International Journal of Control, Automation, and Systems

Institute of Control, Robotics and Systems (제어로봇시스템학회)
권호 표지

Online Probability Density Estimation of Nonstationary Random Signal using Dynamic Bayesian Networks

  • Published : 2008.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

We present two estimators for discrete non-Gaussian and nonstationary probability density estimation based on a dynamic Bayesian network (DBN). The first estimator is for off line computation and consists of a DBN whose transition distribution is represented in terms of kernel functions. The estimator parameters are the weights and shifts of the kernel functions. The parameters are determined through a recursive learning algorithm using maximum likelihood (ML) estimation. The second estimator is a DBN whose parameters form the transition probabilities. We use an asymptotically convergent, recursive, on-line algorithm to update the parameters using observation data. The DBN calculates the state probabilities using the estimated parameters. We provide examples that demonstrate the usefulness and simplicity of the two proposed estimators.

Keywords

References

  1. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, Wiley-Interscience Publication, New York, 2001
  2. E. Parzen, "On estimation of a probability density function and mode," Analysis of Mathematical Statistics, vol. 33, pp. 1065-1076, 1962 https://doi.org/10.1214/aoms/1177704472
  3. B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman & Hall/CRC, 1986
  4. S. Fiori and P. Bucciarelli, "Probability density estimation using adaptive activation function neurons," Neural Processing Letters, vol. 12, pp. 31-42, 2001
  5. A. Sarajedini, R. Hecht-Nielsen, and P. M. Chau, "Conditional probability density function estimation with sigmoidal neural networks," IEEE Trans. on Neural Networks, vol. 10, no. 2, pp. 231-238, 1999 https://doi.org/10.1109/72.750544
  6. Y. Baram and Z. Roth, "Forecasting by density shaping using neural networks," IEEE/IAFE Proc. of Computational Intelligence for Financial Engineering, pp. 57-71, 1995
  7. A. Likas, "Probability density estimation using artificial neural networks," Computer Physics Communications, vol. 135, no. 2, pp. 167-175, 2001 https://doi.org/10.1016/S0010-4655(00)00235-6
  8. D. S. Modha and Y. Fainman, "A learning law for density estimation," IEEE Trans. on Neural Networks, vol. 5, no. 3, pp. 519-523, 1994 https://doi.org/10.1109/72.286931
  9. H. Yin and N. M. Allinson, "Self-organizing mixture networks for probability density estimation," IEEE Trans. on Neural Networks, vol. 12, no. 2, pp. 405- 411, 2001 https://doi.org/10.1109/72.914534
  10. T. Kostiainen and J. Lampinen, "On the generative probability density model in the self-organizing map," Neurocomputing, vol. 48, pp. 217-228, 2002 https://doi.org/10.1016/S0925-2312(01)00649-X
  11. M. Strikanth, H. K. Kesavan, and P. H. Roe, "Probability density function estimation using the MinMax measure," IEEE Trans. on Systems, Man, and Cybernetics-Part C: Applications and Reviews, vol. 30, no. 1, pp. 77-83, 2000 https://doi.org/10.1109/5326.827456
  12. G. Miller and D. Horn, "Maximum entropy approach to probability density estimation," Proc. of Int. Conf. on Knowledge-Based Intelligent Electronic Systems, vol. 1, pp. 225-230, 1998
  13. J. N. Kapur, G. Baciu, and H. K. Kesavan, "The MinMax information measure," Int. J. of System Science, vol. 26, no. 1, pp. 1-12, 1995 https://doi.org/10.1080/00207729508929020
  14. K. Kokkinakis, "Exponent parameter estimation for generalized Gaussian probability density functions with application to speech modeling," Signal Processing, vol. 85, no. 9, pp. 1852-1858, 2005 https://doi.org/10.1016/j.sigpro.2005.02.017
  15. C. Wang and W. Wang, "Links between PPCA and subspace methods for complete Gaussian density estimation," IEEE Trans. on Neural Networks, vol. 17, no. 3, pp. 789-792, 2006 https://doi.org/10.1109/TNN.2006.871718
  16. T. Kohonen, "The self-organization map," Proc. of the IEEE, vol. 78, no. 9, pp. 1464-1480, 1990 https://doi.org/10.1109/5.58325
  17. M. E. Tipping and C. M. Bishop, "Probability principle component analysis," J. Roy. Statist. Soc., Ser. B, vol. 21, no. 3, pp. 611-622, 1999
  18. A. Rodriguez and E. T. Horst, "Dynamic density estimation with financial applications," http://ftp.stat.duke.edu/WorkingPapers/06-21.pdf, 2006
  19. C. M. Bishop, Neural Networks for Pattern Recognition, Oxford University Press, 1995
  20. K. Murphy, Dynamic Bayesian networks: Representation, Inference and Learning, Ph.D. Dissertation, UC Berkeley, 2002
  21. J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control, Prentice Hall, New Jersey, 1995
  22. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw Hill, 2002
  23. R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley & Sons, New York, 1980
  24. W. J. Rugh, Linear System Theory, Prentice Hall, 1996
  25. H. Wang, A. Wang, and Y. Wang, "Online estimation algorithm for the unknown probability density functions of random parameters in auto-regression and exogenous stochastic systems," IEEE Proc.-Control Theory Applications, vol. 153, no. 4, pp. 462-468, 2006 https://doi.org/10.1049/ip-cta:20050312