S. Cambanis and C. Houdr'e, "On the continuous wavelet transform of second order random processes," IEEE Trans. on IT, vol. 41, no. 3, pp. 628--633, May 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....considerable simplification and usefulness for turbulence signals. Other applications where fractal signals have shown promise include biomedical, biochemistry [1] and communications [461] Other multiscale analyses of related processes, such as nonstationary processes with stationary increments [49, 212] 2 have resulted in stationarizing properties, thus allowing the application of classical statistical techniques and equally important a deeper understanding of other nonstationary parametric processes [212] A particularly interesting extension to 2 dimensional signals also 2 Fractional ....

S. Cambanis and C. Houdr'e, "On the continuous wavelet transform of second order random processes," IEEE Trans. on IT, vol. 41, no. 3, pp. 628--633, May 1995.

....The second one proposed the construction of almost 1=f processes, on the basis of uncorrelated wavelet coecients. Both types of results were subsequently considered in more general settings: stationarization was shown to be closely linked to the existence of stationary increments (Masry (1993) Cambanis and Houdr e (1995)) whereas ecient fBm synthesis procedures were further developed by Sellan (1995) Abry and Sellan (1996) and Meyer, Sellan and Taqqu (2000) Linking somewhat the analysis and synthesis viewpoints, the correlation structure of fBm wavelet coecients was studied in greater detail by Tew k and Kim ....

....of (u 1 ; u p ) are zero: P p i=1 i q u i = 0; q = 0; N 1. Initial attempts for applying wavelet based techniques to scaling processes were partly heuristic, and somewhat overlooked the fact that wavelet theory was then essentially aimed at deterministic nite energy signals. Cambanis and Houdr e (1995) and later Averkamp and Houdr e (1998) Cohen, Froment and Istas (1991) and more recently Kato and Masry (1999) addressed general questions regarding both the existence, interpretation and (distributional) properties of wavelet transforms, when applied speci cally to stochastic processes, and ....

Cambanis, S. & Houdre, C. (1995), `On the continuous wavelet transform of second-order random processes', IEEE Transactions on Information Theory 41(3), 628-642.

....h 2 j;l = 1; and are orthogonal to even shifts. Since L j j (2 j Gamma 1) L 1 Gamma 1) 1 T , the wavelet function is compactly supported and with probability one ensures that the wavelet coefficients for a mean square convergent series will be a random process with finite second moments [Cambanis and Houdr e (1995)] Using these filters the wavelet coefficients of a discrete signal x t , t = 0; T Gamma 1, equals W (x) j;k = L j Gamma1 X l=0 h j;l x 2 j k Gammal ; j = 1; J log 2 T ; and k = 0; 1; T=2 j Gamma 1 where one can see that only every other 2 j th filtered ....

Cambanis, S. and C. Houdr'e (1995) "On the continuous wavelet transform of second-order random processes," IEEE Transactions on Information Theory 41, 628-642.

....a smooth differentiation of the signal, with the degree of differentiation equal to the number of vanishing moments. Thus, as already mentioned, means and smooth trends are removed, and non stationary processes which have stationary increments of order N produce stationary wavelet coefficients ([12], see also [33] for the fractional increments) If such nonstationary processes exhibit a scaling of type (3.5) then the corresponding parameter can be estimated by the same procedure as before. INFINITE SOURCE 23 Suppose now X is self similar with Hurst parameter H, but not necessarily long ....

S. Cambanis and C. Houdr'e. On the continuous wavelet transform of second-order random processes. IEEE Trans. Inform. Theory, 41, 628--642, 1995.

.... restricting the time variation of the second order structure of fX t g precisely as in the case of time varying Fourier spectra, see Dahlhaus (1997) von Sachs and Schneider (1996) Neumann and von Sachs (1997) Most existing work on wavelets with stochastic processes (Cambanis and Masry (1994) Cambanis and Houdre (1995), Kawasaki and Shibata (1995) and Cheng and Tong (1996) does not aim to give a decomposition with respect to an (orthogonal) increment process in the time scale plane. These papers focus on probabilistic approximations and do not cover estimation. Morettin and Chang (1995) develop a wavelet ....

Cambanis, S. and Houdre, C. (1995) On the continuous wavelet transform of second-order random processes. IEEE Trans. Inf. Theor., 41, 628--642.

....various attributes of nonlinear wavelet packet estimates. 1 Introduction Research interest in wavelets and their applications have tremendously grown over the last five years. Only, more recently, however, have their applications been considered in a stochastic setting [Fl1, Wo1, BB , CH1] A number of papers which have addressed the optimal representation of a signal in a wavelet wavelet packet basis, have for the most part given a deterministic treatment of the problem. In [Wo1] a Karhunen Lo eve approximation was obtained for fractional Brownian motion with the assumption that ....

Cambanis, S., Houdr'e., C.: On the continuous wavelet transform of second order random processes. preprint (1993)

....various attributes of nonlinear wavelet packet estimates. 1 Introduction Research interest in wavelets and their applications have tremendously grown over the last five years. Only, more recently, however, have their applications been considered in a stochastic setting [Fl1, Wo1, BB , CH1] A number of papers which have addressed the optimal representation of a signal in a wavelet wavelet packet basis, have for the most part given a deterministic treatment of the problem. In [Wo1] a Karhunen Lo eve approximation was obtained for fractional Brownian motion with the assumption that ....

Cambanis, S., Houdr'e., C.: On the continuous wavelet transform of second order random processes. preprint (1993)

.... restricting the time variation of the second order structure of fX t g precisely as in the case of time varying Fourier spectra, see Dahlhaus (1997) von Sachs and Schneider (1996) Neumann and von Sachs (1997) Most existing work on wavelets with stochastic processes (Cambanis and Masry (1994) Cambanis and Houdre (1995), Kawasaki and Shibata (1995) and Cheng and Tong (1996) does not aim to give a decomposition with respect to an (orthogonal) increment process in the time scale plane. These papers focus on probabilistic approximations and do not cover estimation. Morettin and Chang (1995) develop a wavelet ....

Cambanis, S. and Houdre, C. (1995) On the continuous wavelet transform of second-order random processes. IEEE Trans. Inf. Theor., 41, 628--642.

....localized autocovariances (and local variances, also) of the original stochastic process. In the last section we provide some numerical simulations which also indicate the usefulness of the SWT [NaSi] in our approach. Most of the existing work on wavelets with stochastic processes (e.g. CaMa] [CaHo], KaSh] CT] does not aim to give a decomposition with respect to an (orthogonal) increment process in the time scale plane. The first three papers focus on probabilistic approximations and do not cover estimation. Indeed [KaSh] develop a wavelet process representation where they study ....

Cambanis, S. and Houdr'e, C. (1995) On the continuous wavelet transform of second--order random processes. IEEE Trans. Inf. Theory, 41, 628--642.

.... estimation theory is possible without control of an otherwise arbitrary time variation of the spectrum (see Dahlhaus [8] von Sachs and Schneider [33] Neumann and von Sachs [24] Most of the existing work on wavelets with stochastic processes (Cambanis and Masry [3] Cambanis and Houdr e [2], Kawasaki and Shibata [17] and Cheng and Tong [5] does not aim to give a decomposition with respect to an (orthogonal) increment process in the time scale plane. The first three papers focus on probabilistic approximations and do not cover estimation. Indeed, Kawasaki and Shibata [17] develop a ....

S. Cambanis and C. Houdr'e. On the continuous wavelet transform of second-order random processes. IEEE Trans. Inf. Theor. , 41:628--642, 1995.

....the nice properties of wavelets, provided a potential and a framework for an efficient analysis of nonstationary processes. A number of papers have addressed the topic of a wavelet decomposition of random processes [1, 2, 3, 4] and only a few have specifically addressed the nonstationarity issue [5, 6, 7, 8, 9, 10, 11]. Flandrin [5] first presented some fundamental results on the time scale analysis of the fractional Brownian motion (fBm) Other subsequent works [6, 7] provided more insight into the statistical characterization of the wavelet coefficients of the fBm. Masry [8] has generalized these results to a ....

....Other subsequent works [6, 7] provided more insight into the statistical characterization of the wavelet coefficients of the fBm. Masry [8] has generalized these results to a redundant and an orthonormal wavelet decompositions of processes with stationary increments. Recently, Houdr e and Cambanis [9, 10] have derived other fundamental results on the wavelet transform of stochastic processes with stationary increments of an arbitrary order. This class of processes are often used in time series analysis in applied fields such as economics, hydrology, physics and systems modeling. All these ....

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S. Cambanis and C. Houdr'e, "On the continuous wavelet transform of second order random processes." Preprint, 1993.

....The continuous wavelet analysis of stochastic processes has been developed in the last decade, mainly in connection with self similar processes like the fractional Brownian motion (e. g [Fla89] RZ91] Mas93] Recently, many correspondences have been established between the second order ([CH95]) and distributional properties ( AH98] of a process and its wavelet transform. For 1 e mail: caguerin math.chalmers.se our purposes, we will retrieve some of these results, and derive some others which we think are new. The results of the present work can be summarised in the following way. ....

....if for some g 2 S 0 (R) W g X( Delta; a) is stationary for all a 0 then X is stationary. iii) Under the stronger assumption bg( 6= 0 for all , it is enough that W g X( Delta; a) be stationary for some arbitrary a 0 for X to be stationary. A proof of this result has been given in [CH95] using a tauberian theorem. We propose, however, an alternative proof based on simple properties of tempered distributions. The spirit of the proof will be adapted later to the case of locally stationary processes. Proof. Only ii) and iii) need to be proved. ii) Let g 2 S 0 (R) The idea is to ....

[Article contains additional citation context not shown here]

S. Cambanis and C. Houdr'e. On the continuous wavelet transform of second-order random processes. IEEE Trans. Inform. Theory, 41(3):628--642, 1995.

....q = C 0 (q; ff)2 jf2 (q) j 1 j j 2 ; 2) where f 2 (q) f 1 (q) q=2. For mono fractal processes with a single scaling parameter, f 2 (q) reduces to the linear form: f 2 (q) qff=2, and hence to equation (1) when q = 2. ffl P2: Due to F2 and on condition that N is chosen appropriately [12, 8, 13, 2], the fd x (j; Delta)g form a stationary process and in particular are identically distributed. ffl P3: Due to F1 and F2, the d x (j; k) are quasi decorrelated [12] In particular, the long range dependence that may exist in the time domain representation is reduced to residual short range ....

S.Cambanis and C.Houdr'e, On the continuous wavelet transform of second-order random processes. In IEEE Trans. on Info. Theory, Vol. 41, No. 3, May 1995, pp. 628--642.

....fast recursive filter bank based pyramidal algorithm which has a lower computational cost than that of a FFT [11] ffl Discrete Wavelet Transform of Stochastic Processes. It has been clearly demonstrated in the literature that the wavelet transform can be applied to stochastic processes, see e.g. [9, 15]. More specifically, for the second order random processes of interest in the LRD context, it is well known that the wavelet transform is a second order random field, on the condition that the scaling function OE 0 (and hence the wavelet 0 ) satisfy certain mild conditions [9, 15] related to ....

.... see e.g. 9, 15] More specifically, for the second order random processes of interest in the LRD context, it is well known that the wavelet transform is a second order random field, on the condition that the scaling function OE 0 (and hence the wavelet 0 ) satisfy certain mild conditions [9, 15] related to the statistical properties of the analysed process. For instance, if the covariance sum of the random process is bounded, OE 0 and 0 have to be in L 2 ; if the covariance sum diverges (while the covariance remains bounded) OE 0 and 0 must be in L 1 [9] Let us denote by IE ....

[Article contains additional citation context not shown here]

S. Cambanis and C. Houdr'e, On the continuous wavelet transform of second-order random processes. In IEEE Trans. on Info. Theory, Vol. 41, No. 3, May 1995, pp. 628--642.

....periodogram and the Wigner Ville spectrum in the time frequency plane (see also [NvS] In the last section we provide some numerical simulations which also indicate the usefulness of the SWT [NaSi] in our approach. Most of the existing work on wavelets with stochastic processes (e.g. CaMa] [CaHo], KaSh] CT] does not aim to give a decomposition with respect to an (orthogonal) increment process in the time scale plane. The first three papers focus on probabilistic approximations and do not cover estimation. Indeed [KaSh] develop a wavelet process representation where they study ....

Cambanis, S. and Houdr'e, C. (1995) On the continuous wavelet transform of second--order random processes. IEEE Trans. Inf. Theory, 41, 628--642.

....To name a few, we begin with the problem of PC and APC random fields. Alekseev [2] has begun the problem of spectral density estimation of two dimensional Gaussian PC fields. Ramanathan and Zeitouni [79] introduce a connection between wavelets and cyclostationary processes; Cambanis and Houdr e [8] take this idea a little farther. The determination of the presence of the PC or APC property in a time series is another problem of interest; a test based on Goodman s [40] spectral coherence is given in [53] Results for the second order case can also be found in the work of Dandawate and ....

S. Cambanis and C. H. Houdr'e, "On the Continuous Wavelet Transform of Second Order Random Processes," Technical Report No. 390, Center for Stochastic Processes, Dept. of Statistics, UNC at Chapel Hill, 1993.

....so too is w. In all that is to follow, we shall construct probability models directly for w, although it should be noted that if Y ( Delta) is a stationary process, then w J0 and fw j : j = J 0 ; J Gamma 1g are also stationary processes, except for some points near the boundary (Cambanis and Houdr e, 1995). We assume the following Bayesian model: w j fi; oe 2 Gau(fi; oe 2 I) 2) fi j ; Gau Gamma ; Sigma( Delta ; 3) 1. Empirical Bayesian Spatial Prediction Using Wavelets 3 where Sigma( is an n Theta n covariance matrix with structure (depending on parameters ) to be ....

Cambanis, S. and Houdr'e, C. (1995). On the continuous wavelet transform of second-order random processes. IEEE Transactions on Information Theory, 41:628--642.

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S. Cambanis and C. Houdr# (1995): On continuous wavelet transforms of second order random processes. IEEE Trans. Inf. Theory 41, 628642.

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