ANALYSES OF THE CHAOTIC BEHAVIOR OF THE ELECTRICITY PRICE SERIES Radko Kříž University of Hradec Kralove Faculty of Science Radko.kriz@uhk.cz

Content Introduction Input data Methodology Results Conclusions

Introduction Is the world stochastic or deterministic

Input data EPEX (European Power Exchange) Phelix hourly spot prices in EUR/MWh between 8.2.2005 to 31.12.2016 more then 100 000 samples high volatility rate

Electricity spot prices

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

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?

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

Mutual information

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.

Nearest neighbors

The largest Lyapunov exponent Z0 Zt The largest Lyapunov exponent:

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

Slope is the largest Ljapunovuv exponent 0,00022.

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

Logistic equation r=3,55 r=3,97

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

Test 0-1

Test 0-1

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

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)

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

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

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

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

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

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

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