Neumann, M. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian I.I.D. situation. Lect. Notes Stat., 103, 301--329.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....non Gaussian errors and even to non stationary situations. There is a recent and growing literature based on Gaussian approximations of empirical wavelet coefficients in a variety of situations. In addition to the numerous references cited at the end of Section 8 of JS, we wish to mention Neumann and von Sachs (1995). 1.1 Basic definitions and notation We first establish some notation and recall the definition of SURE thresholding for observed data. Let W be a periodic discrete wavelet transform operator (in practice implemented with a fast cascade algorithm) and let Y be the n vector of observations Y 1 ; ....

Neumann, M. H. and von Sachs, R. (1995), Wavelet thresholding: Beyond the gaussian i.i.d. situation, in A. Antoniadis, ed., `Wavelets and Statistics', Springer Verlag Lecture Notes. Volume 103.

....by the negative of the conditional log likelihood of e given w. Thus a modified criterion for non Gaussian noise can be obtained by simply replacing the second term in MDL ind ( f) by Gamma log 2 g( ej w) Previous work on non Gaussian noise can be found, for example, in Moulin (1994) Neumann von Sachs (1995) and Gao (1997) 10 6.2 Correlated Noise Suppose that the noise e i is correlated, and can be adequately modelled by an AR series of an unknown order p: e i = a 1 e i Gamma1 a 2 e i Gamma2 : a p e i Gammap i ; where a = fa 1 ; a p g are unknown AR parameters and i is a ....

Neumann, M. H. & von Sachs, R. (1995), Wavelet thresholding: beyond the gaussian i.i.d. situation, in A. Antoniadis & G. Oppenheim, eds, `Wavelets and Statistics', New York : Springer-Verlag, pp. 301--330.

.... with threshold given by 2 (l; m; j; T ) var (bv lm ) log 2 (T ) for each fixed j, Z 1 0 E i e S j (z) Gamma S j (z) j 2 dz = O i log 2 (T ) T 2 3 j : 21) This theorem is based on existing results on quadratic forms of Gaussian variables, which are 2 distributed (see Neumann and von Sachs (1995), Theorem 3.1 A) For non normality, techniques as in Neumann and von Sachs (1997) could also be applied. In practice some modification might be appropriate such as thresholding the log periodogram. This transform stabilizes the coefficient variance, pulls their distribution closer to normality ....

Neumann, M.H. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian i.i.d.

....in practice, for many applied examples from biological observations over time this model of independent identically distributed observations is no longer valid. Various authors considered wavelet estimation in the case of stationary correlated noise: Brillinger (1994) obtained pointwise results, Neumann and von Sachs (1995) studied the L 2 risk of wavelet threshold estimator for general stationary errors, Wang (1996) considered minimax rates of threshold estimators for signals plus fractional Brownian noise; and Johnstone and Silverman (1997) inspired by a neurophysiological problem, carried out a detailed ....

....2, we introduce the appropriate notation for wavelet thresholding methods, and elaborate the model of locally stationary errors. Then we derive the asymptotic formulae for bias and variance of the empirical wavelet coefficients. The main part of this section is that, inspired by the work of Neumann and von Sachs (1995), we establish an upper bound for the uniform L 2 risk of the wavelet threshold estimator over certain smoothness classes for not necessarily Gaussian locally stationary errors. This is done by showing that in the corresponding sequence space of empirical wavelet coefficients the considered L 2 ....

[Article contains additional citation context not shown here]

Neumann, M. H., and von Sachs, R. (1995). Wavelet thresholding: Beyond the Gaussian i.i.d. situation. in: A. Antoniadis, G. Oppenheim (eds), `'Wavelets and Statistics", LN Statistics 103, Springer Verlag (1995), 301--329.

....assumptions of Theorem 5. 5, with a threshold as given in equation (33) for each fixed j, Z 1 0 E i e S j (z) Gamma S j (z) j 2 dz = O i log 2 (T ) T 2 3 j : 34) This theorem is based on existing results on quadratic forms of Gaussian variables, which are 2 distributed (see Neumann and von Sachs (1995), Theorem 3.1 A) For non normality, techniques as in Neumann and von Sachs (1997) should be applied, replacing log(T ) by p log(T ) in the universal threshold, resulting into a rate of order O (log(T ) T 2 3 ) Neumann and von Sachs (1995) Theorem 3.1 B) In practice some modification might ....

....variables, which are 2 distributed (see Neumann and von Sachs (1995) Theorem 3. 1 A) For non normality, techniques as in Neumann and von Sachs (1997) should be applied, replacing log(T ) by p log(T ) in the universal threshold, resulting into a rate of order O (log(T ) T 2 3 ) (Neumann and von Sachs (1995), Theorem 3.1 B) In practice some modification might be appropriate as for this approach the thresholds depend on the unknown wavelet spectrum. This is exactly as for stationary Fourier spectrum estimation with wavelet thresholding of periodograms and also for the time dependent version of this, ....

Neumann, M.H. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian i.i.d. Lect.

....for, e.g. regression and spectral density estimation, with non Gaussian, dependent, heteroskedastic and even nonstationary errors. This question has been addressed by embedding the respective denoising problem into the paradigm of minimax estimation: By methods described in a general overview in [NvS95] it can be shown that non linearly thresholded wavelet estimators perform uniformly well in a whole range of function classes measured by the worst case L 2 risk between the estimator and the unknown function modelled to be in a ball of one of these function classes. That is, the resulting ....

....the hypothesis of stationarity. A related approach of estimating time varying autoregressive spectra by non linear wavelet estimation of time varying autoregressive parameters is [DNvS] These approaches make all use of the same principle of generalizing from Gaussian i.i.d. data as given by [NvS95] and presented in Section 3.3 below. 3.2 A theorem on near optimal rates for L 2 risks We consider the estimator e f = X k b ff j 0 k OE j 0 k J Gamma1 X j=j0 2 j Gamma1 X k=0 ffi ( b fi jk ; jk ) jk (20) which is an estimator based on individual thresholds jk , i.e. ....

[Article contains additional citation context not shown here]

Neumann, M. H., and von Sachs, R. (1995) Wavelet thresholding: Beyond the Gaussian i.i.d. situation. in: A. Antoniadis, G. Oppenheim (eds), `'Wavelets and Statistics", Lect. Notes Statist., 103, 301--329. Springer-Verlag: New York.

....the assumptions of Theorem 3. 4, with a threshold as given in equation (33) for each fixed j, Z 1 0 E i e S j (z) Gamma S j (z) j 2 dz = O i log 2 (T ) T 2 3 j : 34) This theorem is based on existing results on quadratic forms of Gaussian variables, which are 2 distributed (see Neumann and von Sachs (1995), Theorem 3.1 A) For non normality, techniques as in Neumann and von Sachs (1997) should be applied, replacing log(T ) by p log(T ) in the universal threshold, resulting into a rate of order O (log(T ) T 2 3 ) Neumann and von Sachs (1995) Theorem 3.1 B) In practice some modification ....

....Gaussian variables, which are 2 distributed (see Neumann and von Sachs (1995) Theorem 3. 1 A) For non normality, techniques as in Neumann and von Sachs (1997) should be applied, replacing log(T ) by p log(T ) in the universal threshold, resulting into a rate of order O (log(T ) T 2 3 ) (Neumann and von Sachs (1995), Theorem 3.1 B) In practice some modification might be appropriate as the thresholds depend on the unknown wavelet spectrum. This is exactly as for stationary Fourier spectrum estimation with wavelet thresholding of periodograms and also for the time dependent version of this, see von Sachs ....

Neumann, M.H. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian i.i.d. Lect.

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Neumann, M. H. and von Sachs, R. (1995). Wavelet thresholding: beyond the Gaussian i.i.d. situation, In Lecture Notes in Statistics: Wavelets and Statistics, A. Antoniadis ed., 301--329, 1995.

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Neumann, M. H. and von Sachs, R. (1995). Wavelet thresholding: beyond the Gaussian i.i.d. situation. In LN Statistics 103: Wavelets and Statistics, A. Antoniadis ed., 301--329.

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Neumann, M. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian I.I.D. situation. Lect. Notes Stat., 103, 301--329.

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