TIME SERIES VIA FRACTALS

GEOMETRY

K.BHAVANI(MA14M003)R.HEMA(MA14M012)

Why?Where?How?

Why time series via fractals?

Chaotic systems exhibit rich and surprising mathematical structures.

Deterministic chaos provides a striking interpretation for irregular

temporal behaviour.

The most direct link between chaos theory and the real world is the

analysis of time series of real systems in terms of nonlinear

dynamics. Sometimes time series are so complex that they could not

be examined by traditional methods. In such cases time series data is

analysed via fractals.

Where?

Geo physics time series(recording of temperature, precipitation)

Medical and physiological time series(recording of heartbeat, ECG,

blood pressure, respiration, blood flow, glucose levels.. etc.)

Astrophysical time series

Technical time series(Internet traffic, highway traffic )

Finance and economy.

Application of fractal dimension in ECG data:ECG curve

Above figure defines the shape and size of the P-QRS-T wave of the

electrocardiogram(ECG), this is essential to determine the classification

of heart failure from ECG. The characteristics of all interval (PR, QRS,

QT, RR) can change by specific heart failures. ECG signals are widely

examined signals, thinking the significance in humans life.

Decisions about ECG signals were made through handling the intervals

between P-QRS-T waves..

How…

Three kinds of fractal dimension methods in fractal geometry are used.

1.Correlation method

2.Hausdorff method

3.Box counting method

Fractal dimension:

It is a measure of the space-filling capacity of a pattern

that tells how a fractal scales differently from

the space it is embedded in.

a fractal dimension does not have to be an integer.

Correlation method:

Autocorrelation:

Autocorrelation function is a time dependent function.

the autocorrelation function measures the average magnitude of the

correlation of two points separated in time with a lag t.

When there is no correlation in the time series then the expectation value

of correlation function is zero. Autocorrelation function is computed by

the formula.

A( )=-)(-)/-)^2ᵼ

represents ith element of the series, τ is time delay (Optimal time lag)

and N is equal to length of the series.

τ is determined from the shape of the correlation function, as the

position of its first decrease under 1/e or 1/10, and this value is used to

find the embedded dimension in phase space, which gives the

correlation dimension value.

Functions related to autocorrelation are directly the main way in order

to estimate the fractal dimension by means of power law relations.

Quantities varying with time often turn out to have fractal graphs. One

way in which their fractal nature is often manifested is by a power-law

behaviour of correlation between measurements separated by time h.

A measure of the correlation between f at times separated by h is

provided by the autocorrelation function

C(h) = ) )dt

Autocorrelations provide several methods of estimating the dimension

of the graph of a function or signal f .

The autocorrelation function C(h) or equivalently, the mean-square

change in signal in the time h over a long period could be computed,

and so

2[C(0)-C(h)] ≅ dt

If the power law behaviour is

C(0)-C(h) ≅ c

Then s is said to be dimension.

Hausdorff method:

The direct calculation of the Hausdorff dimension is often intractable.

It is calculated by using Hurst exponent method.

Hurst exponent is calculated by using Rescaled Range analysis.

To do this, a time series of full length is divided into a number of shorter time series and the rescaled range is calculated for each of the smaller time series. A minimum length of eight is usually chosen for the length of the smallest time series. So, for example, if a time series has 128 observations it is divided into:

two chunks of 64 observations each

four chunks of 32 observations each

eight chunks of 16 observations each

16 chunks of eight observations each

Steps for estimating the Hurst exponent after breaking the time series into chunks:

For each chunk of observations, compute:

The mean of the time series,

A mean centered series by subtracting the mean from the series,

The cumulative deviation of the series from the mean by summing up the mean centered values,

The Range (R), which is the difference between the maximum value of the cumulative deviation and the minimum value of the cumulative deviation,

The standard deviation (S) of the mean centered values

The rescaled range by dividing the Range by the standard deviation.

Finally, average the rescaled range over all the chunks. i.e. H

Hausdorff dimension has the advantage of being

defined for any set, and is mathematically convenient,

it is based on measures, which are relatively easy to

manipulate.

The Hausdorff dimension calculate by using the formulae

DH=2-H

where H- hurst exponent(0<H<1)

H can be calculated by the following formulae

where n- no.of data points in time series

R(n)-range value

S(n)-standard deviation

c-constant

Box counting method:

One of the simplest and most intuitive method of fractal dimension is box counting with dimension .

Given a fractal set in a d-dimensional Euclidean space, gives the scaling between the number of d-dimensional ε boxes needed to cover the set completely, and the boxes size ε.

=

For the ECG time series graph given above the different size of the boxes the no. of boxes tabulated below

The data given above in the table will able to plot a regression line.

Taking the general fractal dimension formula, the logarithmic N(E) values and the logarithmic E values have a linear relationship.

=

There is a facility to find the optimal box number to calculate the box-counting dimension.

“simple linear regression, Least Square Method”, which are an useful and popular technique in the field of statistics, to find the box counting dimension via the slope of the regression line.

Fitting Regression line:

The method of least squares is a standard approach to the

approximate solution of overdetermined systems, i.e., sets of

equations in which there are more equations than unknowns. "Least

squares" means that the overall solution minimizes the sum of the

squares of the errors made in the results of every single equation.

The least squares line uses a straight line y=a+bx

To approximate the given set of data, (,),……, (,),where n≥2.

The best fitting curve f(x) has the least square error, .i.e.

where a and b are unknown coefficients while all  and   are given. To

obtain the least square error, the unknown coefficients  a and  b must yield

zero first derivatives.

By expanding above equations we will get

where b gives the slope of the regression line.

In the lightening of this expression, the box-counting dimension value can be obviously seen from the regression equation, because it is a linear equation and the value of b gives the slope of the regression line, exactly being equal to the value of box-counting dimension.

In the below chart 1.4068 is the box dimension of the data.

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

f(x) = − 1.40677198257381 x + 9.43259631641876R² = 0.996559218520085

Chart TitleSeries1 Linear (Series1)

Ln of box size

Ln o

f num

ber

of

boxes

Accuracy of Fractal Dimension Methods

We need to prove the accuracy of these 3 fractal dimension methods.

Therefore a sinusoidal wave was examined using correlation, Hausdorff

and box-counting dimensions, in the lightening of the knowledge that

this sinusoidal wave should have the value of nearly one as dimension.

This plot shows the correlation dimension of the above sinusoidal wave.

The Hurst exponent and Hausdorff Dimension Graph of Sinusoidal

Wave is

Hurst exponent value is calculated from the slope of the line.

Hurst exponent value=0.991829, Hausdorff dim=1.008171.

Box dimension:

SLOPE=1.8314

R 1 2 4 8 16 32 64 128 256

512

1024

N(r) 234550

59010

14805

3763

972

252

63 20 6 2 1

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

f(x) = − 1.83143528976727 x + 12.0944652952774R² = 0.996011221986434

Chart Title

Series1 Linear (Series1)

LN(R)

LN

(N(R

))

CONCLUSION:

According to these results, two techniques indicate high degree of accuracy

apart from Box dimension, because the value of sinusoidal wave dimension

should be nearly 1, Box dimension value is quite high.

FRACTAL DIMENSION TYPE DIMENSION VALUE

CORRELATION 1.002

HAUSDORFF 1.008171

BOX 1.8314

SINUSOIDAL WAVE 1

Applications:

By using 3 fractal dimensions ECG data is analysed. It contains 2700

data points.

Plot of the time series data:

Correlation dimension: Embedded points = 2692

Time lag =6

Oribital lag =56

Correlation dimension

1.924 where embedded

Dimension is 2.

Hausdorff dimension

H value is 0.995760

Hausdorff dim=2-H=1.004240.

Box dimension:

THANK YOU