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2. Regularization Dimension

This dimension is defined is the following way: One first computes smoother and smoother versions of the original signal, obtained simply through convolution with a kernel. Now, if the original signal is "fractal", its graph has infinite length, while all regularized versions have finite length. When the smoothing parameter tends to 0, the smoothed version tends to the original signal, and its length will tend to infinity. The regularization dimension measures the speed at which this convergence to infinity takes place. In many cases, this will coincide with the usual box dimension. In general, it can be shown that the regularization dimension is more precise than the box dimension, in the sense that it is always smaller, but still larger than the Hausdorff dimension. In addition, the regularization dimension lends to more robust estimation procedures for various reasons. One of them is that we may choose the regularization kernel. Also, the smoothed versions are adaptive by construction. Finally, the smoothing parameter can be varied in very small steps, as box sizes have to undergo sudden changes. Another advantage is that, due to the fully analytical definition of the regularization dimension, it is easy to derive an estimator in the presence of noise.

Check first, as usual, the Input data name. The current implementation of the regularization dimension allows to deal with both 1D and 2D signals, in a way which is transparent to the user: Just input your signal, and Fraclab will recognize its type.

Second, you need to decide on the minimum and maximum amount of smoothing. This is done by specifying the corresponding sizes for the kernel, using the Nmin and Nmax parameters. This is expressed in sample units, i.e. a value of 5 means that your kernel will have a "width" of 11 sample points (the precise definition of the width depends on the kernel). You then choose a Kernel shape among Gaussian and Rectangular. The width in the Gaussian case in simply the standard deviation, while it is the number of non zero coefficients in the case of the rectangular kernel. The Voices parameter lets you choose how many smoothed curves you want to compute. A value of, e.g., 64, means that 64 smoothed signals will be generated with smoothing parameters regularly spaced between Nmin and Nmax. As alluded to above, the estimation of the regularization dimension can accommodate for the presence of additive white Gaussian noise in the data. If you want to use this feature, just enter the standard deviation of the noise in the StD box below the Noise heading (estimate your noise using any classical method, or use the built-in estimation available in the Denoising menu). As usual, the dimension will be estimated through regression, and you may choose which type of regression you will use, i.e. Least Square Regression, Weighted Least Square, Penalized Least Square, Maximum Likelihood or Lepskii Adaptive Procedure. All these methods are well-known except the last one, for which you may consult reference (2). If you select Range Specify, you will be able to choose interactively a region where an approximate linear behaviour holds (the next paragraph details how to do this). Otherwise, set the Range to Automatic. It is sometimes instructive to look at the Regularized graphs: when this option is checked, you will get a graphic window displaying all the regularized version of your signal (these will be contour plots in case you are dealing with an image). No regularized graphs is the default. In case you are interested in keeping the regularized versions for further processing, check the Save regularized graphs button. Forget regularized graphs is the default (note that this option is disabled if No regularized graphs has been checked). Be careful that, if you use a large number of voices and you keep the regularized graphs, you will add a large number of variables to your environment (each regularized graph is a distinct structure).

You are now ready to hit the Compute button. If you decided to use the Automatic range, then you will simply get the estimated dimension to the right of the box Regularization Dimension=. If you selected Range Specify, then a graphic window will pop up. In both graphs of this window, abscissa represents the logarithm of the smoothing parameter. In the lower graph, the ordinate represents the logarithm of the length of the regularized versions. If the parameter StD is 0, the upper graph simply represents the increments of the lower graph. This device is useful because it allows to emphasize more clearly a possible linear behaviour (linearity in the lower graph translates into constancy in the upper graph). Otherwise (if StD is positive), the upper graph displays a curve related to the relative strength of the noise and the signal at all scales (see reference (1) for more). Using the cross that appear when you point inside the graphic window, choose a region where approximate linearity holds: Select this region by clicking on its endpoints. A red line showing the regression will appear, and the corresponding estimated dimension will be displayed above the lower graph. Repeat this selection operation until you are satisfied, then hit return on your keyboard. The cross will disappear, and the final estimated dimension will be displayed to the right of the box Regularization Dimension=, at the very bottom of the Regularization Dimension window.

Note that if you select both Range Specify and Regularized graphs, the graphic window displaying the regularized graphs will appear first, then the window for the regression range selection will appear on the top of it and will mask it. Just move it if you want to inspect the regularized graphs.


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