A Prediction Technique for Chaotic Time Series

- By Mr. Suhel S. Mulla

SY M. Tech (Digital System).

MIS No. – 121333009.

College of Engineering, Pune.

Time Series :

A time series a sequence of data points, typically consisting of successive measurements made over a time interval.

e.g. ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

Dynamical System :

A dynamical system is a concept in mathematics where a fixed rule describes how a point in a geometrical space depends on time.

e.g. Mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

12/9/2014 2:24:32 PM 2College of Engineering, Pune.


What is Chaos…??In common usage, chaos means “a state of disorder”

Difference between chaotic and random signal.

Chaos theory studies behavior of dynamical systems which are sensitive to initial conditions.

Chaotic system can be analog or digital. In analog chaos, it is given by differential equation while in digital case, it is given by difference equation

Chaotic systems are predictable for a while, then they appear to become random.


Continued…… In Lorenz words :

“Chaos is when present determines future, but approximate present determines approximate future.”

The approximate present information comes from even minute errors in measurement which are inevitable. Thus, correct measurement of chaotic system is not possible.

This makes the system forecast possible for small period of time.

The term used for prediction interval of chaotic system is called as Lyapunov Exponent.


AttractorsAn attractor is a set of numerical properties toward

which a system tends to evolve, for a wide variety of starting conditions of the system.

Types –

1. Fixed Point

2. Limit Cycle

3. Limit Torous

4. Strange Attractor.


Types of Attractors

Fixed Point Limit Cycle

Torus Strange Attractor

Focus Node


Lyapunov Exponent It is a quantity that characterizes the rate of

separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation δZ0 diverge at a rate given by

where λ is the Lyapunov exponent.

It determines a notion of predictability for a dynamical system.

For the chaotic system it has to be between 0 and 1.


Lyapunov Exponent continued…

=<ln|Rn/R0|>

Rn = R0en


Takens Theorem

It gives a notion through which complete system can be reconstructed from the observed time series.

The key point is that, though we are observing only one time series, it is made by all the state variables present in the system.

Thus, the history of time series can determine the nature of all the state variables in the system.

According to Taken’s theorem, an m-dimensional system requires (2m+1) delay co-ordinated graph of observed time series.

It tells us that we do not have to measure all the state space variables of the system for finding out the internal details of the system.


Prediction Mechanism Reconstruction of Phase Space :


Reconstruction of Sine Wave Generator


State-space Prediction


Determination of Delay time ‘τ’ Delay time can be calculated by finding

autocorrelation function of the data and selecting ‘τ’ as its first zero crossing.

Small value of ‘τ’ gives the reconstructed delay graph approximately linear whereas large value of ‘τ’ gives completely uncorrelated graph.

Along with all the methods, ‘τ’ can also be calculated by trial and error method, such results give satisfactory answers in some cases.


Embedding Dimensions ‘d’ If the co-ordinated graph has less dimensions

provided by Taken’s theorem, then the orbits overlap and a complex path crisis is created.

Sometimes the reconstructed graph with less dimensions than required may help, this occurs only when and n dimensional system can be uniquely represented in (n-m) dimensions.

If the system is reconstructed with less than required dimensions, overlap points occur as shown in the figure.


Continued…………….

The overlap in above figure can be removed by adding extra dimensions to the graph by means of new delay co-ordinates.

This process is repeated till all the overlaps are removed.


Correlation Dimension It gives the dimensionality of the space occupied by a

set of random points, often referred to as a type of fractal dimension.

It is a measure of the extent to which the presence of a data point affects the position of the other point lying on the attractor.

If correlation dimensions value is finite low and non-integer, the system is chaotic.

If correlation exponent increases without bound with increase in the embedding dimensions, the system is stochastic.


Largest Lyapunov Exponent Calculation & Prediction

After the correct reconstructed phase space plot is configured, Lyapunov exponent for each is calculated and the largest among them all is selected for future prediction.

The period limit on accurate prediction of a chaotic system is a function of largest Lyapunov Exponent.

To be chaotic, the largest Lyapunov exponent must be between zero and one.

If Lyapunov exponent is greater than one, the system is stochastic.


Algorithm :

1. Start with unfolded attractor in m-dimensional space and time lag τ.

2. Take the initial vector y(t1).

3. Select the k nearest trajectories on the attractor.

4. Afterwards select the respective k nearest points to y(t1), one on each trajectory.

5. An average of all these trajectories is calculated

6. The determined value is used to point next point on predicted trajectory.

7. The predicted point is then set as a new starting vector and the process is repeated.


References Arslan Basharat, Mubarak Shah, “Time Series

Prediction by Chaotic Modeling of Nonlinear Dynamical Systems,” IEEE cpnference 2009.

Pengjian Shang, Xuewei Li, Santi Kamae, “Chaotic analysis of traffic time series,” IEEE conference 2004.

H. D. I. Abarbanel, “Analysis of Observed Chaotic Data,” Springer, 1995.

Steven H. Strogatz, “Non-linear Dynamics and Chaos with applications to Physics, Biology, Chemistry and Engineering,” Westview Press. Perseus Publishing, 2004.

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Thank You…..
