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248
Multiscale Representations of Markov Random Fields
 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL 41. NO 12. DECEMBER 1993
, 1993
"... Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this pap ..."
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Cited by 102 (27 self)
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Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this paper, we show that this model class is also quite rich. In particular, we describe how 1D Markov processes and 2D Markov random fields (MRF’s) can be represented within this framework. The recursive structure of 1D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2D MRF’s are well known to be very difficult to analyze due to their noncausal structure, and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification. In contrast, our multiscale representations are based on scalerecursive models and thus lead naturally to scalerecursive algorithms, which can be substantially more efficient computationally than those associated with MRF models. In 1D, the multiscale representation is a generalization of the midpoint deflection construction of Brownian motion. The representation of 2D MRF’s is based on a further generalization to a “midline ” deflection construction. The exact representations of 2D MRF’s are used to motivate a class of multiscale approximate MRF models based on onedimensional wavelet transforms. We demonstrate the use of these latter models in the context of texture representation and, in particular, we show how they can be used as approximations for or alternatives to wellknown MRF texture models.
Connectionlevel Analysis and Modeling of Network Traffic
 in ACM SIGCOMM Internet Measurement Workshop
, 2001
"... Abstract — Most network traffic analysis and modeling studies lump all connections together into a single flow. Such aggregate traffic typically exhibits longrangedependent (LRD) correlations and nonGaussian marginal distributions. Importantly, in a typical aggregate traffic model, traffic bursts ..."
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Cited by 98 (5 self)
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Abstract — Most network traffic analysis and modeling studies lump all connections together into a single flow. Such aggregate traffic typically exhibits longrangedependent (LRD) correlations and nonGaussian marginal distributions. Importantly, in a typical aggregate traffic model, traffic bursts arise from many connections being active simultaneously. In this paper, we develop a new framework for analyzing and modeling network traffic that moves beyond aggregation by incorporating connectionlevel information. A careful study of many traffic traces acquired in different networking situations reveals (in opposition to the aggregate modeling ideal) that traffic bursts typically arise from just a few highvolume connections that dominate all others. We term such dominating connections alpha traffic. Alpha traffic is caused by large file transmissions over high bandwidth links and is extremely bursty (nonGaussian). Stripping the alpha traffic from an aggregate trace leaves a beta traffic residual that is Gaussian, LRD, and shares the same fractal scaling exponent as the aggregate traffic. Beta traffic is caused by both small and large file transmissions over low bandwidth links. In our alpha/beta traffic model, the heterogeneity of the network resources give rise to burstiness and heavytailed connection durations give rise to LRD. Queuing experiments suggest that the alpha component dictates the tail queue behavior for large queue sizes, whereas the beta component controls the tail queue behavior for small queue sizes. Keywords—network traffic modeling, animal kingdom I.
Wavelets on graphs via spectral graph theory
, 2009
"... We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. ..."
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Cited by 90 (4 self)
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We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T t g = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
A Wavelet Based Joint Estimator of the Parameters of LongRange Dependence.
, 1998
"... A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased estimator of the scaling parameter [4], as well as ..."
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Cited by 88 (14 self)
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A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased 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...
SelfSimilarity and LongRange Dependence Through the Wavelet Lens
, 2000
"... Selfsimilar and longrange 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, ..."
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Cited by 81 (11 self)
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Selfsimilar and longrange 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 semiparametric estimates of the scaling exponent. We then focus on the case of longrange 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...
Fast, Approximate Synthesis of Fractional Gaussian Noise for Generating SelfSimilar Network Traffic
 ACM SIGCOMM, Computer Communication Review
, 1997
"... Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically selfsimilar processes instead of traditional Poisson processes [LTWW94, PF95]. One difficulty in dealing with selfsimilar models is how to efficiently synthesize traces (sample p ..."
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Cited by 77 (2 self)
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Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically selfsimilar 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 selfsimilar traffic. We present a fast Fourier transform method for synthesizing approximate selfsimilar sample paths for one type of selfsimilar 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 selfsimilar 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 longrange dependence present in a selfsimilar time series. 1
On estimation of the wavelet variance
 Biometrika
, 1995
"... The wavelet variance provides a scalebased decomposition of the process variance for a time series or random field. It has seen increasing use in geophysics, astronomy, genetics, hydrology, medical imaging, oceanography, soil science, signal processing and texture analysis. In practice, however, da ..."
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Cited by 66 (7 self)
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The wavelet variance provides a scalebased decomposition of the process variance for a time series or random field. It has seen increasing use in geophysics, astronomy, genetics, hydrology, medical imaging, oceanography, soil science, signal processing and texture analysis. In practice, however, data collected in the form of a time series or random field often suffer from various types of contamination. We discuss the difficulties and limitations of existing contamination models (pure replacement models, additive outliers, level shift models and innovation outliers that hide themselves in the original time series) for robust nonparametric estimates of secondorder statistics. We then introduce a new model based upon the idea of scalebased multiplicative contamination. This model supposes that contamination can occur and affect data at certain scales and thus arises naturally in multiscale processes and in the wavelet variance context. For this new contamination model, we develop a full Mestimation theory for the wavelet variance and derive its large sample theory when the underlying time series or random field is Gaussian. Our approach treats the wavelet variance as a scale parameter and offers protection against contamination that operates additively on the log of squared wavelet coefficients and acts independently at different scales.
LongRange Dependence: revisiting Aggregation with Wavelets.
 Journal of Time Series Analysis
, 1998
"... The aggregation procedure is a natural way to analyse signals which exhibit longrange 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 ..."
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Cited by 65 (15 self)
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The aggregation procedure is a natural way to analyse signals which exhibit longrange 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 Haarmultiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: "aaggregation". It is shown that aaggregation cannot lead to good estimators of H, and so a new kind of aggregation, "daggregation", is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on daggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.
A WaveletBased Analysis of Fractal Image Compression
 IEEE Trans. Image Processing
, 1997
"... 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 ..."
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Cited by 60 (2 self)
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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 blockbased fractal compression schemes. Within this framework we are able to draw upon insights from the wellestablished 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 Jacquinstyle coding scheme. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergen...