Bruce, A., and Gao, H. (1996). Applied Wavelet Analysis with S-PLUS. Springer, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ones represent mostly noise. The wavelet coefficients are suitably modified and the denoised data is obtained by an inverse wavelet transform of the modified coefficients. In our work, we consider two dimensional versions of methods that were originally developed for one dimensional signals in [1, 2, 3, 4, 5] and compare them to the method proposed for images in [6] Using decimated wavelet transforms, and the mean squared error optimality criterion, we evaluate the different methods on test images corrupted with additive Gaussian white noise. Our goal is to address several issues. First, we want to ....

.... impulse removal filters with local adaptive filtering in the transform domain to remove not only white and mixed noise, but also their mixtures [15] A different class of methods exploits the decomposition of the data into the wavelet basis and shrinks the wavelet coefficients to denoise the data [1, 2, 3, 4, 5, 6, 16, 17]. While this is typically done using the more memory efficient decimated wavelet transforms, it is well known that the use of non decimated transforms minimizes the artifacts in the denoised data [18, 19] Other authors have combined wavelets with Hidden Markov models and spatially adaptive ....

[Article contains additional citation context not shown here]

A. Bruce and H. Gao, Applied Wavelet Analysis with S-PLUS, Springer Verlag, 1996.

....actual and estimated values of the parameters. 4. Discussion 4.1. Justification for Eqs. 4) and (5) Recall that coefficients s j,k , d j,k , d 1,k are the wavelet transform coefficients that can be used to synthesize the function f(t) They are related to the approximating wavelet functions [1], f J,k (t) and y j,k (t) by the following integrals: s J,k ## f J,k (t)f(t)dt# f f(t)dt (18) d j,k ##y j,k (t)f(t)dt= y f(t)dt, j=1,2, J (19) In the above integrals, note that f(t) is fixed, the length of y j 1,k will be two times that of y j,k . Thus with the ....

Bruce A, Gao H-Y. Applied wavelet analysis with S-Plus. Springer-Verlag, 1996.

....process. A.1 Selecting Wavelet Coe#cients for a Single Data Set When using wavelet transforms, one needs first to decide which family of wavelets to use. The default wavelet family used in the popular S PLUS package for the DWT operation is the symmlet, s8; see page 17 of Bruce and Gao [4] for details. The s8 wavelet is an excellent overall choice for representing many functions since it is orthogonal, smooth, nearly symmetric, and nonzero on a relatively short interval. In a study not reported here, we experimented with other wavelet families, such as coiflets and daublets. The ....

....families, such as coiflets and daublets. The results were similar to the results obtained using s8 wavelets. In this article we limit our discussion to the results obtained with the s8 wavelet. One method often used to fit data using wavelets is to compute a set of multiresolution approximations [4, 16]. This method involves first constructing an approximation to the data using the coarsest scale signal and then adding increasingly finer levels of resolution. As more levels of resolution are used, the approximation to the target data set improves. Figures 3 5 below depict multiresolution ....

A. Bruce and H.-Y. Gao, Applied Wavelet Analysis with S-PLUS . New York: Springer-Verlag, 1996.

....in control process. A.1 Selecting Wavelet Coe#cients for a Single Dataset When using wavelet transforms, one needs first to decide which family of wavelets to use. The default wavelet family used in the popular S Plus package for the DWT operation is the symmlet, s8; see page 17 of Bruce and Gao [3] for details. The s8 wavelet is an excellent overall choice for representing many functions since it is orthogonal, smooth, nearly symmetric, and nonzero on a relatively short interval. In a study not reported here, we 6 experimented with other wavelet families, such as coiflets and daublets. The ....

A. Bruce and H.-Y. Gao, Applied Wavelet Analysis with S-Plus, New York: SpringerVerlag, 1996. 24

....A and Population B will be denoted by A i;j and B i;j , respectively. Let w i;j be the ith empirical DWT coecient at scale 2 j for the series we are interested in classifying. The range of j is determined by the length of the series, the boundary condition, and the type of wavelet used (Bruce Gao, 1996, pg. 69) We will represent the number of coecients at scale 2 j by n j . Thus i = 1; n j where n j = n=2 j . If the realization belongs to Population A, then (w i;j A i;j ) iid N(0; 2 A;j ) where 2 A;j denotes the population variance for the scale 2 j DWT coe cients ....

.... a symmlet, this wavelet is orthogonal, smooth, non zero on a relatively short interval, and nearly symmetric (hence the name least asymmetric) Although there are no de nitive rules for choosing a wavelet, the preceding properties make the LA(8) wavelet a good overall choice for many applications (Bruce Gao, 1996, pg. 69) Of course, the choice of a wavelet ultimately depends on the characteristics of the signal being analyzed. Figure 1 depicts the two pure signals f A (x) sin(6x) and f B (x) sin(6x 1) Note that the two signals are identical except for a slight phase shift. Figure 2 shows ....

Bruce, A.G. and Gao, H.-Y. (1996). Applied Wavelet Analysis with S-PLUS. New York: Springer.

....accommodate jumps or discontinuities in s(t) for example if the process changes from one state to another at some point. The local structure allows jumps to be modelled in the signal without a ecting the t to the signal at other times. Further details of the potential of wavelets can be found in [4], 5] and [6] This paper focuses on sampling methods for Bayesian wavelet models though the results are applicable to any orthogonal system of basis functions [7] We highlight how, in this setup, it is possible to draw iid samples from the posterior distribution of the model space using the ....

....the perfect sampling algorithm and how it relates to the models under consideration with Section 5 demonstrating the use of the algorithm on a number of canonical examples. A discussion of the approach is given in Section 6. II. Wavelet bases Wavelet methods (see, for example, the books by [4], 5] 6] estimate s with the usual linear in the parameters form of (2) However, when using wavelet bases it is more helpful to think of the estimate to the signal being given by the truncated series, s(t) 0 J 1 X j=1 n(j) X i=1 ji B ji (t) t 2 f1; 2; Tg; 3) where J = ....

A Bruce and H-Y Gao, Applied wavelet analysis with S-plus, New York: Springer-Verlag, 1996.

....gloss over many technical details, which the reader can find in a book by the current author and A. T. Walden [9] There are several good books that address the statistical application of wavelets and would be useful for a reader who wants to delve more into the subject, including Bruce and Gao [2], Carmona et al. 3] Mallat [7] Ogden [8] Vidakovic [11] and Wornell [12] In addition, anyone interested in wavelets can benefit from perusing the Wavelet Digest athttp: www.wav , which regularly issues newsletters and maintains other useful information for the wavelet community. TH ....

Bruce, A. G. and Gao, H.--Y. (1996), Applied Wavelet Analysis with S-PLUS, Springer, New York.

.... a non decimated DWT called the maximal overlap DWT (see Percival and Guttorp (1994) and Percival and Mofjeld (1997) for details on this transform; other versions of the non decimated DWT are discussed in Shensa (1992) Beylkin (1992) Coifman and Donoho (1995) Nason and Silverman (1995) and Bruce and Gao (1996)) In practice the DWT is implemented via a pyramid algorithm (Mallat 1989) that, starting with the data Y t , filters a series using h 1 and g 1 , subsamples both filter outputs to half their original lengths, keeps the subsampled output from the h 1 filter as wavelet coefficients, and then ....

Bruce, A. and H.-Y. Gao (1996). Applied Wavelet Analysis with S-PLUS. New York: Springer.

....different scales, oscillations, and locations: f(t) X j X b X k w j;b;k W j;b;k (t) where w j;b;k is the wavelet packet coefficient. The range of summations for the levels j and the oscillations b is chosen so that the wavelet packet functions are orthogonal. A fast splitting algorithm [12] which is an adaptation of the pyramid algorithm [13] for discrete wavelet transform is used for finding the wavelet packet table. The splitting algorithm differs from the pyramid algorithm in that low pass and high pass filters are applied to the detailed coefficients in addition to the smooth ....

A. Bruce and H. Gao. Applied Wavelet Analysis with S-Plus. Springer-Verlag, New York, 1996.

....scales, oscillations and locations: f(t) w W t j b k k b j j b k , where w j,b,k is the wavelet packet coefficient. The range of summation for the levels j and the oscillations b is chosen so that the wavelet packet functions are orthogonal. A fast splitting algorithm [14] which is an adaptation of the pyramid algorithm [15] for discrete wavelet transform is used for finding the wavelet packet table. The splitting algorithm differs from the pyramid algorithm in that low pass and high pass filters are applied to the detailed coefficients in addition to the smooth ....

A. Bruce, and H. Gao, Applied Wavelet Analysis with S-Plus, Springer-Verlag, New York, 1996.

....we thank P. Abry and D. Veitch for making available their programs to perform the wavelet based global scaling analysis in Section 2.1. Some of the local scaling analysis techniques presented in this paper use functions that are part of the wavelet package in S Plus, described in detail in [3]. ....

A. Bruce and H.-Y. Gao. Applied Wavelet Analysis with S-Plus. Springer-Verlag, New York, 1996.

....has the same sample mean and variance as X) Other possibilities for dealing with the circularity assumption include zero padding, tapering and polynomial extrapolations. A comprehensive discussion of how to handle circularity is beyond the scope of this paper, but Chapter 14 of the recent book by Bruce and Gao (1996) discusses several di#erent options and gives practitioners guidelines for choosing a technique that is suitable for a given time series. For the Crescent City series, we found that using either a reflected series or zero padding rather than just assuming X to be circular made very little ....

Bruce, A., and Gao, H.-Y. (1996), Applied Wavelet Analysis with S-PLUS, New York: Springer.

....coefficients provide good diagnostics for data analysis since the data or function of interest is completely captured in the 6 RESULTS 13 wavelet coefficients. Time scale (scaleogram) and time frequency plots (spectogram) are useful for studying the tradeoff between time and scale localization (Bruce and Gao 1996). Image querying, multiresolution editing, and multiresolution plots provide alternative means of visually exploring complex relationships in the data. Other standard methods of graphical data analysis such as box plots and Q Q plots can also be applied to wavelet coefficients. We now turn our ....

Bruce, A. and Gao, H., 1996, Applied Wavelet Analysis with S-Plus (New York: Springer).

....of the mathematical aspects of wavelets we refer to Meyer (1992) Daubechies (1992) and Wojtaszczyk (1997) Jawerth Sweldens (1994) provide an excellent overview of wavelet based multiresolution analyses. For a general statistical introduction to wavelets see Nason Silverman (1994) Bruce Gao (1996) and Ogden (1997) 2.2 Besov spaces on the interval In this section we briefly introduce some relevant aspects of the (inhomogeneous) Besov spaces on the interval that we need further. For more details we refer, for example, to DeVore Popov (1988) DeVore, Jawerth Popov (1992) Meyer (1992) ....

Bruce, A., & Gao, H.-Y. (1996). Applied Wavelet Analysis with S-Plus.

....degrees of freedom, shrinkage, smoothing, wavelets, variable selection. 1 Introduction Wavelet analysis has quickly established itself as a standard method for the analysis and smoothing of time series. Of particular importance is the application to the denoising and compression of signals (e.g. Bruce and Gao, 1996). In this paper we consider wavelets within a Bayesian framework. The main motivation is to generalise previous work (Clyde, Parmagiani and Vidavokic, 1998; Muller and Vidakovic, 1995) and to make prior assumptions that can be compared to conventional model selection criteria. In addition we ....

....up to the coeffificient being effectively removed from the model. Note that we do not attempt to perform level dependent shrinkage because this appears to be inconsistent with our knowledge that under the wavelet decomposition, the noise enters additively across all coefficients (sec 6. 2, Bruce and Gao, 1996). 4 Three Samplers The model probability is given by the marginal likelihood and a prior on the degrees of freedom of the model. In this section we consider three methods of sampling from the posterior model space. The first approach is to fix i small for all i so that little shrinkage is ....

Bruce, A. and Gao, H-Y. (1996) Applied Wavelet Analysis with S-Plus. New York: SpringerVerlag.

.... problems, see Donoho (1995) and Abramovich and Silverman (1997) classification, see Saito (1994) Coifman and Saito (1994) Buckheit and Donoho (1995) Learned and Willsky (1995) and Saito and Coifman (1996) For a general statistical introduction to wavelets see Nason and Silverman (1994) and Bruce and Gao (1996), or see Strang (1993) for a more mathematical view. More detailed comprehensive expositions are Daubechies (1992) Meyer (1992) and Chui (1992) Wavelets are a type of building block for constructing functions. More precisely wavelets form bases for function spaces such as L 2 (R) This article ....

....for the last digit which may be odd or even (so for example, 23 is kept, but 32 is not) Figure 3 shows a schematic of the NWT. The computational cost of the NWT is O(N log N) A detailed description of the NWT can be found in Nason and Silverman (1995) See also Percival and Guttorp (1994) Bruce and Gao (1996) and Percival and Mofjeld (1997) Coifman and Donoho (1995) developed a translationinvariant curve estimation procedure using the NWT. See also Nason and Silverman (1995) Lang et al. 1995) and Johnstone and Silverman (1997) for more on curve estimation with the NWT. The NWT appeared earlier in ....

Bruce, A., & Gao, H.-Y. (1996). Applied Wavelet Analysis with S-Plus. New York: Springer-Verlag.

....DEGREES OF FREEDOM; SHRINKAGE; SMOOTHING; WAVELETS; VARIABLE SELECTION. 1. INTRODUCTION Wavelet analysis has quickly established itself as a standard method for the analysis and smoothing of time series. Of particular importance is the application to the denoising and compression of signals (e.g. Bruce and Gao, 1996). In this paper we consider wavelets within a Bayesian framework. The main motivation is to generalise previous work (Clyde, Parmigiani and Vidakovic, 1998; Muller and Vidakovic, 1999) and to make prior assumptions that can be compared to conventional model selection criteria. In addition we ....

....the coefficient being effectively removed from the model ( i large) Note that we do not attempt to perform level dependent shrinkage because this appears to be inconsistent with our knowledge that under the wavelet decomposition, the noise enters additively across all coefficients (Section 6. 2, Bruce and Gao, 1996). 4. THREE SAMPLERS From (4) we see that the model probability is given by the marginal likelihood multiplied by a prior on the degrees of freedom of the model. In this section we consider three MCMC strategies for sampling from the posterior model space to compare their predictive performance. ....

Bruce, A. and Gao, H-Y. (1996). Applied Wavelet Analysis with S-Plus. New York: Springer-Verlag.

.... problems, see Donoho (1995) and Abramovich and Silverman (1998) classification, see Saito (1994) Coifman and Saito (1994) Buckheit and Donoho (1995) Learned and Willsky (1995) and Saito and Coifman (1996) For a general statistical introduction to wavelets see Nason and Silverman (1994) Bruce and Gao (1996), and Ogden (1997) or see Strang (1993) for a more mathematical view. More detailed comprehensive expositions are Daubechies (1992) Meyer (1992) and Chui (1992) Wavelets are building blocks for constructing functions. More precisely wavelets form bases for function spaces such as L 2 (R) This ....

....even (so for example, 23 is kept, but 32 is not) Figure 2 shows a schematic of the NWT. The computational cost of the NWT is O(LN log N) A detailed description of the NWT and some of its statistical applications can be found in Nason and Silverman (1995) See also Percival and Guttorp (1994) Bruce and Gao (1996) and Percival and Mofjeld (1997) Coifman and Donoho (1995) developed a translation invariant curve estimation procedure using the NWT. See also Nason and Silverman (1995) Lang et al. 1995) and Johnstone and Silverman (1997) for more on curve estimation with the NWT. The NWT appeared earlier in ....

Bruce, A., & Gao, H.-Y. (1996). Applied Wavelet Analysis with S-Plus. New York: SpringerVerlag.

No context found.

Bruce, A., and Gao, H. (1996). Applied Wavelet Analysis with S-PLUS. Springer, New York.

No context found.

Bruce, A. & Gao, H-Y. (1997). Applied Wavelet Analysis with S-PLUS, Springer, New York.

No context found.

A. Bruce and Y. Gao, "Applied Wavelet Analysis with S-PLUS", Springer, 1996.

No context found.

Bruce, A.G. & Gao, H.-Y. (1996). Applied Wavelet Analysis with S-Plus. New York: Springer-Verlag.

No context found.

Bruce, A. & Gao, H-Y. (1997). Applied Wavelet Analysis with S-PLUS, Springer, New York.

No context found.

Bruce, A. and H.-Y. Gao (1996). Applied Wavelet Analysis with S-PLUS. New York: Springer.

No context found.

Bruce, A.G. & Gao, H.-Y. (1996). Applied Wavelet Analysis with S-Plus. New York: Springer-Verlag.

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