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Fig. 2.4 Logscale diagrams with biscaling. Internet UDP packet data displaying two scaling regimes, examples of biscaling. Left: Delay series. Regime I on the left (small j's) is related to continuous but nondifferentiable sample paths h P 0; 1 and regime II (large j's) to longrange dependence. Right: Loss series (1 for lost packets, else 0). Regime I corresponds to discontinuous sample paths h < 0 . In regime II there is trivial white noise scaling a 0 , indicating stationary SRD behavior.

the context of delays (left) and losses (right) experienced by consecutive user datagram protocol (UDP) packets sent over the Internet in a regular stream (see Andren et al. [12] for details of such data sets). In both gures the alignment at large scales corresponds to long-range dependence, whereas at small scales it is associated with highly irregular sample paths. Note that when second-order properties are insuf cient to fully describe the scaling nature of the process (an extreme example is afforded by multifractals), then the correct interpretation of each branch of the bisecting will require the examination of logscale diagrams across a range of orders, as discussed in Section 2.3.3. 2.3.2 Estimation Within the Logscale Diagram

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Because Java has separate namespaces into which classes can be loaded, it can't simply have a unified "phone directory" tracking which class corresponds to which classname. Instead, the Java Virtual Machine maintains a separate directory for each class. These independent directories keep track of the names needed by each class. For example, if class A has a reference to a class called B, the directory for A will have an entry for B that points to the actual class the name represents. Figure 5.5 shows a more complicated example with four classes referencing each other. A big applet could consist of more than four classes, but the idea is the same: The applet is a set of classes that reference each other.

In this subsection it is assumed that a scaling range j P j1 ; j2 has been correctly identi ed Sums over j, and regressions, are always taken over this range The estimators to be de ned are semiparametric, as they depend on the range of scales j P j1 ; j2 , where scaling is deemed to be present, and the scaling property P1 valid there, but not on any tightly speci ed parametric model 2321 Estimating the Scaling Exponent a Because of property P1, the measurement of a is reduced to the determination of the slope over the alignment region in the logscale diagram A natural way to achieve this in a statistical estimation context is through linear regression, the de ning hypothesis of which is Eyj a j a, where a is a real constant.

changes on a financial statement, are filling the role of data guardian when working in concert with the service directory. Figure 12-2 shows that these roles and rights would be difficult to do with just inheritance features and must be doled out with the aid of a directory where the identity is known and treated for access rights in a predetermined way after authentication. If you are not the data creator or owner and you have access to or custody of the data, you are considered for this discussion an agent. Agents can receive rights from other agents, but only if the granting agent has those rights.

Because in general E log T log E , this condition is not exactly satis ed, however We therefore introduce small corrective deterministic factors g j , discussed below, and rede ne the yj as yj log mj g j , so that Eyj aj a by de nition Any kind of linear regression of yj on j constitutes an unbiased estimator of a, as the lack of bias does not require decorrelation between the yj , nor knowledge of their variances or distributions A weighted regression where the weights are related to the variances s2 of the yj is preferable, however, as this is the minimum variance j unbiased estimator (MVUE) for the regression problem [48] Intuitively this re nement is signi cant as we know that the s2 are far from equal.

Figure 5.5 Four classes linked together. Each box represents a class. The circled name at the top of each box is the name of the class, and the entries underneath show how the class's namespace is defined.

To exploit the j optimality, however, the correlation factors g j and the variances s2 must be j calculated a dif cult task They can nonetheless be well approximated, provided simplifying idealized properties are adopted The particular idealizations chosen here are the following: ID1a: For each xed j the dX j; are stationary sequences of uncorrelated variables IDIb: The processes dX j; and dX jH ; , j T jH , are uncorrelated ID2: The process X and, hence, the processes dX j; are Gaussian The above conditions may appear unduly restrictive at rst; however, the underlying effectiveness of the method is based on P1 and P2, ID1 ID2 being added.

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