WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA

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Fig. 2.3 LRD and H-sssi behavior in Ethernet traf c data. Left: Logscale diagram for the discrete series of successive interarrival times, showing a range of alignments and an a estimate consistent with long-range dependence. Right: Logscale diagram for the cumulative work process (bytes up to time t), consistent with an asymptotically self-similar (close to exactly self-similar) process with stationary increments.

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If an estimate of the scaling exponent a is found to lie in (0, 1), and the range of scales is from some initial value j1 up to the largest one present in the data, then the scaling could be said to correspond to long-range dependence with a scaling exponent that is simply the measured a. Examples are afforded by the left-hand plots in each of Figs. 2.2, and 2.3. If there were a priori physical reasons to believe that the data were stationary, then long-range dependence would be an especially relevant conclusion. This applies to the left-hand plots in Fig. 2.3, as the series corresponds to successive interarrival times of Ethernet packets, which under steady traf c conditions one would expect to be stationary (the Ethernet data in Fig. 2.3 is from the ``pAug'' Bellcore trace [43]). Another key example, illustrated in the right-hand plots in Figs. 2.2 and 2.3, is a value of a greater than 1 but also measured over a range including the largest scales. Such a value precludes long-range dependence and may indicate that a self-similar or asymptotically self-similar model is required, implying that the data are nonstationary. The exponent would then be reexpressed as H a 1 =2, the Hurst parameter. Again conclusions should be compared with a priori physical reasoning. The right-hand plot in Fig. 2.3 is the analysis of a cumulative work process for Ethernet traf c, that is, the total number of bytes having arrived by time t. Such a series is intrinsically nonstationary, though under steady traf c conditions one would expect it to have stationary increments. Thus a conclusion of nonstationarity is a ^ natural one, and the estimated value of H 0:80, being in (0,1), indeed corresponds to an H-sssi process. It would have been problematic, however, if underlying physical reasons had indicated that in fact stationarity was to be expected. Such an apparent paradox could be resolved in one of two ways. It may be that the underlying process is indeed stationary and exhibits 1=f noise over a wide range of scales, but that the data set is simply not long enough to include the upper cutoff

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2.3 WAVELETS AND SCALING: ESTIMATION

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scale. The alternative is to accept that empirical evidence has shown the physical reasoning concluding stationarity to be invalid. If, on the other hand, the scaling was concentrated at the lowest scales (high frequencies), that is, j1 1 with some upper cutoff j2 , then the scaling may best be understood as indicating the fractal nature of the sample path. The observed a should then be reexpressed as h a 1 =2, the local regularity parameter. Values of h in the range (0, 1], for example, would then be interpreted as indicating continuous but nondifferentiable sample paths (under Gaussian assumptions [28]), as observed in the leftmost alignment region in the Internet delay data in the left plot in Fig. 2.4. The stationarity or otherwise of the data in such a case may then not be relevant. Note that j 1 has been excluded from the leftmost alignment region in each plot in Fig. 2.4. This is not in contradiction with interpretations of fractality, as it is known that the details at j 1 can be considerably polluted due to errors in the initialization of the multiresolution algorithm (see Section 2.2.1.4). If scaling with a > 1 is found over all or almost all of the scales in the data, such as in the right-hand plot in Fig. 2.2, then exact self-similarity could be chosen as a model, again with H a 1 =2 being the relevant exponent. However, in this case one could equally well use the local regularity parameter h a 1 =2, with the interpretation that the fractal behavior at small scales is constant over time and happens to extend right up to the largest scales in the data. Finally, more than one alignment region is certainly possible within a single logscale diagram, a phenomenon that we refer to as biscaling. One could imagine, for example, fractal characteristics leading to an alignment at small scales with one exponent, and long-range dependence resulting in alignment at large scales with a separate scaling exponent. Examples of this phenomenon are shown in Fig. 2.4 in

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