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Conversely, if equation (113) holds, and if f is uniform H lderian, then f C (t0 ) From this theorem and with the simpli cation adj( j (t0 )) = j (t0 ) only, the new estimator is determined At each point t0 of the signal X decomposed on n scales, the regularity is estimated by the following formula: 1 sj log2 max(|x |) W L (n, t0 ) = Kn 2 j=1 1134 Limit inf and limit sup regressions The de nition of the H lder exponent makes use of a lower limit, which allows the exponent to exist without conditions The three estimators presented above, however, calculate the exponent through a linear regression Typically, they will not converge to the true exponent when the resolution tends to in nity if the expression de ning the exponent does not converge, ie when the lower limit is not a plain limit Indeed, if the upper and lower limits are different, the slope given by a linear regression has no relevance However, as shown in [LEG 04b, LEV 04a], it is still possible to use regressions to obtain the exponent The method is general, as it applies to the estimation of upper and lower limits through a modi ed regression scheme, that we proceed to explain now The use of these liminf and limsup regression methods is of great practical importance, as it allows us to estimate, on arbitrary signals, various fractal quantities such as dimensions, exponents, and multifractal spectra Note that, even for fractal signals, the H lder exponents are often obtained as genuine lower limits (ie the limit does not exist)

malicious activities Since the amount of data usually collected by sensors is very large for humans to go through, analysis engines that can examine all of the available data collected by NIDS and HIDS are very important Analysis of IDS data involves consolidating the data possibly at a central location and identifying malicious activities automatically to the extent possible In some cases the HIDS and NIDS sensors themselves contain some preliminary analysis capabilities Such a capability allows the IDS to detect the attack locally at the sensor, increasing detection speed and allowing faster reaction Such a capability also minimizes the need to transfer all of the collected data to a centrally located analysis engine, reducing the amount of overhead data that is generated Analysis engines may utilize a variety of techniques for detecting malicious behavior The two most widely used techniques are:

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j Let (lj )j 1 be an arbitrary sequence of real numbers, and denote uj = j Let a = lim inf j uj In our framework, think for instance of lj as the logarithm of the sizes of balls used in the oscillation calculations De ne, for all n 1:

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0 Vn = {1, , n}

Let (a0 , b0 ) be the parameters of the least square linear regression of L0 with n n n n 0 respect to Vn , ie the real numbers that minimize j=1 (lj aj b)2 over all couples (a, b) We write:

Now let:

.

1 0 Vn = {j Vn , lj a0 j + b0 }, n n 1 L1 = {lj , j Vn }, n 1 (a1 , b1 ) = Reg(Vn , L1 ) n n n

and de ne recursively:

i i 1 Vn = {j Vn , lj ai 1 j + bi 1 }, n n i Li = {lj , j Vn }, n i (ai , bi ) = Reg(Vn , Li ) n n n

The geometric interpretation of the sequence (ai , bi ) is simple: In the rst step, n n 1 we keep in (Vn , L1 ) those points that are below the regression line of L0 with n n 0 1 respect to Vn We then calculate the regression line of L1 with respect to Vn to obtain n (a1 , b1 ), and iterate the process until at most one point remains below the regression n n line The slope of the liminf regression is then de ned as aNn (the method is similar n for the limsup, just keep the point above the regression line) In many cases of interest, aNn will tend to a when n tends to in nity n 1135 Numerical experiments In this section we compare the three methods W CR, W L and OSC on different kinds of signals For more experiments, see [LEG 04b] Figure 111 represents a generalized Weierstrass function with a regularity H(t) = t and the estimations of the H lder function by regression of the wavelet coef cients (W CR), by wavelet leaders (W L) and by oscillation (OSC) This experiment shows that the best results are obtained by the W L method in this case

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