文件列表:

DEMO_EWA_comparison.m (1385, 2011-02-08)
EWA.m (872, 2011-02-01)
Golubev.m (1098, 2011-02-01)
Golubev_Oracle.m (1220, 2011-02-01)
James_Stein.m (513, 2011-02-01)
James_Stein_block.m (1428, 2011-02-01)
MakeSignal.m (8768, 2006-01-03)
aggreg_EWA.m (498, 2011-02-01)
compare_perf.m (346, 2011-02-01)
dct.m (1895, 2010-11-01)
dedouble.m (481, 2011-02-01)
dedouble_mat.m (506, 2011-02-01)
generate_grid_parameter.m (872, 2011-02-01)
generate_signal.m (629, 2011-02-01)
generate_smooth_signal.m (727, 2011-02-01)
idct.m (1895, 2010-11-01)
launching_performance_fft.m (5761, 2011-02-08)
matrix2latex_article.m (5693, 2011-02-01)
pinsker.m (927, 2011-02-01)
plot_article.m (647, 2011-02-08)
scritp_article.m (2351, 2011-02-02)
soft_thresh.m (492, 2011-02-01)
sure_shrink.m (1076, 2011-01-31)
tables_MSE.m (5969, 2011-02-02)

============================================================================= = EWA : Exponentially Weighted Agregates for regression = = README.txt Version 1.0 = ============================================================================= = Copyright 2010, 2011 = = Arnak Dalalyan*, Joseph Salmon** = = * IMAGINE / CERTIS = = Ecole des Ponts- ParisTech = = 6 Av Blaise Pascal -Cite Descartes = = Champs-sur-Marne = = 77455 Marne-la-Vallee Cedex 2- FRANCE = = ** LPMA / UMR 7599 = = Universtite Paris Diderot-Boite courrier 7012 = = 75251 PARIS Cedex 05 (FRANCE) = = = = Corresponding Author: Joseph Salmon = ============================================================================= ========================================================================== ================ Overview ====================== ========================================================================== We present some numerical experiments for regression with homoscedastic Gaussian noise with known variance. We evaluate different estimation routines on 1D signals. We retained 6 signals for our experiments because of their diversity. We have also carried out experiments on their smoothed versions obtained by taking an antiderivative. In both cases, prior to applying estimation routines, we normalize the (true) sampled signal to have an empirical norm equal to one. We use the Discrete Cosine Transform (DCT) and shrink coefficient with several approaches. The estimation routines, including the EWA, used in our experiments are detailed below: -Soft Thresholding (ST), Donoho and Johnstone (1994): This is the soft-thresholding estimator of the vector of DCT coefficients, with the (fixed) universal threshold. -SURE SHRINK (SS-ST) Donoho and Johnstone (1995): We use the soft-threshold method using the threshold minimizing the estimated unbiased risk defined via Stein’s lemma. -Blockwise James-Stein (BJS) shrinkage, Cai (1999): The set of indices {1,..., n} is partitioned into N = [n/ log(n)] non-overlapping blocks B_1, B_2,...,B_N of equal size L. If n is not a multiple of N, the last block may be of smaller size than all the others. The corresponding blocks of true coefficients are estimated by shrinking the blocks of noisy coefficients. The threshold parameter lambda used is lambda = 4.50524 as in Cai (1999). -Unbiased risk estimate (URE) minimization, Golubev (1992), Cavalier et al. (2002): it consists in using a Pinsker filter, with a data-driven choice of parameters alpha and w. This choice is done by minimizing an unbiased estimate of the risk over a suitably chosen grid for the values of alpha and w. Here, we use geometric grids ranging from 0.1 to 100 for alpha and from 1 to n for w. The bi-dimensional grid used in all the experiments has 100 × 100 elements. -EWA on Pinsker’s filters: We consider the same finite family of linear smoothers—defined by Pinsker’s filters—as in the URE routine described above. This leads to an estimator which is nearly as accurate as the best Pinsker’s estimator in the given finite family. -ORACLE risk estimate minimization: it consists in using a Pinsker filter, with an ORACLE choice of parameters alpha and w. This choice is done by minimizing an the TRUE risk over the same grid as for URE. This is the target method with which we want to compare. For more details, see http://arxiv.org/pdf/???? To help users, we provide a few examples of our algorithm. To view a demonstration, execute in MATLAB >> DEMO_EWA_comparaison NOTE: These demonstration uses the idct and dct functions ( Discrete Fourier Transform). We include this two functions in the toolbox. ========================================================================== ================ DEMO_EWA_comparaison ================== ========================================================================== Output: This demonstration automatically displays the following: -The 6 (non Smooth) signals considered -Their corresponding denoised version as described below. This demonstration automatically creates the figure associated with the Non smooth signals for one level of noise (sigma=0.1) and signals of size signal_size=2^9. This is done for the 6 images. Top left are the original signal, top right are the noisy versions, bottom left are the EWA (Pinsker) denoised signals, and bottom right are the Block James Stein method, the URE and the SURE-Shrink Soft thresholding (SS-ST) method. Copyright (2011): A. Dalalyan and J. Salmon

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