Estimation of fractal signals from noisy measurements using wavelets

Wornell, Gregory W.; Oppenheim, Alan V.

doi:10.1109/78.120804

Cited by 335 publications

(209 citation statements)

References 19 publications

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“…An empirical wavelet based application to the cluster Poisson model can be found in Hohn et al [19]. Using the above properties of the wavelet coefficients, in particular (6.48), an alternative estimation procedure was suggested by Moulines et al [26]; see also Wornell and Oppenheim [35]. They propose to estimate α by minimization of a local Whittle contrast, defined as follows: where we choose ∆ as follows: assume that d j,k is observable for J min ≤ j ≤ J max and 0 ≤ k < 2 Jmax−j .…”

Section: Fitting the Model To Internet Traffic Datamentioning

confidence: 99%

Modeling teletraffic arrivals by a Poisson cluster process

et al. 2006

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Abstract. In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γj, a flow of packets is initiated which is modeled as a partial iid sum process Γj + P k i=1 Xji, k ≤ Kj , with a random limit Kj which is independent of (Xji) and the underlying Poisson points (Γj). We study the covariance structure of the increment process of N . In particular, the covariance function of the increment process is not summable if the right tail P (Kj > x) is regularly varying with index α ∈ (1, 2), the distribution of the Xji's being irrelevant. This means that the increment process exhibits long-range dependence. If var(Kj) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N (t)) t≥0 and give conditions on the distribution of Kj and Xji under which the random sumsXji have a regularly varying tail. Using the form of the distribution of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distributional characteristics of Kj , Xji, and of the Poisson process.

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Section: Fitting the Model To Internet Traffic Datamentioning

confidence: 99%

Modeling teletraffic arrivals by a Poisson cluster process

et al. 2006

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“…Kaplan and Kuo [4] apply the Haar wavelet to the incremental process F [k], and Wornell and Oppenheim [11] apply higher order Daubechies wavelets to B [k]. We use the multiscale framework just described to develop a Haar wavelet multiscale stochastic model which applies directly to B [k].…”

Section: Fractal Estimatormentioning

confidence: 99%

“…As was the case with Wornell and Oppenheim [11], we do not construct an exact model of fBm, but rather choose an appropriate approximation -in our case within this multiscale framework. The selection of such a multiscale model is achieved in the next section.…”

Section: Determine the Likelihood [A() G() C() R() ()] Of A Set Omentioning

confidence: 99%

“…In particular the approach in [11] is based on a 1/f process constructed using a wavelet basis in which the wavelet coefficients are independent, with variances that vary geometrically with scale with exponent equal to H. The method in [4] determines the exact statistics of the Haar wavelet coefficients of the discrete fractional Gaussian noise (DFGN) process…”

Section: B[k] = B(kat)mentioning

confidence: 99%

“…Oppenheim [11], Kaplan and Kuo [4], Tewfik and Deriche [10], and Flandrin [3]. The exact maximum likelihood (ML) calculation for the fractal dimension of fBm is computationally difficult (see [10]);…”

Section: B[k] = B(kat)mentioning

confidence: 99%

See 2 more Smart Citations

Fractal Estimation using Models on Multiscale Trees

Fieguth¹,

Willsky²

1995

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In this paper we estimate the fractal dimension of stochastic processes with 1/f-like spectra by applying a recently-introduced multiresolution framework. This framework admits an efficient likelihood function evaluation, allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease. In addition to yielding results that compare well to other proposed methods. and in contrast to other approaches, our method is directly applicable, with at most very simple modification, in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2-D random fields.

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Study of single motor unit discharge patterns using 1/f process model

Fang

Shahani

Bruyninckx³

1997

Muscle Nerve

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We developed a technique to study temporal discharge patterns of single motor units using 1/f process model, whose fractional parameter γ was shown to be a useful indicator for distinguishing between discharge behaviors of single motor units of normal subjects and patients with upper motor neuron lesions. We have studied a total of 47 motor units in 3 normal subjects, and 41 in 3 patients with upper motor neuron lesions from biceps and extensor digitorum communis muscles during steady contraction. The parameter γ was estimated with an algorithm based on wavelet analysis. The mean value of γ in patients was 1.52, and the mean value of γ in normal subjects was −0.06. These results suggest that 1/f process can be used to document the impaired motor control mechanisms at single motor unit level. © 1997 John Wiley & Sons, Inc. Muscle Nerve, 20, 293–298, 1997.

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Estimation of fractal signals from noisy measurements using wavelets

Cited by 335 publications

References 19 publications

Modeling teletraffic arrivals by a Poisson cluster process

Modeling teletraffic arrivals by a Poisson cluster process

Fractal Estimation using Models on Multiscale Trees

Study of single motor unit discharge patterns using 1/f process model

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