Next Previous Contents

7. Homework

We have already used some of the routines of the estimation menu in the homework section of the Overview and main functionalities of this help.

We will here compare the various methods for estimating the local and pointwise exponent functions (i.e. the exponents at all points of the signal) on a test signal, namely a generalized Weierstrass function with linearly increasing regularity. In that view, first synthesize this signal: Go to Synthesis/Functions/Deterministic/Generalized Weierstrass. Choose h(t)=t as a Hölder Function. Set the Sample size to 1024, leave lambda unchanged, and set the Sum Order to 30. Hit Compute to obtain the signal Gwei0. View it in the usual way, and notice how the regularity visibly increases from left to right.

The signal GWei0 has, at all point t in (0,1), both local and pointwise exponent equal to t. We shall see that estimating the regularity exponents is not an easy task in general, and that some methods give an acceptable result on this particular signal. We shall first test the CWT method: Go to 1D Exponents Estimation/Pointwise Holder Exponent/CWT based estimation. In the window that appears, click on Advanced compute. Now compute the CWT of GWei0 as explained above: Check first that the Input Signal is indeed GWei0, otherwise Refresh it in the usual way. Keep the default parameters for fmin, fmax, Voices, Mirror, Size and Type of the analyzing wavelet. Hit Compute WT. The wavelet transform cwt_Wei00 should appear in the Input CWT box. Uncheck the box Single Time Exponent, so that it becomes Hölder Function. Leave the other parameters unchanged and hit Compute. After a short while, you'll get a new signal GWei0_Ht0, that contains the estimated pointwise exponent. View this signal. As you'll see, this is very far from being the expected y=x line. In order to understand why we are so much off, we shall now try and estimate the exponent at a single point. In that view, click on Hölder Function which now turns to Single Time Exponent. In the Time instant box, enter, e.g., 300. Leave the other parameters unchanged, and hit Compute. A graphic window appears: On its left part you can see the points in the scale/space plane that were selected as local maxima. On the right part is displayed a log-log plot of the magnitude of the wavelet coefficients at these points with respect to scale. The slope, currently indicated by the red line, is supposed to give the Hölder exponent. With the cross that appears when you point to this window, select a regression range (by default, this is the whole available range). Selecting the range between points 4 and 7, you'll get an estimated exponent of around 18. Between points 2 and 4, you'll get something like -1. As a matter of fact, you can obtain almost any exponent by clicking in an adequate region. The problem here is that the exponent is governed by the extremal values of the wavelet coefficients. A robust method should then compute the regression only considering those extremal values. Unfortunately, it is not easy to detect which values are indeed extremal.

Once you're finished with testing various regression ranges, hit return on your keyboard and close this window. We shall now try the GIFS based estimation. In the Variables list of the main window, select GWei0, and go to 1D Exponents Estimation/Pointwise Holder Exponent/GIFS based estimation. Verify that the Input is what you want, and hit Compute with the default parameters. The estimation, called alphagifs_GWei00, quickly appears in the Variables list. On viewing it, you'll see that the general trend is what we were expecting, with a number of small oscillations around it. The slope is somewhat smaller than 1, but we do get the impression of a signal with steadily increasing regularity.

Our third method is the oscillation based one. Again, select GWei0 in the Variables list, and go to 1D Exponents Estimation/Pointwise Holder Exponent/oscillation based method. Check your Input data, and hit Compute with the default parameters. You get the estimation in the signal called pht_GWei00: We obtain again a roughly correct general trend, with more oscillations than in the GIFS based estimation, but with a better slope (you can verify this by viewing both pht_GWei00 and alphagifs_GWei00 on the same graph, using the hold facility in the View menu). You can reduce a bit the oscillations by choosing Nmin= 4 and Nmax= 64

In the fourth experiment, we shall compute the local exponent: select GWei0 in the Variables list, and go to 1D Exponents Estimation/Local Holder Exponent/oscillation based method. Check your Input data. Hit Compute with the default parameters. The output is called pht_GWei01. The same remark as above applies, i.e. we get a lot of oscillations with a roughly correct slope. Again, you can minimize the oscillations by choosing Nmin= 4 and Nmax= 64. It is interesting to check that the local estimation is, at (almost...) all points, a lower bound to the pointwise one, as predicted by the theory.

Finally, we'll see what happens when using a more advanced method based on 2-microlocal analysis. Select GWei0 in the Variables list, and go to 1D Exponents Estimation/2-microlocal Exponent/Oscillation based (1). In the window that appears, hit Refresh so that the name of the Input data becomes GWei0. Choose a window size of 100 instead of the default 1024. Since the computations are a bit long with this method, we'll perform the estimation only every tenth point. Thus, we set the step parameter to 10. Finally, click on frontier and select local instead. Hit Compute. You will probably wait for less than a couple of minutes until the outputs, called lenlocd_GWei00 and Gloc_GWei00 appear. The estimated local exponent is Gloc_GWei00 and this is the one we shall view. As above, we get a curve that increases in the mean between 0 and 0.8, although this time there is a larger number of points where the local exponent is grossly underestimated. We shall also estimate the pointwise exponent with this method: click again on local and this time select pointwise instead. Hit Compute. View the output called Gpt_GWei00. This estimate is a little bit better that the local one, with fewer underestimated values. Recall however that, in both cases here, the fact that we estimate only every tenth exponent make the underestimated areas look more important. If you are very patient, you may check this by launching the estimation with a step of 1 instead of 10.


Next Previous Contents