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...ly LRD. The bucket size of the internal randomization or the band of frequencies to be removed depend on how short-range is defined on the specific process. Especially, suggestions such as the one in =-=[15]-=-, where the Hurst exponent is estimated using explicitly only short-range correlations are bound to produce misleading results in any realistic dataset. This approach is based on solving Eqn. 3 for k ...

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... = H − 1/α + 1/2 by the well known aggregational variance method (see, for example, Taqqu, Teverovsky and Willinger [24]), or by a simple but still very useful method introduced in Kettani and Gubner =-=[12]-=-. Both methods are based on the second order selfsimilarity property. Again, our numerical experiments show that estimation via the method of [12] is fairly accurate. The inner measure ρ can also be e...

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...rd deviation and T, time. This leads to the Hurst parameter estimator: ˆH ∼ log( RT )/ log(T ). ST 2. Estimation of the Hurst parameter. While there are many available methods for the estimation of H =-=[2,3,4,5,6,7,8,9,11]-=-, this paper will apply rescaled range (R/S) [2], dispersional analysis (DA) [5], and bridge-detrended scaled window variance (bdSWV) [2,6], because they are well understood and yielded meaningful res...

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... subsequent time intervals between events. Stream of events (5) is asymptotically self-similar with parameter , if (11) The measure of self-similarity is the Hurst parameter introduced by H. E. Hurst =-=[8]-=-. The Hurst parameter for self-similar processes is in the range of . For two identical processes, . Lower values of Hurst parameter indicate larger differences in processes, and for , processes are n...

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...rrelation among data traffic smooths out with longer time scale and the process becomes memoryless. Nonetheless, it is this fractal [10] behaviour that makes the self-similar model to capture the burstiness of internet traffic. There are several properties of a self-similar process which can be summarized as follows [9] - 1. It is a stochastic process that shows the Long Range Dependence. 2. The distribution is fractal like, which means that the process shows the same characteristic at any scale. 3. Mathematically the self-similar process is described by a parameter called the Hurst Parameter [7]. This parameter defines the degree of self-similarity. 4. It has a slowly decaying variance which is the most salient feature of the self-similar process from the statistical point of view. Several studies have shown that the internet traffic follows the Self-similar model [9, 15] and the mathematical model for Self-Similarity can be found here [12, 18]. Although, the self-similar model captures the internet traffic most accurately, the complexity of this model with several parameters to be considered makes it difficult to analyse and computationally expensive to use. 2.3 On/Off Model A model...

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...hy lrd is also called Self-Similarity [26]. III. HEAF: A ‘HURST EXPONENT BY AUTOCORRELATION FUNCTION’ ESTIMATOR A new estimator has been introduced [1] by extending the approach of Kettani and Gubner =-=[27]-=-. As in [27], for a given observed data i X (i.e. X 1 ,......... , X n ), the sample autocorrelation function can be calculated by the following method: 1 n Let ˆμ n = ∑ X i n i= 1 (3.1) 1 n k and γˆ ...

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...a pollutes the wavelet coefficients, if better estimation is attempted through a higher value of a particular parameter (in their analysis), and a practical compromise is in order. Kettani and Gubner =-=[14]-=- propose a simple estimator for the Hurst parameter based on the known relation between the normalized first autocovariance coefficient of exactly self-similar signal and the Hurst parameter. The auto...

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...ess of an equidistant discretization of fBm, and define Y n := 1 ∑n n k=1 Yk for n ∈ N. The HEAF estimator (HEAF meaning ‘Hurst Exponent from Autocorrelation Function’) proposed by Kettani and Gubner =-=[14]-=- puts the estimate ̂ρn = ∑ n−1 k=1 (Yk − Y n)(Yk+1 − Y n) ∑n k=1 (Yk − Y n) 2 , of ρH(1) into the monotonic functional relation ρH(1) = 2 2H−1 − 1 obtained from (7). In order to receive only finite no...