Ieee Sp International Symposium on Time Frequency and Time Scale Analysis Pittsburgh Pa October a Multifractal Wavelet Model for Positive Processes

In this paper we describe a new multiscale model for char acterizing positive valued and long range dependent data The model uses the Haar wavelet transform and puts a con straint on the wavelet coe cients to guarantee positivity which results in a swift O N algorithm to synthesize N point data sets We elucidate our model s ability to capture the covariance structure of real data study its multifrac tal properties and derive a scheme for matching it to real data observations We demonstrate the model s utility by applying it to network tra c synthesis The exibility and accuracy of the model and tting procedure result in a close match to the real data statistics variance time plots and queuing behavior INTRODUCTION Fractals models arise frequently in a variety of scienti c disciplines such as physics chemistry astronomy and bi ology In DSP fractals have long proven useful for applica tions such as computer graphics and texture modeling More recently fractal models have had a major impact on the analysis of data communication networks such as the Internet In their landmark paper Leland et al demon strated that network tra c exhibits fractal properties such as self similarity burstiness and long range dependence LRD that are inadequately described by classical tra c models Characterization of these fractal properties partic ularly LRD has provided exciting new insights into network behavior and performance Fractals are geometric objects that exhibit an irregular structure at all resolutions Most fractals are self similar if we zoom in or out of the fractal we obtain a picture similar to the original Deterministic fractals usually have a highly speci c structure that can be constructed through a few simple steps Real world phenomena can rarely be de scribed using such simple models Nevertheless similarity on all scales can hold in a statistical sense leading to the notion of random fractals As the pre eminent random fractal model fractional Brownian motion fBm has played a central r ole in many elds FBm is the unique Gaussian process with sta tionary increments and the following scaling property for all a B at fd aB t with the equality in nite dimensional distribution The parameter H H is known as the Hurst parameter It rules the LRD of fBm as we will see later but it also governs its local spikiness In particular for all t