P.Abry, P.Goncalv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lecture Notes in Statistics, 105, pp.15--30, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to obtain a 0 (k) and then proceeds upward to calculate all the detail coecients. The analysis then consists of studing their statistical properties, for example the measurement of the exponent can be made by considering the logarithm of the variance of d j; as a function of j [20] [24]. To invert this procedure, that is to begin from the approximation and detail coecients at the coarsest level and to reconstruct the ner level approximations, we need the inverse relation a j;k = X n u k 2n a j 1;n v k 2n d j 1;n (5) illustrated in gure 2, again for nite generating ....

P. Abry, P. Goncalves, and P. Flandrin, Wavelets and Statistics, vol. 105 of Lecture Notes in Statistics, chapter Wavelets, Spectrum estimation, 1=f processes., pp. 15-30, Springer-Verlag, New York, 1995.

....a log 2 scale. Linear regions in these plots indicate selfsimilarities , and the slope of the linear region gives an estimate of the (local) Hurst parameter. The variance of the wavelet coecients measures the energy in the signal in the given scale. If we go back to the analysis presented in [20], 19] we see that this energy is really an estimate of the power spectral density of the process about a frequency determined by the particular scale. The frequency progression in scales is logarithmic, i.e. the j th wavelet space, denoted by the scale 2 j , represents a frequency 2 j ....

P. Abry, P. Goncalves, and P. Flandrin, \Wavelets, spectrum estimation, 1/f processes," Lecture Notes in Statistics, vol. 105, pp. 15-30, 1995.

....For H GammaSS processes with second moments, equation 1 holds for all octaves in the data and H = ff Gamma 1) 2. Note that the exact reproduction of the power law in equation (1) is a non trivial result due to F1 not shared by standard spectral estimates such as the periodogram (see e.g. [1]) The property P2 is that which allows stationary and non stationary forms of scaling to be treated in a single framework, as N is a parameter which can be freely chosen both in theory and in practice. With the coefficients d(j; k) being stationary in time k at each scale, it is meaningful to ....

....g q (j) before the measurement of the slope truly corresponds to the estimate of ff. We write the corrected values as y q j = log( q j ) Gamma g q (j) The exact form of these factors can not be calculated in all cases, especially given the unknowns in real data, however it is known [1, 19, 10] that for large n j they become negligible. Thus the problem is not serious for the analysis of regularity properties at small scales where the n j are large, and for the most important case at large scales, namely the question of LRD, the full solution is known in the Gaussian case [18, 19] ....

P.Abry, P.Gon¸calv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lectures Note in Statistics, Vol.105, pp.15--30, (1995).

....a mixture of long and short range dependent processes. Since network traffic has both short and long range dependence, we need to extend the previous work to a broader class of Gaussian processes in order to study correlation structures. Wavelets were also used to estimate Hurst parameters[2][1][15] The possibility of using wavelets for modeling network traffic was mentioned in [40] 14] but no further investigation was done on wavelet modeling of network traffic. 10 Conclusions In this work, we have proposed wavelet models as a unified approach for complicated network traffic with ....

P. Abry, P. Goncalves, and P. Flandrin. Wavelet, spectrum analysis and 1=f processes. Lecture notes in statistics, 103:15--30, 1995.

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P.Abry, P.Goncalv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lecture Notes in Statistics, 105, pp.15--30, 1995.

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P.Abry, P.Goncalv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lecture Notes in Statistics, 105, pp.15--30, 1995.

.... Recent work based on wavelets however has provided a semi parametric estimator for H which gives unbiased estimates together with signi cant computational advantages, notably a run time complexity of only O(n) Details of this estimator are summarized below, in Section II, and can be found in [3], 4] see also [5] 6] 7] In [8] it was shown how these computational advantages can be exploited to allow H to be estimated in real time simply, inexpensively, and with very low memory requirements. Section III describes the real time implementation of the estimator, which is the subject of ....

....of all three together, is addressed in the next section in the context of Ethernet measurement. The statistical performance of the batch joint AV estimator, and comparisons with other methods of estimating LRD parameters, have been described in detail elsewhere [4] 11] for H only see also [5] [3], 6] Brie y, the estimator o ers excellent statistical performance: negligible bias, close to optimal variance, and robustness of various kinds including with respect to superimposed deterministic non stationarities. It is not the aim of this paper to repeat these studies, but rather to ....

P. Abry, P. Goncalves, and P. Flandrin, Wavelets and Statistics, vol. 105 of Lecture Notes in Statistics, chapter Wavelets, Spectrum estimation, 1=f processes., pp. 15-30, Springer-Verlag, New York, 1995.

....parameter, H, describes the (asymptotic) self similarity of the cumulative trac process corresponding to x(t) which generates the LRD of x(t) itself described by . It is nonetheless common practice to speak of H in relation to LRD. The two are related as H = 1 ) 2. In [8] 9] see also [13], 11] 14] 12] 10] a semi parametric joint estimator of ( c f ) is described based on the Discrete Wavelet Transform. Wavelet transforms in general can be understood as a more exible form of a Fourier transform, where x(t) is transformed, not into a frequency domain, but into a ....

....with con dence intervals, and from it the range of octaves [j 1 ; j 2 ] where scaling occurs is determined. The LRD parameters H and c f are then extracted by performing a weighted linear regression over those scales. Notes: Since the expectations of the details are all identically zero [13], the average of the squares of the details at a given j is an estimate of the variance at that j. In forming y j small corrective terms g(j) are in fact subtracted from log 2 ( j ) to account for the fact that E [log] 6= log(E [ H is related to the slope of the plot, and c f to a ....

P. Abry, P. Goncalves, and P. Flandrin, Wavelets and Statistics, vol. 105 of Lecture Notes in Statistics, chapter Wavelets, Spectrum estimation, 1=f processes., pp. 15-30, Springer-Verlag, New York, 1995.

.... Recent work based on wavelets however has provided a semi parametric estimator for H which gives unbiased estimates together with significant computational advantages, notably a run time complexity of only O(n) Details of this estimator are summarized below and can be found in [7] 6] see also [1], 2] 3] The aim of the present paper is to show how these computational advantages can be exploited in an on line setting to allow H to be estimated in real time simply, rapidly, and with very low memory requirements. The method scales to arbitrary size with respect to both memory and ....

....y j = log 2 ( j ) against j and from it the range of octaves [j 1 ; j 2 ] where scaling occurs is determined. The LRD parameters H and c f are then extracted by performing a weighted linear regression over those scales. Notes: ffl Since the expectations of the details are all identically zero [1], the average of the squares of the details at a given j is an estimate of the variance at that j. ffl In forming y j small corrective terms g(j) are in fact subtracted from log 2 ( j ) to account for the fact that E [log] Delta) 6= log(E [ Delta] ffl H is related to the slope of the ....

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P.Abry, P.Gon¸calv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lectures Note in Statistics, Vol.105, pp.15--30, (1995).

....against the discrete Whittle estimator is that it has been widely used in the analysis of network traffic. The wavelet estimator. A semi parametric wavelet based estimator for the Hurst parameter with excellent statistical, computational, and robustness properties has already been reported in [3], 4] see also [6] The joint estimator described here is based on the same approach, where several properties of the wavelet decomposition combine to reduce LRD in the time domain to short range dependence (SRD) in the wavelet representation. Key properties are the band pass nature of the ....

....(and scaling functions) are generated from the change of scale operator which matches the power law form of LRD spectra, and the fact that the number of vanishing moments of the wavelets can be controlled. In this paper we improve upon the estimator for the scaling exponent ff reported in [3], 4] 6] and extend it to the joint case (ff; c f ) Under reasonable additional technical idealisations, we first show that the related joint estimator (ff; d c f C) is unbiased and asymptotically efficient, and give explicit formulae for the covariance matrix and Cramer Rao bound. It is ....

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P.Abry, P.Gon¸calv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lectures Note in Statistics, Vol. 105, pp. 15--30, 1995.

....in general for statistics of LRD processes shows that H is of central importance. It is vital that it be estimated well, and if joint parameter estimation is impossible or impractical, that it be estimated first. The main aim of the paper is to introduce an estimation tool from wavelet analysis [2, 3] which provides a natural, statistically and computationally efficient, estimator of the Hurst parameter H. It is known [10] that simple traditional estimators can be seriously biased. Asymptotically unbiased estimators derived from Gaussian Maximum Likelihood Estimation are available [10] 33] ....

....scale time wavelet decomposition. As such, very little needs to be assumed about the underlying process. Should evidence of LRD be found, it then offers an unbiased semi parametric estimator which can be very efficiently implemented using techniques from non redundant multi resolution analysis [3]. The wavelet based estimator has the additional virtue of robustness against an important class of nonstationarity, namely the addition of deterministic trends. This is a particularly important advantage in a LRD context where it is very difficult in theory and in practice to distinguish between ....

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P.Abry, P.Gon¸calv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lectures Note in Statistics, Vol. 105, (1995), pp.15-30.

....Transforms of fl x , OE 0 and 0 respectively. The variances can be written as IEja x (j; k)j 2 = R Gamma x ( 2 j j Phi 0 (2 j )j 2 d IEjd x (j; k)j 2 = R Gamma x ( 2 j j Psi 0 (2 j )j 2 d oe (8) These relations, which can be given a spectral estimation interpretation [2], constitute the starting points for deriving equations (10) and (12) As mentioned above, LRD (stationary) processes are closely connected to self similar (non stationary) processes such as fBm. The autocovariance functions of these latter processes are not bounded, however the second order ....

....and the Haar multiresolution analysis: studying x over longer and longer observation periods T simply translates in the MRA vocabulary to increasing the scale of analysis 2 j , or equivalently to lowering the resolution. Amongst others, a consequence is that whenever x is a LRD process, one has [2]: var (a x (j; k) 2 j(1 Gammafi) j 1; 8k : 10) ffl Beyond Haar Aggregation: Changing the wavelet. There exist infinitely many MRAs (see e.g. 5, 1] We selected the Haar only to underline the possibility of exactly reformulating the aggregation procedure as a multiresolution analysis. ....

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P.Abry, P.Gon¸calv`es and P.Flandrin, Wavelets, Spectrum estimation, 1=f processes. Wavelets and Statistics, Lectures Note in Statistics, Vol. 103, (1995), pp.15--30.

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