Modeling Solar Radiation at the Earth’s Surface
New Methodfor Estimating the Fractal Dimension of Discrete Temporal Signals
In order to contribute in improving the accuracy of fractal dimension estimation of the discrete temporal signals we developed a simple method based on a covering by rectangles called Rectangular Covering Method.
Presentation of the Method
The method based on Minkowski-Bouligand approach consists in covering the curve for which we want to estimate fractal dimension by rectangles. The choice of this type of structuring element is due to the discrete character of the studied signals.
From the mathematical point of view, the use of the rectangle as structuring element for the covering is justified. Indeed, Bouligand (1928) showed that DM
(Minkowski-Bouligand dimension) can be obtained by also replacing the disks in the previous covers with any other arbitrarily shaped compact sets that posses a nonzero minimum and maximum distance from their center to their boundary.
 |
 |
Thus, as shown in Fig. 2.4, for different time intervals At, the area S(At) of this covered curve is calculated by using the following relation
where N denotes the signal length, f (tn) is the value of the function representing the signal at the time tn and |f (tn + At) - f (tn) | is the function variation related to the interval At. The fractal dimension is then deduced from Eq. (2.20) where є is replaced by the time interval At. Hence
(2.20)
Thus, to determine the fractal dimension D which represents the slope of the straight line of Eq. (2.20), it is necessary to use various time scales At and to measure the corresponding area S(At). We then obtain several points (At,, S(Aті)) constituting the line.
 |
 |
A good estimation of the fractal dimension D requires a good fitting of the log - log plot defined by Eq. (2.20). Therefore, the number of points constituting the plot is important. This number is fixed by Аттах which is the maximum interval through which the line of the log-log plots is fitted.
< > Jr
Fig. 2.4 An example of temporal curve covered by rectangles
As mentioned above, to estimate the fractal dimension most of methods determine ATmax experimentally. This procedure requires much time and suffers from precision. Also, we developed an optimization technique to estimate ATmax.
Optimization Technique
Experience shows that ATmax required for a good estimation of D depends on several parameters, especially the time length of the signal N. At should not be too weak, in order not to skew the fitting of the line, and it must not exceed N/2. At must also satisfy the condition of linearity of the line.
Our optimization technique (Harrouni et al. 2002) consists first in taking a ATmax initial about 10, because the number of points constituting the plot should not be very small as signaled above; then, A Tmax is incremented by step of 1 until N/2. We hence obtain several straight log-log lines which are fitted using the least squares estimation. The ATmax optimal is the one corresponding to the log-log straight line with the minimum least square error. This later is defined by the following formula
n
I dr
Equad = — (2.21)
n
In this relation n denotes the number of points used for the straight log-log line fitting, di represents the distance between the points (ln(1/At) ,ln (S(At) /At2)) and the fitted straight log-log line.
Validation of the Method
In order to test the validity and the accuracy of the rectangular covering method, we applied it to two different types of parametric fractal signals whose theoretical fractal dimension is known, these test signals are the Weirstrass function (WF) which is a deterministic signal and the random signal of the fractional Brownian motion (FBM). These fractal signals that will be briefly defined below are most commonly used in various applications.
The Weierstrass Function (FW): It is defined as (Hardy 1916; Mandelbrot 1982; Berry and Lewis 1980)
WH (t) = Iy-kHcos (2Л-/Л, as 0 < H < 1 (2.22)
This function is continuous but nowhere differentiable; у is an integer such as у > 1. This parameter is fixed by the experimenter so that he can choose the shape of the signal, the fractal dimension of this function is D = 2 - H. In our experiments, we synthesized discrete time signals from WF’s by sampling t e [0,1] at N +1 equidistant points, using у = 2.1 and truncating the infinite series so that the summation is done only for 0 < к < kmax. The kmax is determined by the inequality 2кук < 1012 established by Maragos and Sun (1993).
Fractional Brownian Motion (FBM): It is one of the most mathematical models used to describe self-affine fractals existing in nature. Mandelbrot and Wallis proposed an extension of this motion: the fractional Brownian motion. The function of the Brownian fractional motion BH (t) with parameter 0 < H < 1 is a time varying random function with stationary, Gaussian distributed, and statically self-affine increments. So
{[Bh (t) - Bh (t0)]2) = 2D (t - t0)2H, as 0 < H < 1 (2.23)
The fractal dimension D of BH (t) is D = 2 - H. To synthesize FBM signals, several methods exist (Mandelbrot and Wallis 1969; Voss 1988; Lundahl and all. 1986) the most known are: Choleski decomposition method, Durbin-Levinson algorithm, FFT method and circulant matrix method. In our experiments we synthesized FBM signals via the Durbin-Levinson method.
To validate our rectangular covering method we applied it to these synthesized test signals. For this purpose, the error between the theoretical fractal dimension and the estimated one is used. The experimental results indicate that, for the two fractal signals WF and FBM, the rectangular covering method performs well in estimating dimensions D є [1.1,1.9], since the estimation error is less than or equal 6 % for the WF signals and 7 % for FBM signals (Harrouni and Guessoum 2006). By varying the signals’ length N є [100,1000] with a step of 100 we have also observed similar performance of this method. Over 99 different combinations of (D, N) the average estimation error of the rectangular covering method was 4 % for both WF’s and FBM’s.
