Radko Kříž University of Hradec Kralove Faculty of Science Radkokrizuhkcz Content Introduction Input data Methodology Results Conclusions Introduction Is the world stochastic or deterministic ID: 634645
Download Presentation The PPT/PDF document "ANALYSES OF THE CHAOTIC BEHAVIOR OF THE ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
ANALYSES OF THE CHAOTIC BEHAVIOR OF THE ELECTRICITY PRICE SERIES
Radko Kříž
University of Hradec Kralove
Faculty of Science
Radko.kriz@uhk.cz
Slide2
Content
IntroductionInput dataMethodologyResultsConclusionsSlide3
Introduction
Is the world stochastic or deterministicSlide4
Input data
EPEX (European Power Exchange) Phelix hourly spot pricesin EUR/MWhbetween 8.2.2005 to 31.12.2016more then 100 000 samples
high volatility rate
Slide5
Electricity spot pricesSlide6
Descriptive statistics of the spot prices
Mean
Median
Std
dev
Skewness
Kurtosis
Min
Max
25%
qtl
75%
qtl
4
2
,
5
39
,
3
25
,
8
-2,43
16
,
67
500
,
0
2436,6
34,3
53,1Slide7
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
?Slide8
Optimal time delay
Very small near-linear reconstructionsVery large
obscure the deterministic structure
the mutual information between
x
n
and
x
n
+
Mutual information function:Slide9
Mutual informationSlide10
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.Slide11
Nearest neighborsSlide12
The largest Lyapunov exponent
Z
0
Z
t
The
largest
Lyapunov
exponent:Slide13
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: =
0,00
05Slide14
Slope is the largest
Ljapunovuv
exponent 0,00022.Slide15
The 0-1 test for chaos
developed by Gottwald & Melbournescalar time series of observations φ
1
, ... ,
φ
N
construct the Fourier transformed
seriesSlide16
Logistic equation
r=3,55 r=3,97Slide17Slide18
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 growthSlide19
Test 0-1Slide20
Test 0-1Slide21
Long memory in time series
Hurst exponent (H)Is between 0 and 1Random walk 0,5Higher values trend without
volatilit
y
self-similarity processSlide22
Long memory in time series
is usually characterized in time or frequency domainRescaled Range Analysis (R/S Analysis)Detrended Fluctuation Analysis (DFA)Geweke et Porter‐
Hudak
Analysis (GPH)
Awerage
Wavelet Coefficients (AWC)Slide23
Rescaled Range analysis
Hurst exponent[R(n)/S(n)] is the rescaled rangeE[y]
is expected value
n
is number of data points in a time series
C
is a constantSlide24
Results
Method
Hurst
koeficient
R/S analýza
0,853
DFA
0,909
GPH
0,917
AWC
0,976
mean
0,914
Std
dev
0,044Slide25
Fractal dimensions
ε – side of
hypercube
N(ε)
–
minimum
numberSlide26
Correlation dimension
Correlation integralwhere Θ is the Heaviside step function
Correlation dimension
Our results:
D
C
=
1
,
3Slide27
Recurrence analysis
based on topological approach, was used to show recurring patterns and non-stationarity in time seriesRecurrence is a fundamental property of dynamical
systemsSlide28Slide29
Conclusions
time delay = 15embedding dimension m = 6
t
he
largest
Lyapunov
exponent
=
0,00
022
chaos
i
s
present according to 0-1 testHurst exponent:
H=0,914
the
electricity
price
series
is chaotic and
contains
long
memorySlide30
THANK YOU FOR YOUR ATTENTION
Any questions or comments?