This document discusses using the Unscented Kalman Filter (UKF) to estimate parameters and forecast chaotic systems. It aims to improve forecast accuracy by identifying the chaotic component of time series processes using nonlinear dynamic models. The UKF is applied to a logistic map model example to check its effectiveness. It is also applied to communication traffic data to forecast values and residues shows good approximation. The UKF method shows potential for parameter estimation and forecasting of real-world chaotic systems.

1. Parameters Estimation for Chaotic Systems Using
the Unscented Kalman Filter (UKF) to Construct a Forecast
Elena I. Malyutina
Federal State State-Financed Educational Institution of High Professional Education
“South Ural State University”
05.13.01 System analysis, management and information processing (Postgraduate)
AIST Conference. April 2014. Yekaterinburg

2. e.mankevich@gmail.com
What is deterministic chaos?
Implementation of irregular oscillation regime similar to a random process but strictly
predictable in the sense of the law of evolution.
Actuality
Complexity of solving the problem of data processing which contain chaotic component is:
- Short time series
- The implementation process is unique
- No information about the probability distribution of errors.
Objective
To improve the forecast accuracy by the identification of the chaotic component of the time
process using deterministic nonlinear dynamic systems with chaotic solutions.
Application
Engineering, cryptography, medicine, economics and other fields of science and technology.
Problem statement
The model of chaotic process
( )
1
, 1,2,...,
n
i
k i k k
i
y a x k N

  
 
 ( )
1 , , 0,1, , 1, 1,2, , ,ii
k i k ix f x k N i n      
   
 ( )
1 1 , 0,1, , , 1,2, ,i ii
k i k kx x x k N i n      
 
   0 0,1 , ,4 , 3.57i
ix      
Fig.1. The objective function

3. e.mankevich@gmail.com
Practical results
Logistic map
In order to check the UKF it is considered a model example:
 1 1
,
1 , 1,2,..., .
k k k
k k k
y ax
x x x k N

  
 
  
0 0.3, 3.69, 10 , 10%, (8 ) 10%x SNR dB REE RPE steps    
Fig. 2. (a) The investigated process kx (–) and approximation (- -), (b) The estimation (- -), true value (–) of the
parameter 
Fig. 3. (а) The forecast of the initial process, (b) Relative prediction error
The communication traffic prediction
Fig.4. (а) The forecast, (b) The residues and approximation of residues