1. ANALYSES OF THE CHAOTIC BEHAVIOR OF THE ELECTRICITY PRICE SERIES
Radko Kříž
University of Hradec Kralove
Faculty of Science

2. Content
Introduction
Input data
Methodology
Results
Conclusions

3. Introduction
Is the world stochastic or deterministic

4. Input data EPEX (European Power Exchange) Phelix hourly spot prices
in EUR/MWh
between to
more then samples
high volatility rate

5. Electricity spot prices

6. Descriptive statistics of the spot prices
Mean
Median
Std dev
Skewness
Kurtosis
Min
Max
25% qtl
75% qtl
42,5
39,3
25,8
-2,43
16,67
500,0
2436,6
34,3
53,1

7. Phase space reconstruction
A point in the phase space is given as:
 is the time delay
m is the embedding dimension
How can we determine optimal  and m?

8. Optimal time delay
Very small   near-linear reconstructions Very large   obscure the deterministic structure the mutual information between xn and xn+ Mutual information function:

9. Mutual information

10. Optimal embedding dimension
false nearest neighbors (FNN)
This method measures the percentage of close neighboring points in a given dimension that remain so in the next highest dimension.

11. Nearest neighbors

12. The largest Lyapunov exponentZ0 Zt
The largest Lyapunov exponent:

13. Rosenstein algorithm
dj(i) is distance from the j point to its nearest neighbor after i time steps
M is the number of reconstructed points.
Our results:  = 0,0005

14. Slope is the largest Ljapunovuv exponent 0,00022.

15. The 0-1 test for chaos developed by Gottwald & Melbourne
scalar time series of observations φ1, ... , φN
construct the Fourier transformed series

16. Logistic equation
r=3, r=3,97

18. The 0-1 test for chaos the output is 0 or 1 Our result: 1
compute the smoothed mean square displacement
estimate correlation coefficient to evaluate the strength of the linear growth

19. Test 0-1

20. Test 0-1

21. Long memory in time series
Hurst exponent (H)
Is between 0 and 1
Random walk 0,5
Higher values  trend without volatility
self-similarity process

22. Long memory in time series
is usually characterized in time or frequency domain
Rescaled Range Analysis (R/S Analysis)
Detrended Fluctuation Analysis (DFA)
Geweke et Porter‐Hudak Analysis (GPH)
Awerage Wavelet Coefficients (AWC)

23. Rescaled Range analysis
Hurst exponent
[R(n)/S(n)] is the rescaled range
E[y] is expected value
n is number of data points in a time series
C is a constant

24. Results Method Hurst koeficient R/S analýza 0,853 DFA 0,909 GPH 0,917
AWC
0,976
mean
0,914
Std dev
0,044

25. Fractal dimensions
ε – side of hypercube N(ε) – minimum number

26. Correlation dimension
Correlation integral where Θ is the Heaviside step function Correlation dimension Our results: DC=1,3

27. Recurrence analysis
based on topological approach, was used to show recurring patterns and non-stationarity in time series
Recurrence is a fundamental property of dynamical systems

29. the electricity price series is chaotic and
Conclusions
time delay  = 15
embedding dimension m = 6
the largest Lyapunov exponent  = 0,00022
chaos is present according to 0-1 test
Hurst exponent: H=0,914 
the electricity price series is chaotic and
contains long memory

30. THANK YOU FOR YOUR ATTENTION
Any questions or comments?