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traf cs that circulate between regional networks, in other words on the Internet, we often nd scale invariance, and almost never nd traf c obeying the traditional rules Certain types of digitized video traf c also exhibited the phenomenon (see Beran et al [BER 95]), though over a less spectacular range than for Ethernet In 1995, Willinger et al [WIL 95] returned to Ethernet data for a more re ned analysis, examining not only the traf c itself but also its components, and once again found scale invariance We will consider this example and its theoretical offshoots in section 123 Numerous discoveries followed (for a more complete list, see the bibliography in [WIL 96]) and almost always scale invariance was found, as well as heavy tails for many quantities such as the length of bursts By this quali er heavy , we understand a slow decrease in probability density at in nity, which gives rise to in nite variances or even means Despite the weight of abundant empirical evidence, resistance against this new wave in traf c was not slow in showing itself Essentially, this resistance was divided between those who preferred to believe that the evidence was in fact merely the artifact of corruptive non-stationarities, and others who thought that a continuum of scales could be effectively approximated by a suf ciently large number of discrete scales, obviating any need to talk about fractals Although it is true that, in many cases, such objections were merely a re ex against the shock of new ideas, it is nevertheless true that non-stationarities can very well be confused with the increased variability of scale invariant processes One of the rst methods used to detect scale invariance and to estimate its exponent was based on equation (123) From X (m) , we estimate v (m) using the standard variance estimator, and then plot its logarithm against log(m), the time-variance plot The slope in this plot corresponds to an estimate of This method, although simple and with a low calculation cost, suffers from many statistical defects: a notable bias, a quite large variance, and in particular a poor robustness with respect to non-stationarity This last defect is shared, at least partly, by many other methods (see [TAQ 95] for a comparison of several methods) The need to measure in a reliable way stimulated further contributions to the already sizable statistical literature In fact, the high volume of teletraf c data excluded the use of the estimators typically in use, which had good statistical performance, but very high computational costs A semi-parametric method based on wavelets, introduced into the eld by Abry and Veitch [ABR 98], solved these problems thanks to its low complexity, only O(n), without sacri cing robustness, and excellent statistical performance due to the natural match between wavelet bases and scale invariant processes Wavelet analysis operates jointly in time and scale It replaces the time signal X(t) with a set of coef cients, the details dX (j, k), j, k Z, where 2j denotes the scale, and 2j k the instant, around which the analysis is carried out In the wavelet domain, equation (123) is replaced with var[dX (j, k)] = cf C 2(1 )j , where the role of m is played by scale, of which j is the logarithm, cf is the frequency analog of c and is proportional to it, and where C is independent of j The analog of the variance-time plot, the graph of the estimates of log(var[dX (j, )]) against j, is called the logscale diagram and constitutes a spectrum estimate of the process, where low frequencies correspond to large scales (on the
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