Acta Mathematica Sinica

, Volume 22, Issue 4, pp 1175–1182 | Cite as

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

  • J. J. Egozcue
  • J. L. Díaz–Barrero
  • V. Pawlowsky–Glahn


The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison's ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre–Hilbert space. A Hilbert space of densities, whose logarithm is square–integrable, is obtained as the natural completion of the pre–Hilbert space.


Bayes' theorem Fourier coefficients Haar basis Aitchison distance Simplex Least squares approximation 

MR (2000) Subject Classification

46C15 42B05 


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  1. 1.
    Aitchison, J.: The statistical analysis of compositional data (with discussion). Journal of the Royal Statistical Society, Series B (Statistical Methodology), 44, 139–177 (1982)MATHMathSciNetGoogle Scholar
  2. 2.
    Aitchison, J., The Statistical Analysis of Compositional Data: Monographs on Statistics and Applied Probability, Chapman & Hall Ltd., London (UK), 1986, (Reprinted in 2003 with additional material by The Blackburn Press)Google Scholar
  3. 3.
    Billheimer, D., Guttorp, P., Fagan, W. F.: Statistical Interpretation of Species Composition. Journal of the American Statistical Association, 96, 1205–1214 (2001)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Pawlowsky-Glahn, V., Egozcue, J. J.: Geometric approach to statistical analysis on the simplex. Stochastic Enviromental Research and Risk Assessment, 15, 384–398 (2001)MATHCrossRefGoogle Scholar
  5. 5.
    Pawlowsky-Glahn, V., Egozcue, J. J.: BLU Estimators and Compositional Data. Mathematical Geology, 34, 259–274 (2002)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Aitchison, J., Barceló-Vidal, C., Egozcue, J. J., Pawlowsky–Glahn, V.: A concise guide to the algebraic-geometric structure of the simplex, the sample space for compositional data analysis, Proceedings of IAMG’02, The Seventh Annual Conference of the International Association for Mathematical Geology, Berlin, Germany, 2002Google Scholar
  7. 7.
    Burbea, J., Rao, R.: Entropy differential metric, distance and divergence measures in probability spaces: aunified approach. Journal of Multivariate Analysis, 12, 575–596 (1982)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Egozcue, J. J., Díaz-Barrero, J. L.: Hilbert space of probability density functions with Aitchison geometry, Proceedings of Compositional Data Analysis Workshop, CoDaWork’03, Girona (Spain) 2003, (ISBN 84-8458-111-X)Google Scholar
  9. 9.
    Egozcue, J. J., Pawlowsky-Glahn, V., Mateu–Figueras, G., Barceló–Vidal, C.: Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35, 279–300 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berberian, S. K.: Introduction to Hilbert Space, University Press, New York, 1961Google Scholar
  11. 11.
    Haar, A.: Zur Theorie der Ortogonalen Funktionen–Systeme. Math. Ann., 69, 331–371 (1910)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions, Dover, New York, 1972Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. J. Egozcue
    • 1
  • J. L. Díaz–Barrero
    • 1
  • V. Pawlowsky–Glahn
    • 2
  1. 1.Applied Mathematics IIIUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Informatics and Applied MathematicsUniversitat de GironaGironaSpain

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