Long‐range Dependence: Revisiting Aggregation with Wavelets
Patrice Abry, Darryl Veitch, Patrick Flandrin · Journal of Time Series Analysis · 1998
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 generalization 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.