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Directional splitting of Gaussian density in non-linear random variable transformation

Directional splitting of Gaussian density in non-linear random variable transformation

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Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving non-linear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the non-linearly transformed Gaussian variable by approximating the to-be-transformed random variable distribution by a Gaussian mixture (GM) distribution. The stress is laid on an analysis and selection of design parameters of the approximate GM distribution to minimise the error imposed by the non-linear transformation such as the location and number of the GM terms. A special attention is devoted to the definition of the novel GM splitting directions based on the measures of non-Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.

References

    1. 1)
      • 1. Anderson, B.D.O., Moore, J.B.: ‘Optimal filtering’ (Prentice Hall, New Jersey, 1979).
    2. 2)
      • 2. Sorenson, H.W., Alspach, D.L.: ‘Recursive Bayesian estimation using Gaussian sums’, Automatica, 1971, 7, pp. 465479.
    3. 3)
      • 3. Williams, J.L., Maybeck, P.S.: ‘Cost-function-based Gaussian mixture reduction for target tracking’. Proc. 6th Int. Conf. Information Fusion, Queensland, Australia, July 2003.
    4. 4)
      • 4. Faubel, F., McDonough, J., Klakow, D.: ‘The split and merge unscented Gaussian mixture filter’, IEEE Signal Process. Lett., 2009, 16, (9), pp. 786789.
    5. 5)
      • 5. Söderström, T.S., Stoica, P.G.: ‘System identification’ (Prentice Hall, 1989).
    6. 6)
      • 6. Král, L., Šimandl, M.: ‘Neural network based bicriterial dual control with multiple linearization’. Proc. 10th IFAC Workshop on Adaptation and Learning in Control and Signal Processing, Antalya, Turkey, August 2010.
    7. 7)
      • 7. Flídr, M., Straka, O., Šimandl, M.: ‘Pruning and merging strategies in receding horizon bicriterial dual controller with multiple linearization’. Proc. 19th Mediterranean Conf. Control & Automation, Corfu, Greece, June 2011.
    8. 8)
      • 8. Ding, S.X.: ‘Model-based fault diagnosis techniques: design schemes, algortihms and tools’ (Springer, London, 2008).
    9. 9)
      • 9. Gizda, D.R., Singla, P., Jah, M.K.: ‘An approach for nonlinear uncertainty propagation: application to orbital mechanics’. Proc. AIAA Guidance, Navigation, and Control Conf., Chicago, Illinois, August 2009.
    10. 10)
      • 10. DeMars, K.J., Bishop, R.H., Jah, M.K.: ‘Entropy-based approach for uncertainty propagation of nonlinear dynamical systems’, J. Guid. Control Dyn., 2013, 36, (4), pp. 10471057.
    11. 11)
      • 11. Reif, K., Günther, S., Yaz, E., et al: ‘Stochastic stability of the discerete-time extended kalman filter’, IEEE Trans. Autom. Control, 1999, 44, (4), pp. 714728.
    12. 12)
      • 12. Duník, J., Straka, O., Šimandl, M.: ‘Nonlinearity and non-Gaussianity measures for stochastic dynamic systems’. Proc. 16th Int. Conf. Information Fusion, Istanbul, 2013.
    13. 13)
      • 13. Liu, Y., Li, X.R.: ‘Measure of nonlinearity for estimation’, IEEE Trans. Signal Process., 2015, 63, (9), pp. 23772388.
    14. 14)
      • 14. Duník, J., Straka, O., Mallick, M., et al: ‘Survey of nonlinearity and non-Gaussianity measures for state estimation’. Proc. 19th Int. Conf. Information Fusion, Heidelberg, Germany, July 2016.
    15. 15)
      • 15. Raitoharju, M., Piché, R., Ala-Luhtala, J., et al: ‘Partitioned update Kalman filter’, J. Adv. Inf. Fusion, 2016, 11, (1), pp. 314.
    16. 16)
      • 16. Raitoharju, M., García-Fernández, A.F., Piché, R.: ‘Kullback – Leibler divergence approach to partitioned update Kalman filter’, Signal Process., 2017, 130, pp. 289298.
    17. 17)
      • 17. Terejanu, G., Singla, P., Singh, T., et al: ‘Uncertainty propagation for nonlinear dynamics systems using Gaussian sum mixture models’, J. Guid. Control Dyn., 2008, 31, (6), pp. 16231633.
    18. 18)
      • 18. Terejanu, G., Singla, P., Singh, T., et al: ‘Adaptive Gaussian sum filter for nonlinear Bayesian estimation’, IEEE Trans. Autom. Control, 2011, 56, (9), pp. 21512156.
    19. 19)
      • 19. DeMars, K.J., Cheng, Y., Jah, M.K.: ‘Collision probability with Gaussian mixture orbit uncertainty’, J. Guid. Control Dyn., 2014, 37, (3), pp. 979985.
    20. 20)
      • 20. Straka, O., Duník, J., Šimandl, M.: ‘Structure adaptation of nonlinear filters based on non-Gaussianity measures’. Proc. 2015 American Control Conf., Chicago, IL, USA, 2015.
    21. 21)
      • 21. Raitoharju, M., Ali-Löytty, S., Piché, R.: ‘Binomial Gaussian mixture filter’, EURASIP J. Adv. Signal Proc., 2015, 36, pp. 118.
    22. 22)
      • 22. Straka, O., Duník, J., Punčochář, I.: ‘Direcional splitting for structure adaptation of Bayesian filters’. Proc. American Control Conf., Boston, MA, USA, July 2016.
    23. 23)
      • 23. Sorenson, H.W., Alspach, D.L.: ‘Approximation of density function by a sum of gaussians for nonlinear Bayesian estimation’. Proc. 1st Symp. Nonlinear Estimation Theory and Its Applications, San Diego, 1970, pp. 8890.
    24. 24)
      • 24. Schön, T., Gustafsson, F., Nordlund, P.: ‘Marginalized particle filters for mixed linear/nonlinear state-space models’, IEEE Trans. Signal Process., 2005, 53, (7), pp. 22792289.
    25. 25)
      • 25. Šmídl, V., Gašperin, M.: ‘Rao-Blackwellized point mass filter for reliable state estimation’. 16th Int. Conf. Information Fusion, Istanbul, Turkey, 2013.
    26. 26)
      • 26. Horwood, J.T., Aragon, N.D., Poore, A.B.: ‘Gaussian sum filters for space surveillance: theory and simulations’, J. Guid. Control Dyn., 2011, 34, (6), pp. 18391851.
    27. 27)
      • 27. Hanebeck, U.D., Briechle, K., Rauh, A.: ‘Progressive Bayes: a new framework for nonlinear state estimation’. Proc. SPIE, AeroSense Symp., Orlando, Florida, 2003, vol. 5099, pp. 256267.
    28. 28)
      • 28. Couvreur, C.: ‘The EM algorithm: a guided tour’. Proc. Second IEEE European Workshop on Computer-Intensive Methods in Control and Signal Processing, Prague, Czech Republic, 1997, pp. 115120.
    29. 29)
      • 29. Faubel, F., Klakow, D.: ‘Further improvement of the adaptive level of detail transform: splitting in direction on the nonlinearity’. Proc. 18th European Signal Processing Conf., Aalborg, Denmark, August 2010.
    30. 30)
      • 30. Huber, M.F.: ‘Adaptive Gaussian mixture filter based on statistical linearisation’. Proc. of the 14th Int. Conf. Information Fusion, Chicago, IL, USA, July 2011.
    31. 31)
      • 31. Raitoharju, M., Ali-Löytty, S.: ‘An adaptive derivative free method for Bayesian posterior approximation’, IEEE Signal Process. Lett., 2012, 19, (2), pp. 8790.
    32. 32)
      • 32. Havlak, F., Campbell, M.: ‘Discrete and continuous probabilistic anticipation for autonomous robots in urban environments’, IEEE Trans. Robot., 2013, 30, (2), pp. 461474.
    33. 33)
      • 33. Cardoso, J.-F.: ‘Dependence, correlation and Gaussianity in independent component analysis’, J. Mach. Learn. Res., 2003, 4, pp. 11771203.
    34. 34)
      • 34. Stroud, A.H.: ‘Approximate calculation of multiple integrals’ (Prentice-Hall, Englewood Cliffs, NJ, 1971).
    35. 35)
      • 35. Nørgaard, M., Poulsen, N.K., Ravn, O.: ‘New developments in state estimation for nonlinear systems’, Automatica, 2000, 36, (11), pp. 16271638.
    36. 36)
      • 36. Julier, S.J., Uhlmann, J.K.: ‘Unscented filtering and nonlinear estimation’, IEEE Rev., 2004, 92, (3), pp. 401421.
    37. 37)
      • 37. Wu, Y., Hu, D., Wu, M., et al: ‘A numerical-integration perspective on Gaussian filters’, IEEE Trans. Signal Process., 2006, 54, (8), pp. 29102921.
    38. 38)
      • 38. Steinbring, J., Hanebeck, U.D.: ‘LRKF revisited: The smart sampling Kalman filter (S2KF)’, J. Adv. Inf. Fusion, 2014, 9, (2), pp. 106123.
    39. 39)
      • 39. Duník, J., Straka, O., Šimandl, M., et al: ‘Random-point-based filters: analysis and comparison in target tracking’, IEEE Trans. Aerosp. Electron. Syst., 2015, 51, (2), pp. 303308.
    40. 40)
      • 40. Carreire-Perpinan, M.A.: ‘Mode-finding for mixtures of Gaussian distributions’, IEEE Trans. Pattern Anal. Mach. Intell., 2000, 22, (11), pp. 13181323.
    41. 41)
      • 41. Golub, G.H., Van Loan, C.F.: ‘Matrix computations’ (The Johns Hopkins University Press, Baltimore and London, 1996, 3rd edn.).
    42. 42)
      • 42. Mallick, M.: ‘Differential geometry measures of nonlinearity with applications to ground target tracking’. Proc. 7th Int. Conf. Information Fusion, Stockholm, Sweden, June 2004.
    43. 43)
      • 43. Groeneveld, R.A., Meeden, G.: ‘Measuring skewness and kurtosis’, J. R. Stat. Soc. D, Stat., 1984, 33, (4), pp. 391399.
    44. 44)
      • 44. von Hippel, P.: ‘Skewness’, in Lovric, M. (Ed.): ‘International encyclopedia of statistical science’ (Springer, 2010).
    45. 45)
      • 45. Hyvärinen, A., Oja, E.: ‘Independent component analysis: algorithms and applications’, Neural Netw., 2000, 13, pp. 411430.
    46. 46)
      • 46. Miller, M.B.: ‘Mathematics and statistics for financial risk management’ (Wiley, Hoboken, New Jersey, 2014).
    47. 47)
      • 47. Kollo, T., Von Rosen, D.: ‘Advanced multivariate statistics with matrices’, (Springer, The Netherland, 2005).
    48. 48)
      • 48. Strang, G.: ‘Introduction to linear algebra’ (Wellesley-Cambridge Press, 2009, 4th edn.).
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