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Long‐range Dependence: Revisiting Aggregation with Wavelets
Author(s) -
Abry Patrice,
Veitch Darryl,
Flandrin Patrick
Publication year - 1998
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00090
Subject(s) - estimator , wavelet , generalization , mathematics , range (aeronautics) , hurst exponent , basis (linear algebra) , haar wavelet , algorithm , wavelet transform , multiresolution analysis , haar , mathematical optimization , discrete wavelet transform , computer science , artificial intelligence , statistics , mathematical analysis , geometry , materials science , composite material
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.