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4. Homework

Try experimenting on some simple curves, such as the graph of a Weierstrass function or a path of a fractional Brownian motion. Check that the regularization dimension estimator always gives superior results as compared to the box dimension estimator.

Try also the following: synthesize a deterministic Weierstrass function with the default parameters, except that you put Sample Size = 4096. Then add to the output signal, Wei0, a white Gaussian noise (type first "x= randn(4096,1);" in your matlab window, then "y = Wei0 +0.2*x;". Import y to Fraclab's workspace by clicking on Scan Workspace and selecting y in the window titled Import Data from Matlab Workspace that appears). Compare visually Wei0 and y, then compute the regularized dimensions of those two signals: Use first the default options. In the case of Wei0, the points in the graph showing the logarithms of the length of the regularized versions are reasonably well aligned at least between abscissa 2 and 4, and, by selecting this region with the cross, you get an estimated dimension of around 1.52, which is not too bad. If you analyze y, however, you see that there is not significant region in the graph showing the logarithms of the length of the regularized versions where a linear behaviour holds. Set then StD = 0.2, and estimate again on y. You'll see as previously on the graph the small black circles corresponding to y, and, in addition, red stars that show the estimator corrected to take into account the noise. The red stars are close to the circles to the right of the graph, then depart from them significantly as we move to the left. Here is why: Points on the right correspond to a large amount or smoothing, or "low frequencies", for which the signal will dominate over the noise. In this region, there is not much to compensate for, as the lengths of the smoothed original and the smoothed noisy signal should not be too different. On the extreme left, however, we are looking at high frequencies (i.e. almost no smoothing), and we mainly analyze noise: The length of the original regularized signal is here significantly smaller that the observed one: this is why the estimated "true" length (the red stars) are well below the measured length (the black circles). The red stars in the region between abscissa 2 and 4 should again be roughly aligned, and selecting this range with the cross should still give you an estimated dimension not too far from 1.5.


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