Abstract
This paper derives recurrent expressions for the maximum attainable estimation accuracy calculated using the Cramér–Rao inequality (Cramér–Rao lower bound) in the discretetime nonlinear filtering problem under conditions when generating noises in the state vector and measurement error equations depend on estimated parameters and the state vector incorporates a constant subvector. We establish a connection to similar expressions in the case of no such dependence. An example illustrates application of the obtained algorithms to lowerbound accuracy calculation in a parameter estimation problem often arising in navigation data processing within a model described by the sum of a Wiener sequence and discrete-time white noise of an unknown variance.
Similar content being viewed by others
References
Yarlykov, M.S., Statisticheskaya teoriya radionavigatsii (Statistical Theory of Radio Navigation), Moscow: Radio i Svyaz’, 1985.
Yarlykov, M.S. and Mironov, M.A., Markovskaya teoriya otsenivaniya sluchainykh protsessov (Markov Theory of Random Processes Estimation), Moscow: Radio i Svyaz’, 1993.
Stepanov, O.A., Primenenie teorii nelineinoi fil’tratsii v zadachakh obrabotki navigatsionnoi informatsii (Application of Nonlinear Filtering Theory in Navigational Data Processing Problems), St. Petersburg: GNTs RF—TsNII “Elektropribor”, 1998.
Dmitriev, S.P. and Stepanov, O.A., Nonlinear Filtering and Navigation, Proc. 5th St. Petersburg Int. Conf. on Integrated Navigation Syst., St. Petersburg, 1998, pp. 138–149.
Bergman, N., Recursive Bayesian Estimation. Navigation and Tracking Applications, in Linkoping Studies in Science and Technology. Dissertations-no. 579, Department of Electrical Engineering, Linkoping University, SE-581-83, Linkoping, Sweden, 1999.
Ivanov, V.M., Stepanov, O.A., and Korenevski, M.L., Monte Carlo Methods for a Special Nonlinear Filtering Problem, Proc. 11th IFAC Int. Workshop Control Applications of Optimization, 2000, vol. 1, pp. 347–353.
Doucet, A., de Freitas, N., and Gordon, N., Sequential Monte Carlo Methods in Practice, New York: Springer-Verlag, 2001.
Rozov, A.K., Nelineinaya fil’tratsiya signalov (Nonlinear Filtering of Signals), St. Petersburg: Politekhnika, 2002, 2nd ed.
Gustafsson, F., Bergman, N., Forssel, U., et al., Particle Filters for Positioning, Navigation and Tracking, IEEE Trans. Signal Process., 2002, vol. 50, no. 2, pp. 425–437.
Ristic, B., Arulampalam, S., and Gordon, N., Beyond the Kalman Filter. Particle Filters for Tracking Applications, London: Artech House, 2004.
Dmitriev, S.P. and Stepanov, O.A., Multiple-alternative Filtering in Navigational Data Processing Problems, Radiotekhn., 2004, no. 7, pp. 11–17.
Daum, F., Nonlinear Filters: Beyond the Kalman Filter, IEEE Aerospace Electron. Syst., 2005, vol. 8, pp. 57–71.
Bar-Shalom, Y. and Li, X.R., Estimation and Tracking: Principles, Techniques, and Software, Norwood: Artech House, 1993. Translated under the title Traektornaya obrabotka. Printsipy, sposoby i algoritmy, tom 1–2, Moscow: Gos. Tekhn. Univ. im. Baumana, 2011.
Bar-Shalom, Y., Willett, P.K., and Tian, X., Tracking and Data Fusion: A Handbook of Algorithms, Storrs: YBS, 2011.
Stepanov, O.A., Osnovy teorii otsenivaniya s prilozheniyami k zadacham obrabotki navigatsionnoi informatsii. Chast’ 1. Vvedenie v teoriyu otsenivaniya (Foundations of Estimation Theory with Application to Navigation Data Processing Problems. Part 1. An Introduction to Estimation Theory), St. Petersburg: GNTs RF—TsNII “Elektropribor,” ITMO, 2010, 2012.
Toropov, A.B. and Stepanov, O.A., Usage of Sequential Monte Carlo Methods in the Correlation- Extremal Navigation Problem, Izv. Vyssh. Uchebn. Zaved., Priborostr., 2010, vol. 53, no. 10, pp. 49–54.
Parakhnevich, A.V., Solonar, A.S., and Gorshkov, S.A., Approaches to Significant Probability Density Choice in Particle Filters, Dokl. Beloruss. Gos. Univ. Informatik. Radioelektr., 2012, no. 4 (66), pp. 68–74.
Kosachev, I.M., A Methodology of High-Accuracy Nonlinear Filtering of Random Processes in Stochastic Dynamic Fixed-Structure Systems, Vestn. Voenn. Akad. Resp. Belarus’, 2014, no. 4 (66), pp. 125–160.
Snyder, C., Bengtsson, T., Bickel, P., et al., Obstacles to High-Dimensional Particle Filtering, Monthly Weather Rev., 2008, vol. 136, no. 12, pp. 4629–4640.
Berkovskii, N.A. and Stepanov, O.A., Error of Calculating the Optimal Bayesian Estimate Using the Monte Carlo Method in Nonlinear Problems, J. Comp. Syst. Sci. Int., 2013, vol. 52, no. 3, pp. 342–353.
Cramér, H., Mathematical Methods of Statistics, Princeton: Princeton Univ. Press, 1946. Translated under the title Matematicheskie metody statistiki, Kolmogorov, A.N., Ed., Moscow: Inostrannaya Literatura, 1948.
Van Trees, H.L., Detection, Estimation, and Modulation Theory, Part 1, New York: Wiley, 1968. Translated under the title Teoriya obnaruzheniya, otsenok i modulyatsii. Tom 1. Teoriya obnaruzheniya, otsenok i lineinoi modulyatsii, Moscow: Sovetskoe Radio, 1972.
Galdos, J.I., A Cramer–Rao Bound for Multidimensional Discrete–Time Dynamical Systems, IEEE Trans. Automat. Control, 1980, vol. AC-25, no. 1, pp. 117–119.
Koshaev, D.A. and Stepanov, O.A., Application of the Rao–Cramer Inequality in Problems of Nonlinear Estimation, J. Comp. Syst. Sci. Int., 1997, vol. 36, no. 2, pp. 220–227.
Tichavsky, P., Muravchik, C., and Nehorai, A., Posterior Cramer–Rao Bounds for Discrete–Time Nonlinear Filtering, IEEE Trans. Signal Process., 1998, no. 46, pp. 1386–1398.
Simandl, M., Kralovec, J., and Tichavsky, P., Filtering, Predictive and Smoothing Cramer–Rao Bounds for Discrete–Time Nonlinear Dynamic Systems, Automatica, 2001, no. 37, pp. 1703–1716.
Van Trees, H.L. and Bell, K.L., Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking, San-Francisco: Wiley–IEEE Press, 2007.
Stepanov, O.A., Proceeding Cramer–Rao Bounds for Special Nonlinear Filtering Problems, Proc. Eur. Control Conf. ECC’99, Karlsruhe, Germany, 1999.
Bergman, N., Poster Cramer–Rao Bounds for Sequential Estimation. See 1999 [7, pp. 321–338].
Batista, P., Silvestre, C., and Oliveira, P., Preliminary Results on the Estimation Performance of Single Range Source Localization, Proc. 21st Mediterranean Conf. on Control & Automation, Platanias-Chania, Crete, Greece, 2013, pp. 421–424.
Koshaev, D.A., A Comparison of Lower Bounds of Accuracy in Problems of Nonlinear Estimation, J. Comp. Syst. Sci. Int., 1998, vol. 37, no. 2, pp. 222–224.
Dmitriev, S.P. and Sokolov, A.I., Frequency Shift Estimation in a Doppler Log via Identification of the Received Signal Model, Giroskop. Navigats., 2006, no. 1, pp. 21–29.
Stepanov, O.A. and Motorin, A.V., A Comparison of Identification Methods for the Error Models of Sensors Based on Allan Variations and Nonlinear Filtering Algorithms, Mater. 21 Sankt-Peterburg. konf. “Integrirovannye navigatsionnye sistemy” (Proc. 21st St. Petersburg Conf. “Integrated Navigation Systems”), 2014.
Stepanov, O.A., Vasilyev, V.A., and Dolnakova, A.S., Cramer–Rao Lower Bound for Parameters of Random Processes in Navigation Data Processing, Proc. 21st Mediterranean Conf. on Control & Automation, Platanias-Chania, Crete, Greece, 2013, pp. 1214–1221.
Stepanov, O.A., Dolnakova, A.S., and Sokolov, A.I., Analysis of Potential Estimation Accuracy of Random Process Parameters in Navigation Data Processing Problems, Tr. XXII Vseross. soveshchan. po problemam upravleniya (Proc. XXII All-Russia Meeting on Control Problems), Moscow: Inst. Probl. Upravlen., 2014, pp. 3730–3740.
Stepanov, O.A., Priblizhennye metody analiza potentsial’noi tochnosti v nelineinykh navigatsionnykh zadachakh (Approximate Analysis Methods for Potential Accuracy in Nonlinear Navigation Problems), Leningrad: Tsentr. Nauchno-Issled. Inst. Rumb, 1986.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © O.A. Stepanov, V.A. Vasil’ev, 2016, published in Avtomatika i Telemekhanika, 2016, No. 1, pp. 104–133.
Rights and permissions
About this article
Cite this article
Stepanov, O.A., Vasil’ev, V.A. Cramér–Rao lower bound in nonlinear filtering problems under noises and measurement errors dependent on estimated parameters. Autom Remote Control 77, 81–105 (2016). https://doi.org/10.1134/S0005117916010057
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916010057