| Hilbert Space of Probability Density Functions Based on Aitchison Geometry |
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| 作者姓名： | J. J. EGOZCUE J. L. DIAZ-BARRERO V. PAWLOWSKY-GLAHN |
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| 作者单位： | [1]Applied Mathematics Ⅲ, Universitat Politecnica de Catalunya, Jordi Girona 1-3, C2, 08034, Barcelona, Spain [2]Informatics and Applied Mathematics, Universitat de Girona, Campus Montilivi, P4, 17071, Girona, Spain |
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| 基金项目： | This research has received financial support from the Direccion General de Investigacion of the Spanish Ministry for Science and Technology through the project BFM2003-05640/MATE and from the Departament d＇Universitats, Recerca i Societat de la Informacio of the Generalitat de Catalunya through the project 2003XT 00079 Acknowledgements The critical and constructive comments and suggestions of an anonymous referee have contributed significantly to the improvement of the paper. |
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| 摘 要： | 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.
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| 关 键 词： | Bayes定理 Fourier系数 Haar基础 单纯性 最小平方逼近 |
| 收稿时间： | 2004-03-18 |
| 修稿时间： | 2004-03-182004-10-18 |
Hilbert Space of Probability Density Functions Based on Aitchison Geometry |
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| J. J. EGOZCUE J. L. DIAZ-BARRERO V. PAWLOWSKY-GLAHN.Hilbert Space of Probability Density Functions Based on Aitchison Geometry[J].Acta Mathematica Sinica,2006,22(4):1175-1182. |
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| Authors: | J J Egozcue J L Díaz–Barrero V Pawlowsky–Glahn |
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| Institution: | (1) Applied Mathematics III, Universitat Politècnica de Catalunya, Jordi Girona 1–3, C2, 08034 Barcelona, Spain;(2) Informatics and Applied Mathematics, Universitat de Girona, Campus Montilivi, P4, 17071 Girona, Spain |
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| Abstract: | 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 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.
This research has received financial support from the Dirección General de Investigación of the Spanish Ministry for Science and Technology through the project BFM2003–05640/MATE and from the Departament d'Universitats,
Recerca i Societat de la Informació of the Generalitat de Catalunya through the project 2003XT 00079 |
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| Keywords: | Bayes' theorem Fourier coefficients Haar basis Aitchison distance Simplex Least squares approximation |
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