Fractal Analysis and Its Application for Investigating Time Series ...

7/30/2019 Fractal Analysis and Its Application for Investigating Time Series ...

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FRACTAL ANALYSIS AND ITS APPLICATION FOR INVESTIGATING TIME SERIES.

In this presentation Id like to give a brief overview of fractal analysis, one of the new

mathematical methods in the modern science. Firstly, Id like to give the definition of the fractal,discuss its characteristics and then move on to the notion of fractal dimension and the application

of fractal methods for the investigation of time series.

Let me start with the definition of the fractal and explain how it appeared in the modernscience.

The notion of fractals was introduced by Benoit Mandelbrot in 1970s. The term fractal isderived from the Latin verb frangere meaning to break into pieces. As one of the definitions of

the fractal runs, FRACTAL is a multitude, which parts are similar to the whole.

The classic example of a natural fractal object is the coastline. In the beginning of the 20 th

century, Richardson, the English hydro mechanical engineer, faced certain difficulties when hetried to measure the length of the Great Britain coastline. He tried to substitute the coastline by thebroken line. It turned out that when the scale was decreased, the calculated length of the broken

line increased greatly.

Mandelbrot offered to apply the law of power dependence to approximate the degree of theincrease of the coastline length.

The main fractal characteristic is the fractal dimension.

It is calculated in the following way: The multitude is covered with the series of cells withvariance step DELTA. Then we calculate the number of cells occupied by the elements of themultitude, and that is N (DELTA).

Then the fractal dimension is calculated using the following expression:

=

1ln

)(ln

lim0N

D

(1)Accordingly, if the object under study is close to the fractal, then the dependence of the

number of squares on the size of an elementary figure will increase by power dependence. In the

double logarithmic coordinates this dependence will tend to a straight line. (we can see it in

Picture 1). The fractal dimension is calculated as the tangent of the angle of inclination of thisline.


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Pic1.

In the real world we usually cannot find examples of pure ordered fractals. In terms of the

real world, one can only talk about fractal phenomena. Such phenomena should be viewed asmodels, statistically approximating to fractals. Recently the fractal theory has been developed

further to the multi-fractal theory. Multi-fractal is a quasi-fractal object with the VARYING(alternating) fractal dimension. It is obvious that the real life objects and processes are betterdescribed in terms of multi-fractals.

The main metaphor of fractal dimensions is the metaphor of the fractal copier. Lets assume

that we have the original multitude. The special machine, the fractal copier, reflects the multitudeand then adds this reflected multitude to the original one. Thus after several simple operations of

this kind, we can get a relatively complicated picture. (SEE EX. In the picture). In the process of

building a fractal there are two important things the original multitude and the mechanism of thetransformation. Such machine is shown in Pic. 2.

LnN

Ln ()


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Pic 2.

With fractal approach, the pre-historical influence on the systems today behavior isimportant. It is natural that the pre-historic influence works through the prism of influencing

factors, this means, through the same copier.

At this point let us look at the methods of geometric fractal formation. They are shown

in the following pictures.

Depending on the algorithm of formation, fractals are divided into linear [linea] and non-

linear. Non-linear fractals, due to their complexity, are so far applied to arts only. The examples of

such fractals are shown in Picture

All the fractal discussed above are non-linear.

Pic 3.

Pic. 4.

At this point let us turn to the application of fractal methods to the analysis of the time

series.


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Pic. 5.

Time series is an aggregation of the systems parameters observed in time.

A lot of experimental data have fractal statistics. The analysis and modeling of this statisticscan be made by means of fractal analysis methods. One of the most prospective trends of fractalanalysis is the study of the time dynamics of the fractal dimension (D).

Pic. 6.

Picture 8 represents the stages in time series modeling based on its trend.


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In the following picture you can see the stages in fractal modeling of time series on the basis

of its trend.

There are several methods of calculating the fractal dimension for the time series.

Time series fractal analysis is especially significant as it considers the systems behaviorboth during the period of change and during the period before the change, or its pre-history. It

corresponds rather well to the metaphor of the fractal copier.

For fractal time series within the interval t0 < t < T the amplitude of the value of parameterR depends on time t by power dependence:

D

t

ttRtR

=

2

0

0 )()( (3)

(D is the fractal dimension of the time series).

According to this equation we can predict the probable amplitude of the parameter inquestion for the future period.

Fractal dimension proves the complexity of the curve. By analyzing the alternation of the

segments with different fractal dimensions and the way the system is influenced by internal and

external factors, we can predict the systems behavior. What is more important, It is also possibleto predict and diagnose the non-stable states of the system.

There is the critical value of fractal dimension of the time series curve when D = 1.6 1.7.Approximating to this value, the system becomes unstable and its parameters may either increaseor decrease rapidly, depending on the todays trend.

This may be clearly seen from the graph which analyzes the dynamics of the US Dollar-

Russian Ruble exchange rate fluctuations during the financial crisis of 1998 in Russia. We may

see that there is A sharp increase of the fractal dimension immediately before the dollar exchangerate against the ruble rises considerably.


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Pic 7.

That means that the fractal dimension with a certain value may be used as an indicator, orthe so-called flag of the catastrophe.

The analysis of the experimental data shows that the trend line for the time series is well

described by the expression:

)(

))(()()(

0

00

0

__

DD

tttKtyty

f

+= (4)

(where y [WAI BAR] (t0) the parameter average for the period before the predicted

period; Kfand [BETA] - the coefficients, t0 the period of time before the predicted period; t the time period for which the forecast is made; d0 the fractal dimension for the period before theforecasted period).

The value of the fractal dimension may also help to determine the number of factors that

currently influence the system. If the fractal dimension is less than 1.4, then there is only one oronly a few factors that influence the system and move it in one direction. If the fractal dimensionis about 1.5, then the forces that influence the system are differently directed, but still they

compensate each other more or less. The systems behavior in this case is stochastic, and it is well

described by the traditional statistical methods. If the fractal dimension is much higher than 1.6,the system becomes unstable and it is ready to shift to the new stage.


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In this respect we may have a look at the Dow Jones Stock Index dynamics, shown in the

following picture. In this picture we may see the average annual fractal dimension of the time

series for Dow Jones Index. During quite stable periods and smooth rises the fractal dimension ofthe time series remains relatively low. During the crises, the total fractal dimension rises sharply.

Pic 8.Of course, the patterns described above are rather general. More specific patterns and rules

of influencing factors must be determined for each system.

The traditional financial models show that the crises should occur very seldom. They areusually based on the probability, calculated by Gauss and Puasson. In this case the probability ofcrises is often diminished. The corner stone of finance is the modern portfolio theory, whichattempts to maximize the profit for this risk level. Mathematics that underlies the basis of portfolio

theory, handles the emergency situations a bit too lightly. It considers drastic market changes to be

highly unlikely. In our opinion, it underestimates the forming influence of crises for the system.

Fractal is the geometrical figure that can be divided into parts. Each part then is the reducedcopy of the whole. In finance, this concept is not just baseless abstractness. It is the theoretical

reformulation of the market saying (which runs as follows) the movements of stock and currencyare externally alike, disregarding the time and price scale. The observer cannot tell by the externallook of the graph, whether these data are weekly, daily or hourly data. There is also a well-knownrule of the First Month. Some stock exchange analysts say that according to this rule, as the things

develop during the first month, so they will approximately do during the rest of the year. All these

features allow us to define the diagrams as fractal curves. It also makes many powerfulinstruments of mathematical and computational analysis available for finance predictions.

DJIA

Fractal

Dimension

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