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3. Local Hölder Exponent

The canonical example that highlights the difference between the two exponents is the so-called chirp (x^a)sin(x^(-b)), with a,b positive: In this case, the pointwise exponent is a and the local one is a/(b+1). The local exponent is always smaller than the pointwise one. The situation where both exponents coincide is favorable, since it usually ease the estimation (for instance, it is a necessary condition for the basic GIFS and wavelet estimators to be valid). Such is the case for instance for the Weierstrass function, the generalized Weierstrass function with smooth h(t), the fractional Brownian motion, or the multifractional Brownian motion with smooth H(t). On the contrary, lacunary wavelet sequences such as the ones available in the synthesis menu are examples of signals which have almost everywhere almost surely different exponents. Although a wavelet-based method could easily be developed for estimating local exponents, only a oscillation-based one is implemented in Fraclab, as it gives generally better results. The main difference with the estimation of the pointwise exponent is that one does not compare the oscillation with the size of the neighbourhood, but with the distance between the two points where the oscillation is attained.

Again, you first check the Input data name box and Refresh it if needed. You then need to choose what are the minimal and maximal sizes of the neighbourhood that will be used to investigate the behaviour of the oscillations. Enter the appropriate values in the Nmin and Nmax boxes, either directly or using the menus. Any integer will do, as long as Nmin is smaller than Nmax and Nmax is compatible with the size of the signal. These values should be understood as follows: Nmax = 8, for instance, means that the largest neighbourhood will be composed of 8 points to each side of the point where one wishes to perform the estimation. In other words, the neighbourhood will be a window of size 17 sample points centered at the point of interest. The same remarks about the choice of values for Nmin and Nmax as in the case of the pointwise exponent apply. Compared to the case of the estimation of the pointwise exponent, there is an additional parameter here, called Neighbourhood size. The Neighbourhood size parameter mainly acts as a smoothing filter.

Finally, you may choose a Regression Type from the usual choice of Least Square Regression, Weighted Least Square, Penalized Least Square, Maximum Likelihood and Lepskii Adaptive Procedure (see reference (6)). On hitting Compute, you get the output data pht_signal#, which contains the estimated exponent at each point.

Note finally that, as in the case of the pointwise exponent, more robust estimates of the local exponent are available when one computes the 2-microlocal exponents. See the help corresponding to this topic below.

A known bug in this sub-menu is that choosing a value for Nmin larger than the default of 1 often results in an error. It is not understood yet in which situations this occurs. You might want to restart the whole process, i.e. close the window and open it again, as this often solves the problem.

It is interesting to compare the results obtained by the oscillation-based methods for computing the pointwise and local exponents: One verifies that the latter is always smaller, and that it varies much more smoothly in general.


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