G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University, Statistics Department, March 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is identical with the wavelet basis. Therefore we choose the second best result, the entropy cost function, for the parameter study. In the following the relation between denoising threshold (and alenoising scale S0 respectively) and the alenoising performance is of interest. As described in Nason [7], an often used, universal threshold can be deter mined, only depending on the signal length and the noise standard deviation o . In our case, h 2log(2Jmax)5 0.0839 for normally distributed noise. How ever, investigations reveal that h lies above the optimal threshold, suppresses too ....

Nason G.P., Wavelet regression by cross-validation, Dept. of Mathematics, Univ. of Bristol, 1994

....thresholding. The coordinates of d are replaced by 0 if they are smaller in absolute value than a xed threshold : The threshold is a tuning parameter of the wavelet shrinkage. Donoho and Johnstone propose several thresholds (i.e. universal, SURE) as well as several thresholding policies. Nason (1994) adjusted the well known crossvalidation method for use with wavelets. The threshold is selected by minimizing a cross validatory estimator of integrated square error (ISE) A few other references in threshold selection and wavelet shrinkage applications are Gao (1997) and Vidakovic (1998, 1999) ....

Nason, G. (1994). Wavelet regression by cross-validation, Tech. Report 447, Statistics, Stanford University.

....shrinkage has recently emerged as a powerful tool for extracting signals from noisy data [1] A fundamental issue in wavelet shrinkage signal de noising is the choice of the shrinkage threshold . Many data dependent threshold selection techniques in statistical function estimation (e.g. 2] 3] [4], 5] and [6] consider only the magnitudes of the empirical wavelet coefficients in determining threshold values. In this paper, we present a novel data analytic technique that considers both the magnitude and location of empirical wavelet coefficients when determining and demonstrate its use ....

G. P. Nason, "Wavelet regression by cross-validation", Technical Report 447, Stanford University Department of Statistics, Stanford, California, 1994.

....one has to determine at least one threshold parameter (possibly several depending on the complexity of the thresholding rule and whether or not thresholding should be scale level adaptive) from the noisy data. Donoho and Johnstone [12, 11] Johnstone and Silverman [26] Donoho et al. 15] Nason [32], Weyrich and Warhola [43] Saito [37, 38] and Vidakovic [42] have proposed various methods for estimating the threshold from the observed noisy data. Each of these papers discusses one or several methods for picking the optimal threshold (for both hard and soft thresholding rules as well as ....

G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University, Statistics Department, March 1994.

....the minimax rates over Besov classes when p 2, so the wavelet estimates significantly outperform linear estimates. The wavelet estimates with best selection of level dependent threshold achieve the minimax risk over a wide range of spaces, so selection of the optimal threshold is very important. Nason (1994) reported that the Universal and Sure methods and Nason s cross validation method (removing half of the data each time) don t work well for correlated data. We propose a cross validation method to select thresholds for dependent data. To reduce correlation the cross validation method deletes more ....

....t j (w j;k ) sign(w j;k ) jw j;k j Gamma t j ) or hard thresholded rule ffi t j (w j;k ) w j;k 1 fjw j;k jt j g ; with level dependent threshold (t j ) In practice the wavelet coefficients w j;k at the first three levels are often left untouched. See Donoho and Johnstone (1994) and Nason (1994)] The wavelet estimates are the reconstruction of the thresholded w j;k . 7.3 Threshold Selection Wavelet estimates with best selection of level dependent threshold achieve the minimax risk over a wide range of spaces, so it is very crucial for wavelet estimates to select the threshold (t j ) ....

[Article contains additional citation context not shown here]

NASON, G. P. (1994). Wavelet regression by cross-validation. To appear in J. Roy. Statist.

....shrinkage has recently emerged as a powerful tool for extracting signals from noisy data [1] A fundamental issue in wavelet shrinkage signal de noising is the choice of the shrinkage threshold . Many data dependent threshold selection techniques in statistical function estimation (e.g. 2] 3] [4], 5] and [6] consider only the magnitudes of the empirical wavelet coefficients in determining threshold values. In this paper, we present a novel data analytic technique that considers both the magnitude and location of empirical wavelet coefficients when determining and demonstrate its use ....

G. P. Nason, "Wavelet regression by cross-validation", Technical Report 447, Stanford University Department of Statistics, Stanford, California, 1994.

....soft thresholding operation [9, 8, 30] j (x) 8 : x Gamma x 0 jxj x x Gamma 8 1. DENOISING FMRI DATA to each coefficient in the detail signals of W f . A fundamental issue in signal recovery is the choice of the threshold , and a variety of methods have been proposed [9, 11, 10, 26, 25, 23, 1]. This chapter will consider both a global approach and a data driven approach for choosing . The first method is the VisuShrink universal threshold due to Donoho and Johnstone [10] oe p 2 log(n) 1.2) where n is the number of data samples. This approach has been well developed in [9, 12] ....

G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University Department of Statistics, Stanford, California, 1994.

....has been presented by Kumar and Foufoula Georgiou (1994) We additionally recommend the very readable paper by Vidakovic and M ller (1996) non threateningly entitled Wavelets for Kids. 3 Wavelets have also found their way into statistics, particularly in non parametric regression (e.g. Nason, 1994) and non parametric function and density estimation (e.g. Donoho and Johnstone, 1994; Donoho et al. 1995) It is with these recent statistical insights that we hope to demonstrate how wavelet analysis can be used in providing measures to compare images and scores to assist in field forecast ....

....a process known as thresholding. We detail recent advances in wavelet statistics that show how to threshold in an objective manner. Thresholding can also be done on one or two dimensional data. For example, thresholding can be employed in nonparametric regression problems or time series analysis (Nason, 1994). 3.1 Mechanics of the discrete transform n r n c Let an image (or grid, or field) be written as the real valued matrix G consisting of n r rows and n c columns. It is possible to construct an orthogonal operator Y , called the mother wavelet, such that the discrete wavelet transform (WT) is ....

[Article contains additional citation context not shown here]

Nason, G.P., 1994: Wavelet regression by cross-validation. Technical Report 447, Stanford University, http://playfair.stanford.edu.

....information by both level and location. These points suggest that we direct future efforts in some or all of the following directions. Future Efforts in Denoising in Conjunction with the WVD ) Adaptive thresholds, perhaps using (a) SURE (Donoho and Johnstone [17] b) Cross validation (Nason [38]) Level and location dependent thresholds. Edge enhancing filters (Richardson [46] such as (a) Mean curvature partial differential equation filters (b) Weighted Majority Minimum Range filters Despite the overall encouraging tone of our work, we were still disappointed with the general ....

G.P Nason (1994) Wavelet regression by cross-validation. Technical Report 447, Department of Statistics, Stanford University.

....thresholding. The coordinates of d are replaced by 0 if they are smaller in absolute value than a fixed threshold : The threshold is a tuning parameter of wavelet shrinkage. Donoho and Johnstone propose several thresholds (i.e. universal, SURE) as well as several thresholding policies. Nason (1994) adjusted the well known cross validation method for use with wavelets. The threshold is selected by minimizing a cross validatory estimator of integrated square error (ISE) A few other references in threshold selection and wavelet shrinkage applications are Gao (1993) and Vidakovic (1994, ....

Nason, G. (1994). Wavelet regression by cross-validation, Tech. Report 447, Statistics, Stanford University.

....the problem, however, the equations are nonlinear, and we have to rely on optimization routines. If the optimal thresholds are found, we can get the lower bound of the risks. Unknown x In the real world, x is unknown, any many methods have been proposed. For example, the cross validation scheme [60], minimum description length (MDL) 69] and the SURE shrink [22] 4.4.4 Denoising in an ON Basis The Method Let y i = x i n i ; i = 1; N (4:40) where x i is the original signal, n i is i.i.d. white Gaussian noise with unit variance, and y i is the observation. The goal is to get a ....

G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University, Statistics Department, March 1994. 121

.... at each level using the data values themselves to divide the data set into big coefficients (representing significant signal ) and small coefficients (corresponding to random noise ) It should be noted that other data dependent thresholding techniques (e.g. Donoho and Johnstone (1995a) Nason (1994), Weyrich and Warhola (1994) Ogden and Parzen (1994) Vidakovic (1994) Wang (1994) and others) consider only the magnitude of the empirical coefficients. In addition to the relative magnitude of coefficients, it is desirable also to include the information contained in the position of the large ....

Nason, G. P. (1994). Wavelet regression by cross-validation. Technical report 447, Stanford University Department of Statistics, Stanford, California.

....is obtained by the following 3 steps: 1] Transform data y into wavelet domain ; 2] Shrink the empirical wavelet coefficients towards zero; 3] Transform the shrunk coefficients back to the data domain. Applications of wavelets in statistics can be found, for example, in McCoy and Walden (1996) Nason and Silverman (1994); Wang (1995) Shrinkage of the empirical wavelet coefficients works best when underlying set of the true coefficients of f is sparse. That is, the overwhelming majority of these coefficients are small, and the remaining few large ones explain most of the functional form in f . Therefore, an ....

Nason, G. P. (1994). Wavelet regression by cross-validation. Technical report, University of Bristol, Bristol BS8 1TW, United Kingdom.

....limited experiments have shown that this method tends to give too large threshold values which result in oversmoothed images. However, a more in detail study is necessary for this method and in particular one needs to consider in details the best basis selection proposed by Saito [12] Nason [9] recently proposed a method for threshold selection based on cross validation. This method for selecting Artifacts occur if one use hard thresholding (i.e. one keep all values above a threshold and sets all other to zero) 0 2 4 6 3600 3650 3700 3750 3800 3850 Thresholding factor f u (a) 0 ....

....in the JPEG data stream can be applied intelligently for improving the performance. In future work we will consider the usage of various wavelet analyses, continue the investigation of the method of minimal description length [12] investigate methods for threshold selection by cross validation [9] and investigate several error measures based on perceptual criteria. ....

G. P. Nason. Wavelet regression by cross-validation. Technical report, Department of Mathematics, University of Bristol, Bristol, U.K., March 1994. Also Tech. Report.

....it. Donoho and Johnstone [4] propose a threshold proportional to the noise level. But in many practical cases the actual amount of noise is not known. Instead of estimating the noise level, we try to find a good threshold directly, only using the input data. Weyrich and Warhola [18] and Nason [15] applied the idea of Cross Validation [7, 17, 1] and obtained excellent results. This Cross Validation is a function of the threshold value only based on the input data. Its minimum is a good approximation for the optimal threshold. Wahba [17] uses the same idea to find an optimal smoothing ....

....causes a bias. with: b 2 (ffi) 1 N kv ffi Gamma vk 2 2 N h(v ffi Gamma v) E(w ffi Gamma v ffi )i; 13) and: 2 (ffi) kE(w ffi Gamma v ffi )k 2 N oe 2 : 14) Figure 3 shows a typical form of the function R(ffi) For more information about this function, we refer to Nason [15]. 4 A first estimator for R(ffi) 4.1 The effect of the threshold operation We are looking for an estimator for R(ffi) which is based on known variables. Therefore we first investigate the effect of the threshold operation on the input data. Define T (ffi) P N i=1 (w ffii Gamma w i ) 2 N ....

G. P. Nason. Wavelet regression by cross validation. Preprint, Department of Mathematics, University of Bristol, UK, 1994.

No context found.

Nason, G.P.: Wavelet regression by cross-validation. Technical Report 447, Department of Statistics, Stanford University, Stanford, (1994).

....all cases is the simple golden section search as mentioned in Press et al. 1992) The algorithm works extremely well in practice. This is mainly because the function M is very nearly convex (to the eye on a large scale it looks convincingly 8 NASON convex) Detailed investigation of M by Nason (1994) shows that the first derivative of M(t) is continuous and linear increasing on intervals defined by increasing fjw jk jg where fw jk g are the noisy wavelet coefficients formed from the transform of g 1 ; g n . At the points t = jw jk j the derivative may experience a discontinuity. ....

....shows that the first derivative of M(t) is continuous and linear increasing on intervals defined by increasing fjw jk jg where fw jk g are the noisy wavelet coefficients formed from the transform of g 1 ; g n . At the points t = jw jk j the derivative may experience a discontinuity. Nason (1994) provides heuristics that indicate that although these jumps may be negative they are usually small (only negative jumps cause nonconvexity of M ) and therefore the zero derivative point of M is usually well determined. Since the first derivative is known it would be possible to use a ....

[Article contains additional citation context not shown here]

Nason, G. P. (1994) Wavelet regression by cross-validation. Technical Report 447. Department of Statistics, Stanford University, Stanford.

....variables. The classic wavelet paradigm for estimating f is to find the discrete wavelet transform of fY i g, apply a soft or hard thresholding rule, and then to invert to yield an estimate of f . See Donoho and Johnstone [DJ2, DJ1] Donoho, Johnstone, Kerkyacharian and Picard [DJKP] Nason [Na2, Na3], Abramovich and Benjamini [AB1] Fan, Hall, Martin and Patil [FHMP] Johnstone and Silverman [JS] Neumann and Spokoiny [NS2] Ogden [Og1] Vidakovic [Vi1] Wang [Wa1] and Weyrich and Warhola [WW1] In this paper we only mention regression in passing, but note that the two inverse methods set out ....

Nason, G.P.: Wavelet regression by cross-validation. Technical Report 447, Department of Statistics, Stanford University, Stanford, (1994).

No context found.

G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University, Statistics Department, March 1994.

No context found.

Nason, G.P.: Wavelet regression by cross-validation. TR 447(1994), Dept.

No context found.

Nason, G. (1994). Wavelet regression by cross-validation. Technical Report 447. Department of Statistics, Stanford University.

No context found.

G. P. Nason, "Wavelet regression by cross-validation", Tech. Report 447. Dept. of Statistic, Stanford (April 1994), and references therein.

No context found.

G. P. Nason. Wavelet regression by cross-validation. Technical Report 447, Stanford University, Statistics Department, March 1994.

No context found.

Nason, G. (1994). Wavelet regression by cross-validation. Technical Report 447. Department of Statistics, Stanford University.

No context found.

G. P. Nason, "Wavelet regression by cross-validation", Tech. Report 447. Dept. of Statistic, Stanford (April 1994), and references therein.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC