filtering Theory, Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. doi: 10.1007/978-1-4757-6592-2.  Google Scholar

[25]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[26]

D. T. B. KellyK. J. H. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579.  Google Scholar

[27]

Koro, Uniform expansivity., Available from: https://planetmath.org/UniformExpansivity. Google Scholar

[28]

K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.  Google Scholar

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.  Google Scholar

[30]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[31]

E. N. Lorenz, Predictability - A problem partly solved, in Predictability of Weather and Climate, Cambridge University Press, 2006, 40-58. doi: 10.1017/CBO9780511617652.004.  Google Scholar

[32]

C. McDonald and S. Yüksel, Converse results on filter stability criteria and stochastic non-linear observability, preprint, arXiv: 1812.01772. Google Scholar

[33]

B. Ni and Q. Zhang, Stability of the Kalman filter for continuous time output error systems, Systems Control Lett., 94 (2016), 172-180.  doi: 10.1016/j.sysconle.2016.06.006.  Google Scholar

[34]

D. Ocone and E. Pardoux, Asymptotic stability of the optimal filter with respect to its initial condition, SIAM J. Control Optim., 34 (1996), 226-243.  doi: 10.1137/S0363012993256617.  Google Scholar

[35]

D. L. Ocone, Asymptotic stability of Beneš filters, Stochastic Anal. Appl., 17 (1999), 1053-1074.  doi: 10.1080/07362999908809648.  Google Scholar

[36]

T. N. Palmer, Stochastic weather and climate models, Nature Reviews Physics, 1 (2019), 463-471.  doi: 10.1038/s42254-019-0062-2.  Google Scholar

[37]

A. S. Reddy, A. Apte and S. Vadlamani, Asymptotic properties of linear filter for deterministic processes, Systems Control Lett., 139 (2020), 8 pp. doi: 10.1016/j.sysconle.2020.104676.  Google Scholar

[38] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.  Google Scholar
[39]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[40]

R. van Handel, Filtering, Stability, and Robustness, Ph.D thesis, California Institute of Technology, 2007.  Google Scholar

[41]

R. van Handel, Nonlinear filtering and systems theory, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS Semi-Plenary Paper), 2010. Available from: https://web.math.princeton.edu/rvan/mtns10.pdf. Google Scholar

[42]

R. van Handel, Observability and nonlinear filtering, Probab. Theory Related Fields, 145 (2009), 35-74.  doi: 10.1007/s00440-008-0161-y.  Google Scholar

[43]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[44]

D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511813658.  Google Scholar

[45]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.  Google Scholar

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, New results in linear system stability, SIAM J. Control, 7 (1969), 398-414.  doi: 10.1137/0307029.  Google Scholar

[2]

A. Apte, M. Hairer, A. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50–64. doi: 10.1016/j.physd.2006.06.009.  Google Scholar

[3]

M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974546.pt1.  Google Scholar

[4]

R. Atar, Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain, Ann. Probab., 26 (1998), 1552-1574.  doi: 10.1214/aop/1022855873.  Google Scholar

[5]

R. Atar and O. Zeitouni, Exponential stability for nonlinear filtering, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 697-725.  doi: 10.1016/S0246-0203(97)80110-0.  Google Scholar

[6]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, 60, Springer, New York, 2009. doi: 10.1007/978-0-387-76896-0.  Google Scholar

[7]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics, 148, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/gsm/148.  Google Scholar

[8]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[9]

A. N. Bishop and P. Del Moral, On the stability of Kalman–Bucy diffusion processes, SIAM J. Control Optim., 55 (2017), 4015-4047.  doi: 10.1137/16M1102707.  Google Scholar

[10]

M. BocquetK. S. GurumoorthyA. ApteA. CarrassiC. Grudzien and C. K. R. T. Jones, Degenerate Kalman filter error covariances and their convergence onto the unstable subspace, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 304-333.  doi: 10.1137/16M1068712.  Google Scholar

[11]

A. Budhiraja, Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 919-941.  doi: 10.1016/S0246-0203(03)00022-0.  Google Scholar

[12]

A. Budhiraja and D. Ocone, Exponential stability in discrete-time filtering for non-ergodic signals, Stochastic Process. Appl., 82 (1999), 245-257.  doi: 10.1016/S0304-4149(99)00032-0.  Google Scholar

[13]

A. Budhiraja and D. Ocone, Exponential stability of discrete-time filters for bounded observation noise, Systems Control Lett., 30 (1997), 185-193.  doi: 10.1016/S0167-6911(97)00012-1.  Google Scholar

[14]

A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, WIREs Clim. Change, 9 (2018). doi: 10.1002/wcc.535.  Google Scholar

[15]

F. Cérou, Long Time Asymptotics for Some Dynamical Noise Free Non-Linear Filtering Problems, New Cases, Research Report, RR-2541, INRIA, 1995. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.57.1616&rep=rep1&type=pdf. Google Scholar

[16]

F. Cérou, Long time behavior for some dynamical noise free nonlinear filtering problems, SIAM J. Control Optim., 38 (2000), 1086-1101.  doi: 10.1137/S0363012995290124.  Google Scholar

[17]

P. Chigansky, Stability of nonlinear filters: A survey, Mini-Course Lecture Notes, Petropolis, Brazil, 2006. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.169.1819&rep=rep1&type=pdf. Google Scholar

[18]

P. Chigansky, R. Liptser and R. Van Handel, Intrinsic methods in filter stability, in Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 319-351.  Google Scholar

[19]

J. M. C. ClarkD. L. Ocone and C. Coumarbatch, Relative entropy and error bounds for filtering of Markov processes, Math. Control Signals Systems, 12 (1999), 346-360.  doi: 10.1007/PL00009856.  Google Scholar

[20]

A. D'Aristotile, P. Diaconis and D. Freedman, On merging of probabilities, Sankhyā Ser. A, 50 (1988), 363–380.  Google Scholar

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.  Google Scholar

[22]

H. E. Gollwitzer, A note on a functional inequality, Proc. Amer. Math. Soc., 23 (1969), 642-647.  doi: 10.1090/S0002-9939-1969-0247016-9.  Google Scholar

[23]

K. S. GurumoorthyC. GrudzienA. ApteA. Carrassi and C. K. R. T. Jones, Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution, SIAM J. Control Optim., 55 (2017), 741-759.  doi: 10.1137/15M1025839.  Google Scholar

[24]

G. Kallianpur, Stochastic Filtering Theory, Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. doi: 10.1007/978-1-4757-6592-2.  Google Scholar

[25]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[26]

D. T. B. KellyK. J. H. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579.  Google Scholar

[27]

Koro, Uniform expansivity., Available from: https://planetmath.org/UniformExpansivity. Google Scholar

[28]

K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.  Google Scholar

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.  Google Scholar

[30]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[31]

E. N. Lorenz, Predictability - A problem partly solved, in Predictability of Weather and Climate, Cambridge University Press, 2006, 40-58. doi: 10.1017/CBO9780511617652.004.  Google Scholar

[32]

C. McDonald and S. Yüksel, Converse results on filter stability criteria and stochastic non-linear observability, preprint, arXiv: 1812.01772. Google Scholar

[33]

B. Ni and Q. Zhang, Stability of the Kalman filter for continuous time output error systems, Systems Control Lett., 94 (2016), 172-180.  doi: 10.1016/j.sysconle.2016.06.006.  Google Scholar

[34]

D. Ocone and E. Pardoux, Asymptotic stability of the optimal filter with respect to its initial condition, SIAM J. Control Optim., 34 (1996), 226-243.  doi: 10.1137/S0363012993256617.  Google Scholar

[35]

D. L. Ocone, Asymptotic stability of Beneš filters, Stochastic Anal. Appl., 17 (1999), 1053-1074.  doi: 10.1080/07362999908809648.  Google Scholar

[36]

T. N. Palmer, Stochastic weather and climate models, Nature Reviews Physics, 1 (2019), 463-471.  doi: 10.1038/s42254-019-0062-2.  Google Scholar

[37]

A. S. Reddy, A. Apte and S. Vadlamani, Asymptotic properties of linear filter for deterministic processes, Systems Control Lett., 139 (2020), 8 pp. doi: 10.1016/j.sysconle.2020.104676.  Google Scholar

[38] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.  Google Scholar
[39]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[40]

R. van Handel, Filtering, Stability, and Robustness, Ph.D thesis, California Institute of Technology, 2007.  Google Scholar

[41]

R. van Handel, Nonlinear filtering and systems theory, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS Semi-Plenary Paper), 2010. Available from: https://web.math.princeton.edu/rvan/mtns10.pdf. Google Scholar

[42]

R. van Handel, Observability and nonlinear filtering, Probab. Theory Related Fields, 145 (2009), 35-74.  doi: 10.1007/s00440-008-0161-y.  Google Scholar

[43]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[44]

D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511813658.  Google Scholar

[45]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.  Google Scholar

Figure 1.  Dependence of $ \frac{D_N(x, y)}{\sum_{i = 0}^N \rho_{i\tau}} $ vs $ t = N\tau $ with $ \tau = 0.01 $ for $ 100 $ samples. We have $ t = N\tau $ with $ \tau = 0.01 $. We chose $ 100 $ different pairs of $ (x, y) $ for five different choices of $ \rho_t = 1000, \; t+1000, \; \log(t+1000); t^2+1000; t^3 + 1000 $. The initial conditions for the samples are randomly chosen from uniform distribution on $ [-10, 10]^{p} $ where the dimension $ p = 3 $ for Lorenz 63 model (left panel) and $ p = 36 $ for the Lorenz 96 model (right panel). The insets show the plots for large $ t $. (Note that the Lyapunov time scale for both these models is $ O(1) $.)
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