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Hilbert space of probability density functions based on Aitchison geometry. (English) Zbl 1113.46016
Summary: The set of probability functions is a convex subset of \(L^1\) 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, generalized perturbation and power transformation are used, 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.

46C99 Inner product spaces and their generalizations, Hilbert spaces
Full Text: DOI
[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) · Zbl 0491.62017
[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) · Zbl 0688.62004
[3] Billheimer, D., Guttorp, P., Fagan, W. F.: Statistical Interpretation of Species Composition. Journal of the American Statistical Association, 96, 1205–1214 (2001) · Zbl 1073.62573 · doi:10.1198/016214501753381850
[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) · Zbl 0987.62001 · doi:10.1007/s004770100077
[5] Pawlowsky-Glahn, V., Egozcue, J. J.: BLU Estimators and Compositional Data. Mathematical Geology, 34, 259–274 (2002) · Zbl 1031.86007 · doi:10.1023/A:1014890722372
[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, 2002
[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) · Zbl 0526.60015 · doi:10.1016/0047-259X(82)90065-3
[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) · Zbl 1113.46016
[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) · Zbl 1302.86024 · doi:10.1023/A:1023818214614
[10] Berberian, S. K.: Introduction to Hilbert Space, University Press, New York, 1961 · Zbl 0121.09302
[11] Haar, A.: Zur Theorie der Ortogonalen Funktionen–Systeme. Math. Ann., 69, 331–371 (1910) · JFM 41.0469.03 · doi:10.1007/BF01456326
[12] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions, Dover, New York, 1972 · Zbl 0543.33001
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