We propose a general partition-based strategy to estimate conditional density with candidate densities that are piecewise constant with respect to the covariate. Capitalizing on a general penalized maximum likelihood model selection result, we prove, on two specific examples, that the penalty of each model can be chosen roughly proportional to its dimension. We first study a classical strategy in which the densities are chosen piecewise conditional according to the variable. We then consider Gaussian mixture models with mixing proportion that vary according to the covariate but with common mixture components. This model proves to be interesting for an unsupervised segmentation application that was our original motivation for this work.
Mots-clés : conditional density estimation, partition, penalized likelihood, piecewise polynomial density, gaussian mixture model
@article{PS_2013__17__672_0,
author = {Cohen, S. X. and Le Pennec, E.},
title = {Partition-based conditional density estimation},
journal = {ESAIM: Probability and Statistics},
pages = {672--697},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012017},
mrnumber = {3126157},
zbl = {1284.62250},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2012017/}
}
TY - JOUR AU - Cohen, S. X. AU - Le Pennec, E. TI - Partition-based conditional density estimation JO - ESAIM: Probability and Statistics PY - 2013 SP - 672 EP - 697 VL - 17 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2012017/ DO - 10.1051/ps/2012017 LA - en ID - PS_2013__17__672_0 ER -
Cohen, S. X.; Le Pennec, E. Partition-based conditional density estimation. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 672-697. doi : 10.1051/ps/2012017. http://archive.numdam.org/articles/10.1051/ps/2012017/
[1] , Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection. Math. Meth. Stat. 21 (2012) 1-28. | MR
[2] and , Inhomogeneous and anisotropic conditional density estimation from dependent data. Electon. J. Statist. 5 (2011) 1618-1653. | MR | Zbl
[3] , and , A multiscale approach for statistical characterization of functional images. J. Comput. Graph. Stat. 18 (2008) 216-237. | MR
[4] , , and , MDL Principle, Penalized Likelihood, and Statistical Risk, in Festschrift in Honor of Jorma Rissanen on the Occasion of his 75th Birthday. Tampere University Press (2008).
[5] and , Bandwidth selection for kernel conditional density estimation. Comput. Stat. Data Anal. 36 (2001) 279-298. | MR | Zbl
[6] , , , , , , , , , and , European research platform IPANEMA at the SOLEIL synchrotron for ancient and historical materials. J. Synchrotron Radiat. 18 (2011) 765-772.
[7] , , and , Model-based cluster and discriminant analysis with the MIXMOD software. Comput. Stat. Data Anal. 51 (2006) 587-600. | MR | Zbl
[8] and , Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 (1998) 329-375. | MR | Zbl
[9] and , Minimal penalties for gaussian model selection. Probab. Theory Related Fields 138 (2007) 33-73. | MR | Zbl
[10] , , and , Optimal dyadic decision trees. Mach. Learn. 66 (2007) 209-241.
[11] , and , Adaptive estimation of the conditional density in presence of censoring. Sankhy 69 (2007) 734-763. | MR | Zbl
[12] and , Conditional density estimation by penalized likelihood model selection and applications. Technical report, RR-7596. INRIA (2011). arXiv:1103.2021.
[13] and , Conditional density estimation by penalized likelihood model selection. Submitted (2012).
[14] and , Unsupervised segmentation of hyperspectral images with spatialized Gaussian mixture model and model selection. Submitted (2012).
[15] and , On conditional density estimation. Stat. Neerlandica 57 (2003) 159-176. | MR | Zbl
[16] , CART and best-ortho-basis: a connection. Ann. Stat. 25 (1997) 1870-1911. | MR | Zbl
[17] , Conditional density estimation in a regression setting. Ann. Stat. 35 (2007) 2504-2535. | MR | Zbl
[18] , Oracle inequality for conditional density estimation and an actuarial example. Ann. Inst. Stat. Math. 62 (2010) 249-275. | MR
[19] , and , Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 (1996) 189-206. | MR | Zbl
[20] and , Rates of convergence for the Gaussian mixture sieve. Ann. Stat. 28 (2000) 1105-1127. | MR | Zbl
[21] and , Nonparametric estimation of conditional distributions. IEEE Trans. Inform. Theory 53 (2007) 1872-1879. | MR
[22] , and , Methods for estimating a conditional distribution function. J. Amer. Stat. Assoc. 94 (1999) 154-163. | MR | Zbl
[23] , Probabilistic latent semantic analysis, in Proc. of Uncertainty in Artificial Intelligence (1999).
[24] , , and , Fast search for best representations in multitree dictionaries. IEEE Trans. Image Process. 15 (2006) 1779-1793.
[25] and , Nonparametric estimation and symmetry tests for conditional density functions. J. Nonparam. Stat. 14 (2002) 259-278. | MR | Zbl
[26] , and , Estimating and visualizing conditional densities. J. Comput. Graphical Stat. 5 (1996) 315-336. | MR
[27] and , Nonlinear piecewise polynomial approximation beyond besov spaces. Appl. Comput. Harmonic Anal. 15 (2003) 177-223. | MR | Zbl
[28] and , Multiscale generalised linear models for nonparametric function estimation. Biometrika 92 (2005) 119-133. | MR | Zbl
[29] , and , Multiscale, multigranular statistical image segmentation. J. Amer. Stat. Assoc. 100 (2005) 1358-1369. | MR | Zbl
[30] and , Nonparametric Econometrics: Theory and Practice. Princeton University Press (2007). | MR | Zbl
[31] , Divergence measures based on the Shannon entropy. IEEE Trans. Inform. Theory 37 (1991) 145-151. | MR | Zbl
[32] , Concentration inequalities and model selection, vol. 1896 of Lecture Notes in Mathematics (2007). Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour (2003), With a foreword by Jean Picard. | MR | Zbl
[33] and , A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: PS 15 (2012) 41-68. | Numdam | MR
[34] and , Data-driven penalty calibration: a case study for Gaussian mixture model selection. ESAIM: PS 15 (2012) 320-339. | Numdam | MR
[35] , Conditional probability density and regression estimators, in Multivariate Analysis II, Proc. of Second Internat. Sympos., Dayton, Ohio, 1968. Academic Press (1969) 25-31. | MR
[36] and , Adjusting mixture weights of gaussian mixture model via regularized probabilistic latent semantic analysis, in Advances in Knowledge Discovery and Data Mining (2005) 218-252.
[37] , The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Stat. 22 (1994) 118-171. | MR | Zbl
[38] , Metric entropy of homogeneous spaces, in Quantum Probability (Gdansk 1997) (1998) 395-410. | MR | Zbl
[39] , The method of sieves and minimum contrast estimators. Math. Methods Stat. 4 (1995) 20-38. | MR | Zbl
[40] and , Weak Convergence. Springer (1996). | MR
[41] and , Density and hazard estimation in censored regression models. Bernoulli 8 (2002) 607-625. | MR | Zbl
[42] and , Multiscale poisson intensity and density estimation. IEEE Trans. Inform. Theory 53 (2007) 3171-3187. | MR
[43] and , Mixtures of regressions with predictor-dependent mixing proportions. Comput. Stat. Data Anal. 54 (2010) 2253-2266. | MR | Zbl
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