Gaussian Mixture Filters and Hybrid Positioning

This paper presents, develops and compares Gaussian Mixture Filter (GMF) methods for hybrid positioning. The key idea of the developed method is to approximate the prior density as a Gaussian mixture with a small number of mixture components. We show why it is sometimes reasonable to approximate a Gaussian prior with a multicomponent Gaussian mixture. We also present both simulated and real data tests of different filters in different scenarios. Simulations show that GMF gives better accuracy than Extended Kalman Filter with lower computational requirements than Particle Filter, making it a reasonable algorithm for the hybrid positioning problem.

[1]  Tor Steinar Schei,et al.  A finite-difference method for linearization in nonlinear estimation algorithms , 1997, Autom..

[2]  R. Martin,et al.  Robust bayesian estimation for the linear model and robustifying the Kalman filter , 1977 .

[3]  Robert Grover Brown,et al.  Introduction to random signal analysis and Kalman filtering , 1983 .

[4]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[5]  Marco Piras,et al.  The Robust Statistics method applied in the Kalman filter: theory and application , 2005 .

[6]  François Marx,et al.  Map-aided indoor mobile positioning system using particle filter , 2005, IEEE Wireless Communications and Networking Conference, 2005.

[7]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[8]  R. Bucy,et al.  Digital synthesis of non-linear filters , 1971 .

[9]  Robert Piche,et al.  Consistency of three Kalman filter extensions in hybrid navigation , 2005 .

[10]  Niels Kjølstad Poulsen,et al.  New developments in state estimation for nonlinear systems , 2000, Autom..

[11]  Jussi Collin,et al.  Investigations of self-contained sensors for personal navigation , 2006 .

[12]  Ward Cheney,et al.  A course in approximation theory , 1999 .

[13]  Kazufumi Ito,et al.  Gaussian filters for nonlinear filtering problems , 2000, IEEE Trans. Autom. Control..

[14]  Jukka Saarinen,et al.  An Evaluation of Motion Model Structures within the IMM Frame Using Range-Only Measurements , 1999, IC-AI.

[15]  T. Singh,et al.  The higher order unscented filter , 2003, Proceedings of the 2003 American Control Conference, 2003..

[16]  Robert Piché,et al.  Robust Extended Kalman Filtering in Hybrid Positioning Applications , 2007, 2007 4th Workshop on Positioning, Navigation and Communication.

[17]  Henry Tirri,et al.  A Statistical Modeling Approach to Location Estimation , 2002, IEEE Trans. Mob. Comput..

[18]  Dimitrios Hatzinakos,et al.  An adaptive Gaussian sum algorithm for radar tracking , 1997, Proceedings of ICC'97 - International Conference on Communications.

[19]  Herman Bruyninckx,et al.  Kalman filters for non-linear systems: a comparison of performance , 2004 .

[20]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[21]  Alexander A. Mikhalev,et al.  Comparison of Hough Transform and Particle Filter Methods of Emitter Geolocation using Fusion of TDOA Data , 2007, 2007 4th Workshop on Positioning, Navigation and Communication.

[22]  Amir Averbuch,et al.  Interacting Multiple Model Methods in Target Tracking: A Survey , 1988 .

[23]  S. Ali-Loytty,et al.  MAT-45806 Mathematics for Positioning TKT-2546 Methods for Positioning , 2008 .

[24]  Ali H. Sayed,et al.  Linear Estimation (Information and System Sciences Series) , 2000 .

[25]  Changlin Ma.  Integration of GPS and Cellular Networks to Improve Wireless Location Performance , 2003 .

[26]  Siliang Wu,et al.  Angle-Parameterizations Range-only Target Tracking for Scalar Miss Distance Measurement System , 2006, 2006 8th international Conference on Signal Processing.

[27]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

[28]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[29]  S. Ali-Loytty,et al.  Efficient Gaussian mixture filter for hybrid positioning , 2008, 2008 IEEE/ION Position, Location and Navigation Symposium.

[30]  Niilo Sirola.  A Versatile Algorithm for Local Positioning in Closed Form , 2004 .

[31]  James Ting-Ho Lo,et al.  Finite-dimensional sensor orbits and optimal nonlinear filtering , 1972, IEEE Trans. Inf. Theory.

[32]  Henry Tirri,et al.  Topics in probabilistic location estimation in wireless networks , 2004, 2004 IEEE 15th International Symposium on Personal, Indoor and Mobile Radio Communications (IEEE Cat. No.04TH8754).

[33]  Yoshiaki Shirai,et al.  Hand gesture estimation and model refinement using monocular camera-ambiguity limitation by inequality constraints , 1998, Proceedings Third IEEE International Conference on Automatic Face and Gesture Recognition.

[34]  Simo Ali-Loytty,et al.  Gaussian Mixture Filter in Hybrid Navigation , 2007 .

[35]  Hugh F. Durrant-Whyte,et al.  A new method for the nonlinear transformation of means and covariances in filters and estimators , 2000, IEEE Trans. Autom. Control..

[36]  N. Bergman.  Bayesian Inference in Terrain Navigation , 1997 .

[37]  Petar M. Djuric,et al.  Gaussian sum particle filtering , 2003, IEEE Trans. Signal Process..

[38]  Demetrios G. Lainiotis,et al.  A new class of efficient adaptive nonlinear filters (ANLF) , 1998, IEEE Trans. Signal Process..

[39]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[40]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[41]  Raphael Andreas Hauser,et al.  Kalman Filtering with Equality and Inequality State Constraints , 2007, 0709.2791.

[42]  W. Rudin.  Real and complex analysis , 1968 .

[43]  J. W. Chaffee,et al.  The GPS filtering problem , 1992, IEEE PLANS 92 Position Location and Navigation Symposium Record.

[44]  Demetrios G. Lainiotis,et al.  Efficient algorithms of clustering adaptive nonlinear filters , 1999, IEEE Trans. Autom. Control..

[45]  Robert Piche,et al.  Closed-Form Solutions for Hybrid Cellular/GPS Positioning , 2003 .

[46]  Harold W. Sorenson,et al.  Recursive Bayesian estimation using piece-wise constant approximations , 1988, Autom..

[47]  Rudolph van der Merwe,et al.  Sigma-point kalman filters for probabilistic inference in dynamic state-space models , 2004 .

[48]  Irwin Guttman,et al.  Optimal collapsing of mixture distributions in robust recursive estimation , 1989 .

[49]  Nando de Freitas,et al.  The Unscented Particle Filter , 2000, NIPS.

[50]  H. Sorenson,et al.  Recursive bayesian estimation using gaussian sums , 1971 .

[51]  Torsten Söderström,et al.  Advanced point-mass method for nonlinear state estimation , 2006, Autom..

[52]  Fabio Dovis,et al.  Navigation and Communication Hybrid Positioning with a Common Receiver Architecture , 2004 .

[53]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[54]  J. Syrjärinne,et al.  Studies of Modern Techniques for Personal Positioning , 2001 .

[55]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[56]  H.F. Durrant-Whyte,et al.  A new approach for filtering nonlinear systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[57]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[58]  Simo Ali-Loytty,et al.  Benchmarking Nonlinear Filters , 2006, 2006 IEEE Nonlinear Statistical Signal Processing Workshop.

[59]  Y. Ho,et al.  A Bayesian approach to problems in stochastic estimation and control , 1964 .

[60]  Dan Simon,et al.  Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches , 2006 .

[61]  S. Bancroft.  An Algebraic Solution of the GPS Equations , 1985, IEEE Transactions on Aerospace and Electronic Systems.

[62]  P. Maybeck,et al.  Multiple model tracker based on Gaussian mixture reduction for maneuvering targets in clutter , 2005, 2005 7th International Conference on Information Fusion.

[63]  Maurizio A. Spirito,et al.  Experimental performance of methods to estimate the location of legacy handsets in GSM , 2001, IEEE 54th Vehicular Technology Conference. VTC Fall 2001. Proceedings (Cat. No.01CH37211).

[64]  Jing Liu,et al.  Survey of Wireless Indoor Positioning Techniques and Systems , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[65]  Rudolph van der Merwe,et al.  Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[66]  T. Ferguson.  A Course in Large Sample Theory , 1996 .

[67]  Henry Tirri,et al.  A Probabilistic Approach to WLAN User Location Estimation , 2002, Int. J. Wirel. Inf. Networks.

[68]  Jeffrey K. Uhlmann,et al.  Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[69]  Simon J. Julier,et al.  Skewed approach to filtering , 1998, Defense, Security, and Sensing.

[70]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice , 1993 .

[71]  Simon J. Julier,et al.  The scaled unscented transformation , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[72]  Simo Ali-Loytty,et al.  A Modified Kalman Filter for Hybrid Positioning , 2006 .

[73]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[74]  D. Salmond.  Mixture reduction algorithms for target tracking , 1989 .

[75]  Robert Grover Brown,et al.  Introduction to random signals and applied Kalman filtering : with MATLAB exercises and solutions , 1996 .