A Novel Parameter Estimation Algorithm for the Multivariate t-Distribution and Its Application to Computer Vision

  • Chad Aeschliman
  • Johnny Park
  • Avinash C. Kak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6312)


We present a novel algorithm for approximating the parameters of a multivariate t-distribution. At the expense of a slightly decreased accuracy in the estimates, the proposed algorithm is significantly faster and easier to implement compared to the maximum likelihood estimates computed using the expectation-maximization algorithm. The formulation of the proposed algorithm also provides theoretical guidance for solving problems that are intractable with the maximum likelihood equations. In particular, we show how the proposed algorithm can be modified to give an incremental solution for fast online parameter estimation. Finally, we validate the effectiveness of the proposed algorithm by using the approximated t-distribution as a drop in replacement for the conventional Gaussian distribution in two computer vision applications: object recognition and tracking. In both cases the t-distribution gives better performance with no increase in computation.


Approximate Algorithm Multivariate Gaussian Distribution Incremental Algorithm Scale Matrix Parameter Estimation Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aeschliman, C., Park, J., Kak, A.C.: A Probabilistic Framework for Joint Segmentation and Tracking. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2010)Google Scholar
  2. 2.
    Chen, F., Lambert, D., Pinheiro, J.C.: Incremental quantile estimation for massive tracking. In: Proceedings of the Sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 516–522. ACM, New York (2000)CrossRefGoogle Scholar
  3. 3.
    Geusebroek, J.M., Burghouts, G.J., Smeulders, A.W.M.: The Amsterdam library of object images. International Journal of Computer Vision 61(1), 103–112 (2005)CrossRefGoogle Scholar
  4. 4.
    Iscaps, C.: Pets2006 (2006),
  5. 5.
    Khan, Z., Balch, T., Dellaert, F.: MCMC-based particle filtering for tracking a variable number of interacting targets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1805–1918 (2005)Google Scholar
  6. 6.
    Kotz, S., Nadarajah, S.: Multivariate t distributions and their applications. Cambridge Univ. Pr., Cambridge (2004)zbMATHCrossRefGoogle Scholar
  7. 7.
    Lange, K.L., Little, R.J.A., Taylor, J.M.G.: Robust statistical modeling using the t distribution. Journal of the American Statistical Association, 881–896 (1989)Google Scholar
  8. 8.
    Liu, C., Rubin, D.B.: ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statistica Sinica 5(1), 19–39 (1995)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Meng, X.L., van Dyk, D.: The EM algorithm–an old folk-song sung to a fast new tune. Journal of the Royal Statistical Society. Series B (Methodological), 511–567 (1997)Google Scholar
  10. 10.
    Nadarajah, S., Kotz, S.: Estimation Methods for the Multivariate t Distribution. Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications 102(1), 99–118 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Nguyen, H.T., Ji, Q., Smeulders, A.W.M.: Spatio-temporal context for robust multitarget tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(1), 52 (2007)CrossRefGoogle Scholar
  12. 12.
    Peel, D., McLachlan, G.: Robust mixture modelling using the t distribution. Statistics and Computing 10(4), 339–348 (2000)CrossRefGoogle Scholar
  13. 13.
    Rothenberg, T.J., Fisher, F.M., Tilanus, C.B.: A note on estimation from a Cauchy sample. Journal of the American Statistical Association 59(306), 460–463 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Simoncelli, E.P.: Statistical modeling of photographic images. In: Handbook of Image and Video Processing, pp. 431–441 (2005)Google Scholar
  15. 15.
    Tierney, L.: A space-efficient recursive procedure for estimating a quantile of an unknown distribution. SIAM Journal on Scientific and Statistical Computing 4, 706 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tipping, M., Bishop, C.M.: Probabilistic principal component analysis. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 61(3), 611–622 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Zhao, J., Jiang, Q.: Probabilistic PCA for t distributions. Neurocomputing 69(16-18), 2217–2226 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Chad Aeschliman
    • 1
  • Johnny Park
    • 1
  • Avinash C. Kak
    • 1
  1. 1.Purdue University 

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