Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically self-similar processes instead of traditional Poisson processes [LTWW94, PF95]. One difficulty in dealing with selfsimilar models is how to efficiently synthesize traces (sample paths) corresponding to self-similar traffic. We present a fast Fourier transform method for synthesizing approximate self-similar sample paths for one type of self-similar process, Fractional Gaussian Noise, and assess its performance and validity. We find that the method is as fast or faster than existing methods and appears to generate close approximations to true self-similar sample paths. We also discuss issues in using such synthesized sample paths for simulating network traffic, and how an approximation used by our method can dramatically speed up evaluation of Whittle's estimator for H, the Hurst parameter giving the strength of long-range dependence present in a self-similar time series.

A joint estimator is presented for the two parameters that define the long-range dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed wavelet-based estimator of the scaling parameter [4], as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well justified technical idealisations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed for...

Abstract—We present an algorithm for fast elastic multidimensional intensity-based image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard real-world problems, it is capable of accepting expert hints in the form of soft landmark constraints. Much fewer landmarks are needed and the results are far superior compared to pure landmark registration. Particular attention has been paid to the factors influencing the speed of this algorithm. The B-spline deformation model is shown to be computationally more efficient than other alternatives. The algorithm has been successfully used for several two-dimensional (2-D) and three-dimensional (3-D) registration tasks in the medical domain, involving MRI, SPECT, CT, and ultrasound image modalities. We also present experiments in a controlled environment, permitting an exact evaluation of the registration accuracy. Test deformations are generated automatically using a random hierarchical fractional wavelet-based generator. Index Terms—Elastic registration, image registration, landmarks, splines. I.

Why does fractal image compression work? What is the implicit image model underlying fractal block coding? How can we characterize the types of images for which fractal block coders will work well? These are the central issues we address. We introduce a new waveletbased framework for analyzing block-based fractal compression schemes. Within this framework we are able to draw upon insights from the well-established transform coder paradigm in order to address the issue of why fractal block coders work. We show that fractal block coders of the form introduced by Jacquin[1] are a Haar wavelet subtree quantization scheme. We examine a generalization of this scheme to smooth wavelets with additional vanishing moments. The performance of our generalized coder is comparable to the best results in the literature for a Jacquin-style coding scheme. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergen...

Abstract: Even in the absence of an experimental effect, functional magnetic resonance imaging (fMRI) time series generally demonstrate serial dependence. This colored noise or endogenous autocorrelation typically hasdisproportionatespectralpoweratlowfrequencies,i.e.,itsspectrumis 1 f-like.Variouspre-whiteningand pre-coloringstrategieshavebeenproposedtomakevalidinferenceonstandardisedteststatisticsestimatedby time series regression in this context of residually autocorrelated errors. Here we introduce anew method based on random permutation after orthogonal transformation of the observed time series to the wavelet domain. This scheme exploits the general whitening or decorrelating property of the discrete wavelet transformandisimplementedusingaDaubechieswaveletwithfourvanishingmomentstoensureexchangeability of wavelet coefficients within each scale of decomposition. For 1-like or fractal noises, e.g., realisations f of fractional Brownian motion (fBm) parameterised by Hurst exponent 0�H�1, this resampling algorithm exactly preserves wavelet-based estimates of the second order stochastic properties of the (possibly nonstationary) time series. Performance of the method is assessed empirically using 1-like noise simulated by f

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The aggregation procedure is a natural way to analyse signals which exhibit long-range dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar-multiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: "a-aggregation". It is shown that a-aggregation cannot lead to good estimators of H, and so a new kind of aggregation, "d-aggregation", is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.

A wavelet based statistical test is described for distinguishing true time variation of the scaling exponent describing scaling behaviour, from statistical fluctuations of estimates across time of a constant exponent. The test is applicable to diverse scaling phenomena including long range dependence and exactly selfsimilar processes in a uniform framework, without the need for prior knowledge of the type in question. It is based on the special properties of wavelet-based estimates of the scaling exponent over adjacent blocks of data, strongly motivating an idealised inference problem: the equality or otherwise of means of independent Gaussian variables with known variances. A uniformly most powerful invariant test exists for this problem and is described. A separate UMPI test is also described for when the scaling exponent undergoes a level change. The power functions of the two tests are given explicitly and compared. Using simulation the effect in practice of deviations from the ide...

In this paper, we develop a simple and powerful multiscale model for the synthesis of nonGaussian, long-range dependent (LRD) network traffic. Although wavelets effectively decorrelate LRD data, wavelet-based models have generally been restricted by a Gaussianity assumption that can be unrealistic for traffic. Using a multiplicative superstructure on top of the Haar wavelet transform, we exploit the decorrelating properties of wavelets while simultaneously capturing the positivity and "spikiness" of nonGaussian traffic. This leads to a swift O(N) algorithm for fitting and synthesizing N-point data sets. The resulting model belongs to the class of multifractal cascades, a set of processes with rich statistical properties. We elucidate our model's ability to capture the covariance structure of real data and then fit it to real traffic traces. Queueing experiments demonstrate the accuracy of the model for matching real data. Our results indicate that the nonGaussian nature of traffic has a significant effect on queuing.

Self-similar and long-range dependent processes are the most important kinds of random processes possessing scale invariance. We describe how to analyze them using the discrete wavelet transform. We have chosen a didactic approach, useful to practitioners. Focusing on the Discrete Wavelet Transform, we describe the nature of the wavelet coefficients and their statistical properties. Pitfalls in understanding and key features are highlighted and we sketch some proofs to provide additional insight. The Logscale Diagram is introduced as a natural means to study scaling data and we show how it can be used to obtain unbiased semi-parametric estimates of the scaling exponent. We then focus on the case of long-range dependence and address the problem of defining a lower cutoff scale corresponding to where scaling starts. We also discuss some related problems arising from the application of wavelet analysis to discrete time series. Numerical examples using many discrete time models are th...