Results 1  10
of
21
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 171 (30 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
Fractal Estimation using Models on Multiscale Trees
 IEEE Transactions on Signal Processing
, 1996
"... In this paper we estimate the Hurst parameter H of fractional Brownian motion (or, by extension, the fractal exponent ' of stochastic processes having 1=f ' like spectra) by applying a recentlyintroduced multiresolution framework. This framework admits an efficient likelihood function evaluation ..."
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Cited by 24 (16 self)
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In this paper we estimate the Hurst parameter H of fractional Brownian motion (or, by extension, the fractal exponent ' of stochastic processes having 1=f ' like spectra) by applying a recentlyintroduced 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 2D random fields. 1 Introduction Many natural and human phenomena have been found to possess 1=f like spectral properties, which has led to considerable study of 1=f processes. One class of such processes that is frequently used because...
A Comparison of Estimators for 1/f Noise
, 1997
"... We use a MonteCarlo approach to investigate the performance of five different timeseries estimators of the exponent ff in 1=f ff noise. We find that a maximumlikelihood estimator is markedly superior to Fourier regression methods and Hurst exponent methods. The results indicate that useful estim ..."
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Cited by 16 (1 self)
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We use a MonteCarlo approach to investigate the performance of five different timeseries estimators of the exponent ff in 1=f ff noise. We find that a maximumlikelihood estimator is markedly superior to Fourier regression methods and Hurst exponent methods. The results indicate that useful estimates of ff can be made from time series that are much shorter than generally presumed. PACS codes: 72:70:+m, 73:50:Td, 74:40:+k Keywords: noise, noise parameter estimation, noise generation 1 Introduction Longterm correlations have been observed in many types of time series from physical, biological, physiological, economic, technological and sociological systems. Examples include geophysical data [1, 2, 3] such as rainfall, temperature measurements, sunspot numbers, earthquake frequencies, and river flows, frequency fluctuations in electrical oscillators [4], rate of traffic flow [1], voltage or current fluctuations in metal films and semiconductor devices [4], loudness fluctuations i...
Statistical Significance of periodicity and logperiodicity with heavytailed correlated Noise
, 2001
"... We estimate the probability that random noise, of several plausible standard distributions, creates a false alarm that a periodicity (or logperiodicity) is found in a time series. The solution of this problem is already known for independent Gaussian distributed noise. We investigate more general s ..."
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Cited by 12 (7 self)
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We estimate the probability that random noise, of several plausible standard distributions, creates a false alarm that a periodicity (or logperiodicity) is found in a time series. The solution of this problem is already known for independent Gaussian distributed noise. We investigate more general situations with nonGaussian correlated noises and present synthetic tests on the detectability and statistical significance of periodic components. A periodic component of a time series is usually detected by some sort of Fourier analysis. Here, we use the Lomb periodogram analysis which is suitable and outperforms Fourier transforms for unevenly sampled time series. We examine the falsealarm probability of the largest spectral peak of the Lomb periodogram in the presence of powerlaw distributed noises, of shortrange and of longrange fractionalGaussian noises. Increasing heavytailness (respectively correlations describing persistence) tends to decrease (respectively increase) the falsealarm probability of finding a large spurious Lomb peak. Increasing antipersistence tends to decrease the falsealarm probability. We also study the interplay between heavytailness and longrange correlations. In order to fully determine if a Lomb peak signals a genuine rather than a spurious periodicity, one should
Analysis, synthesis, and estimation of fractalrate stochastic point processes
 Fractals
, 1997
"... Fractal and fractalrate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computernetwork traffic, and sequences of neuronal action potentials. A particularly useful stati ..."
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Cited by 10 (5 self)
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Fractal and fractalrate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computernetwork traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes nontrivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α. The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the
Polyharmonic smoothing splines and the multidimensional Wiener filtering of fractallike signals
 IEEE TRANS. IMAGE PROCESSING
, 2006
"... Motivated by the fractallike behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensiona ..."
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Cited by 6 (6 self)
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Motivated by the fractallike behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensional data. He proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions (RBFs). Unfortunately, this is tedious to implement for images because of the poor conditioning of RBFs and their lack of decay. Here, we present a much more efficient method for the special case of a uniform grid. The key idea is to express Duchon’s solution in a fractional polyharmonic Bspline basis that spans the same space as the RBFs. This allows us to derive an algorithm where the smoothing is performed by filtering in the Fourier domain. Next, we prove that the above smoothing spline can be optimally tuned to provide the MMSE estimation of a fractional Brownian field corrupted by white noise. This is a strong result that not only yields the best linear filter (Wiener solution), but also the optimal interpolation space, which is not bandlimited. It also suggests a way of using the noisy data to identify the optimal parameters (order of the spline and smoothing strength), which yields a fully automatic smoothing procedure. We evaluate the performance of our algorithm by comparing it against an oracle Wiener filter, which requires the knowledge of the true noiseless power spectrum of the signal. We find that our approach performs almost as well as the oracle solution over a wide range of conditions.
2D Shape Classification Using Multifractional Brownian Motion
"... Abstract. In this paper a novel approach to contourbased 2D shape recognition is proposed. The main idea is to characterize the contour of an object using the multifractional Brownian motion (mBm), a mathematical method able to capture the local self similarity and longrange dependence of a signal ..."
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Cited by 6 (4 self)
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Abstract. In this paper a novel approach to contourbased 2D shape recognition is proposed. The main idea is to characterize the contour of an object using the multifractional Brownian motion (mBm), a mathematical method able to capture the local self similarity and longrange dependence of a signal. The mBm estimation results in a sequence of Hurst coefficients, which we used to derive a fixed size feature vector. Preliminary experimental evaluations using simple classifiers with these feature vectors produce encouraging results, also in comparison with the state of the art. 1
Invariances, Laplacianlike wavelet bases, and the whitening of fractal processes
, 2009
"... In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fraction ..."
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Cited by 6 (4 self)
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In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBmlike processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.
Classification of texture images using multiscale statistical estimators of fractal parameters
 in British Machine Vision Conference, M. Mirmehdi and
, 2000
"... We present a new method of fractalbased texture analysis, using the multiscale fractional Brownian motion texture model, and a new parameter, intermittency. The intermittency parameter Ô describes a degree of presence of the textural information: a low value of Ô implies a very lacunar texture. The ..."
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Cited by 4 (2 self)
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We present a new method of fractalbased texture analysis, using the multiscale fractional Brownian motion texture model, and a new parameter, intermittency. The intermittency parameter Ô describes a degree of presence of the textural information: a low value of Ô implies a very lacunar texture. The multiscale fractional Brownian motion model allows to construct multiregime textures in the frequency domain. Adding intermittency to this model, we compose the intermittent multiscale fractional Brownian motion model: the Hurst and intermittency parameters of such processes are functions À Ð and Ô Ð depending on a scale Ð. The texture is thereby seen as the fusion of structures and details. The structure of the texture is analyzed with the large values of Ð, corresponding to the low frequency content of the texture. The details of the texture are analyzed with the small values of Ð, related to the high frequency content of the texture. The texture is then characterized by all the estimated values of À Ð and Ô Ð, for all the scales Ð of analysis. The method allows a multifrequency analysis, permitting the choice of significant scales in a classification task. An application to the classification of corn silage texture images, for which the low frequency content is determining, is proposed. Texture, fractal, multiscale and multifrequency analysis, multiscale fractional Brownian motion, intermittency, Hurst parameter, statistical estimators, selfsimilarity, image classification. 1