A statistical test for the time constancy of scaling exponents

A statistical test is described for determining if scaling exponents vary over time. It is applicable to diverse scaling phenomena including long-range dependence and exactly self-similar processes in a uniform framework without the need for prior knowledge of the type in question. It is based on the special properties of wavelet-based estimates of the scaling exponent, strongly motivating an idealized inference problem: the equality or otherwise of means of independent Gaussian variables with known variances. A uniformly most powerful invariant test exists for this problem and is described. A separate uniformly most powerful invariant test is also given for when the scaling exponent undergoes a level change. The power functions of both tests are given explicitly and compared. Using simulation, the effect, in practice, of deviations from the idealizations made of the statistical properties of the wavelet detail coefficients are analyzed and found to be small. The tests inherit the significant robustness and computational advantages of the underlying wavelet-based estimator. A detailed methodology is given, describing its use in practical situations. The use and benefits of the test are illustrated on the Bellcore Ethernet data sets.

[1]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[2]  W. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[3]  Rudolf H. Riedi,et al.  An Improved Multifractal Formalism and Self Similar Measures , 1995 .

[4]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[5]  Patrick Flandrin,et al.  Wavelet analysis and synthesis of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[6]  Anja Feldmann,et al.  Data networks as cascades: investigating the multifractal nature of Internet WAN traffic , 1998, SIGCOMM '98.

[7]  O. Bunke.  Feller, F.: An Introduction to Probability Theory and its Applications, Vol. II, John Wiley & Sons, Inc., New York‐London‐Sydney, 1966. XVIII + 626 S., 3 Abb., 2 Tab., Preis $ 12,00 , 1969 .

[8]  Vern Paxson,et al.  Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic , 1997, CCRV.

[9]  Patrice Abry,et al.  Wavelet Analysis of Long-Range-Dependent Traffic , 1998, IEEE Trans. Inf. Theory.

[10]  Patrice Abry,et al.  Wavelets for the Analysis, Estimation, and Synthesis of Scaling Data , 2002 .

[11]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[12]  P. Abry,et al.  Wavelets, spectrum analysis and 1/ f processes , 1995 .

[13]  Patrice Abry,et al.  Long‐range Dependence: Revisiting Aggregation with Wavelets , 1998 .

[14]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[15]  Matthew Roughan,et al.  On-line estimation of the parameters of long-range dependence , 1998, IEEE GLOBECOM 1998 (Cat. NO. 98CH36250).

[16]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[17]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[18]  Ravi Mazumdar,et al.  On the correlation structure of the wavelet coefficients of fractional Brownian motion , 1994, IEEE Trans. Inf. Theory.

[19]  Patrice Abry,et al.  Wavelet based estimator for the self-similarity parameter of /spl alpha/-stable processes , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[20]  Ilkka Norros,et al.  A storage model with self-similar input , 1994, Queueing Syst. Theory Appl..

[21]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[22]  Patrice Abry,et al.  Wavelet based estimator for the self-similarity parameter of α-stable processes. , 1999 .

[23]  N. Terrin,et al.  ESTIMATION OF THE LONG‐MEMORY PARAMETER, BASED ON A MULTIVARIATE CENTRAL LIMIT THEOREM , 1994 .

[24]  Matthew Roughan,et al.  Measuring long-range dependence under changing traffic conditions , 1999, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320).

[25]  D. Veitch.  Wavelet Analysis of Long Range Dependent Traac , 1998 .

[26]  Gregory W. Wornell,et al.  Estimation of fractal signals from noisy measurements using wavelets , 1992, IEEE Trans. Signal Process..

[27]  Walter Willinger,et al.  Self-Similar Network Traffic and Performance Evaluation , 2000 .

[28]  Guitton Henri,et al.  Duncan Luce (R.), Howard Raiffa- Games and decisions ; Feller (W.) - An introduction to probability theory and its applications. Vol. I. Second edition. , 1960 .