Long-Range Dependence: revisiting Aggregation with Wavelets

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 deened, which is related to the details rather than the approximations of a multires-olution 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. It has now been amply demonstrated 14, 12, 16, 8] that long-range dependence (LRD) is present in telecommunications traac data recorded from high speed networks, and that this phenomenon is likely to have major consequences for the performance of the said networks. A full analysis of LRD is therefore of considerable importance. By LRD we understand not only that correlations persist over very long time scales, but also that they possess a certain asymptotically self similar structure. It is therefore very natural to analyze, or even deene, LRD by performing a rescaling operation at diierent scales, and then to observe how measured properties vary as a function of scale. This is the essence of the aggregation procedure 14], an idea which has served as a basis for much of the study of the LRD phenomenon. In particular, a major concern is to accurately estimate the parameters characterising LRD, which basically amounts to the determination of the second-order properties of the data. The aggregation procedure has been used as a natural starting point for the design of such estimators, however the inherent diiculties associated with LRD has prevented them from having desirable statistical properties. The aim of this paper is to show that the aggregation procedure has close connections to the mul-tiresolution analysis underlying the wavelet transform, and therefore can be naturally rephrased in this framework. We then present the extensions made possible by the use of wavelets and show how we arrive at an estimator for the LRD parameter which is particularly simple to use and which enjoys excellent statistical properties, whilst preserving the …