NEW METHOD FOR ESTIMATING THE TIME SERIES FRACTAL DIMENSION:APPLICATION TO SOLAR IRRADIANCES SIGNALS

In: Solar Energy: New Research ISBN 1-59454-630-4
Editor: Tom P. Hough, pp.277-307 © 2006 Nova Science Publishers, Inc.
Chapter 9
NEW METHOD FOR ESTIMATING THE TIME
SERIES FRACTAL DIMENSION: APPLICATION
TO SOLAR IRRADIANCES SIGNALS
S. Harrouni,
∗
Solar Instrumentation & Modeling Group / LINS – Faculty of Electronics and Computer
University of Science and Technology H. Boumediene (USTHB)
P.O Box 32 - El-Alia - 16111 Algiers – Algeria
A. Guessoum ∗∗
Signal Processing and Imagery Laboratory, University of Blida
P. O Box 720 – Blida – Algeria
ABSTRACT
To electrify remote areas, the use of solar energy is the best economical and
technological solution. The choice of the sites for the installation of photovoltaic systems
and analysis of their performance requires the knowledge of the solar irradiation source.
The evolution of the solar irradiation obeys to seasonal variations and random
fluctuations. These latter considerably affects the behavior of the PV systems, therefore it
is necessary to quantify them. The concept of fractals may then be used for modeling the
solar irradiances fluctuations. For daily solar irradiances, the fractal dimension ranges
from 1 to 2. D close to 1 describes a clear sky state without clouds while a value of D
close to 2 reveals a perturbed sky state with clouds. Indeed, a straight line has a fractal
dimension of one, just like its Euclidean dimension, once the line shows curls, its fractal
dimension increases. The curling line will fill the plane more and more, and once the
plane is filled up, it has a fractal dimension of two. Thus, the fractal dimension of a
temporal signal has a fractal dimension between 1 and 2.
∗ [email protected]
∗∗ [email protected]

S. Harrouni and A. Guessoum 278
To measure the fractal dimension of time series, several methods and algorithms based on
various coverings has been elaborated. In order to improve the complexity and the
precision of the fractal dimension estimation of the discreet time series, we developed a
simple method based on a multi-scale covering using the rectangle as a structuring
element of covering. An optimization technique has been associated to this method in
order to determine the optimal time interval
∆τ
max through which the line log-log is fitted
whose slope represents the fractal dimension. This optimization technique is permitted to
improve the precision and the time computing of the method.
In order to measure the performance and the robustness of the proposed method, we
applied it to fractal parametric signals whose theoretical fractal dimension is known,
namely: the Weierstrass function (WF) and the fractional Brownian motion (FBM).
Experimental results show that the proposed method presents a good precision since the
estimation error averaged over 180 tests for the two types of signals is 3.7 %.
The proposed method is then applied to estimate the fractal dimension of solar
irradiances of two sites: Boulder and Golden located at Colorado. A good concordance
has been found between the fractal dimension estimated and the shapes of the solar
irradiances curves.
INTRODUCTION
Fractals are objects presenting a high degree of geometrical complexity, their description
and modeling is carried out using a powerful index called fractal dimension. Indeed, the
fractal dimension is an important characteristic of fractals that contains information about
their geometrical irregularities over multiple scales. The fractal dimension of a curve, for
instance, will lie between 1 and 2, depending on how much area it fills. The fractal dimension
can than be used to compare the complexity of two curves [1]. In a solar field, the fractal
dimension is directly related to the roughness of the irradiance signals. We can then quantify
the irradiance fluctuations in order to establish a classification according to the atmospheric
state [2], [3], [4], [5].
This chapter is devoted to this mathematical tool that has important practical applications.
We start in Section II with generalities on the concept of dimension. Then, we deal in the
Section III with the problem of the theoretical estimation of this index used for the Euclidean
objects or self-similar objects. Section IV focuses on experimental estimation of the fractal
dimension used in the case of natural fractals; a survey of existing methods is presented.
Thereafter, we will be interested in the Section V in the fractal dimension estimation of
curves or one-dimensional profiles e.g. temporal signals, which quantifies the degree of their
fluctuations. The fractal dimension can thus be used to compare the complexity of two curves,
and therein lies its importance for applications. Also, we present a new method to evaluate the
fractal dimension of discrete temporal signals or curves with an optimization technique: the
rectangular covering method. To evaluate its accuracy, the proposed method is applied to
fractal signals whose theoretical fractal dimensions are known: Weierstrass function (WF)
and fractional Brownian motion (FBM).The application of the proposed method to solar
irradiance signals is presented in the Section VI. Finally, in Section VII, we give a conclusion
and discuss experimental results.

New Method for Estimating the Time Series Fractal Dimension 279
PRELIMINARIES
Mathematically, any metric space has a characteristic number associated with it called
dimension, the most frequently used is the so-called topological or Euclidean dimension. The
usual geometrical figures have integer Euclidean dimensions. Thus, points, segments,
surfaces and volumes have dimensions 0, 1, 2 and 3, respectively.
But what for the fractals objects, it is more complicated. The coastline is an extremely
irregular line in such a way that it would seem to have a surface, it is thus not really a line
with a dimension 1, nor completely a surface with dimension 2 but, an object whose
dimension is between 1 and 2. In the same way, we can meet fractals whose dimension ranges
between 0 and 1 (Like the cantor set which will be seen later) and between 2 and 3 (surface
which tends to fill out a volume), etc. So, fractals have dimensions which are not integers but
fractional numbers, called fractal dimension.
In the classical geometry, an important characteristic of objects whose dimensions are
integers is that any curve generated by these elements contours has finite length. Indeed, if we
have to measure a straight line of 1 m long with a rule of 20 cm, the number of times that one
can apply the rule to the line is the number 5. If a rule of 10 cm is used, the number of
applications of the rule will be 10 times, for a rule of 5 cm, the number will be 20 times and
so on. If we multiply the rule length used by the number of its utilizations we will find the
value 1 m for any rule used.
This result, if it is true for the traditional geometry objects, it is not valid for the fractals
objects. Indeed, let us use the same method to measure a fractal curve, with a rule of 20 cm,
the measured length will be underestimated but with a rule of 10 cm, the result will be more
exact. More the rule used is short more the measure will be precise. Thus, the length of a
fractal curve depends on the rule used for the measurement: the smaller it is, the larger the
length is found.
It is the conclusion reached by Mandelbrot when he tried to measure the length of the
coast of Britain. He found that the measured length depends on the details taken into account.
Indeed, the coast can be traversed by car or with foot to better follow the turnings of it, or
with the step of ants in order to circumvent all its sand grains. With each time one will obtain
a larger length, according to the inverse of the step used for the course.
With the question "What it is the length of the coast of Britain?", Mandelbrot answered,
"It is infinite length" since the length of the coast grows as the step of course decreases.
Thus, fractal shapes cannot be measured with a single characteristic length, because of
the repeated pattern we continuously discover at different scale levels.
This growth of the length follows a power law found empirically by Richardson
()
α
η
η
−
∝L (1)
Where L is the length of the coast,
η is the length of the step used and the exponent α
represents the fractal dimension of the coast.
Another main property of fractals is the self-similarity. This characteristic means that an
object is composed of sub-units and sub-sub-units on multiple levels that resemble the
structure of the whole object. So fractal shapes do not change even when observed under a
different scale, this nature is also called scale-invariance. Mathematically, this property

S. Harrouni and A. Guessoum 280
should hold on all scales. However, in the real world the self-similarity is only observed over
some scales, the objects are then statistically self-similar or self-affine.
THE THEORETICAL DETERMINATION OF THE FRACTAL DIMENSION
Fractal dimension being a measurement in the way in which the fractal occupies space, to
determine it we have to draw up the relationship between this way of occupation of space and
its variation of scale.
Let N denote the number of the measurement stallion of n length needed to cover an
object of L length and D the fractal dimension of the considered object. We have
D
n
L
N = (2)
From where:
()
=
n
L
N
D
ln
ln (3)
For the self-similar fractals, L/n represents the magnification factor and n/L the reduction
factor, n corresponds for these objects to the number of objects similar to the original which
are contained in the initial object.
This dimension is often called the Hausdorff−Besicovitch dimension. This is true in the
case of the lines and usual surfaces where fractal dimensions and Hausdorff−Besicovitch
dimensions are equal. But in the general case of the fractal objects, the dimension of
Hausdorff−Besicovitch has a more complicated expression.
Let X denote the set of sequences of subsets {Xi, i=1, 2, 3,…}, such that
∪
∞
=
⊆
1i
i
XX and
()
ε
≤< i Xdiam0 for all i, where
()
i
Xdiam is the largest distance
between any two points of Xi.
For each p ∈[ 0, ∞] the Hausdorff p
− dimensional measure of X is defined by
() ( )
[]
()
0
inflim
→
=∑
ε
ε
H
i
p
ip XdiamXH (4)
There is a critical real number DH ≥ 0 such that Hp(x) = 0 for P > DH whereas Hp(x) = ∞
for p < DH. This critical DH is the Hausdorff dimension of X also called
Hausdorff−Besicovitch dimension [6], [7].

New Method for Estimating the Time Series Fractal Dimension 281
We note through this complex definition that the Hausdorff −Besicovitch dimension is
very abstract. Although mathematically it is the most studied, and in a general way the best
known, due to the fact that it is applicable to any element, it is often too difficult to compute.
For traditional geometry objects, the Hausdorff−Besicovitch dimension is equal to its
topological dimension which is not the case for the fractals objects. This allows Mandelbrot
to state that:
An object is fractal if its Hausdorff − Besicovitch dimension strictly exceeds its topological
dimension.
For better understanding of this, we will compute the fractal dimension of some
Euclidean and fractal objects.
Determination of the Fractal Dimension of the Usual Euclidean Objects
We mean by Euclidean object any usual object of the traditional geometry which is not
fractal. This object can be a segment, a plane or a volume.
In this section we will apply the formula (3) to calculate the fractal dimension of these
Euclidean objects and to check that their fractal dimension is equal to their topological
dimension.
A straight line
Figure 1. Determination of a straight line fractal dimension
Let's consider a segment of a straight line of length L (Figure 1), if one uses a segment of
length n=L/3 to measure the length L, it will be necessary for us to use this segment 3 times
(N = 3). According to formula (3) the dimension of this object will be
()
1
3ln
3ln
ln
ln ==
=
n
L
N
D (5)
This dimension is the topological dimension of a straight line.
For a square of side L (Figure 2), we need to use 9 squares of side n=L/3 to cover the
initial square of side L, from where N = 9. Hence
()
2
3ln
9ln
ln
ln ==
=
n
L
N
D (6)
This is nothing other than the topological dimension of the plan or a surface.
L
n