function ePE = PEeq( x, delay, order, windowSize )
efficiently [UK13] computes permutation entropy [BKP02] for the case of ordinal patterns with tied ranks [BQM2012,UK13,U15] in maximally overlapping sliding windows (see more ordinal-patterns based measures at http://www.mathworks.com/matlabcentral/fileexchange/63782-ordinal-patterns-based-analysis--beta-version-) for orders=1...7 of ordinal patterns with tied ranks.
1 Order of ordinal patterns is defined as in [1,3,7,8], i.e. order = n-1 for n defined as in [UK13]
2 The values of permutation entropy are normalised by log((order+1)!) so that they are from [0,1] as proposed in the original paper [BKP02].
CITING THE CODE
 Unakafova, V.A., Keller, K., 2013. Efficiently measuring complexity on the basis of real-world data. Entropy, 15(10), 4392-4415.
 Unakafova, V.A. (2015). Fast permutation entropy, MATLAB Central File Exchange. Retrieved Month Day, Year.
- indata - considered time series
- delay - delay between points in ordinal patterns with tied ranks (delay = 1 means successive points)
- order - order of the ordinal patterns with tied ranks (order+1 - number of points in ordinal patterns with tied ranks)
- windowSize - size of sliding window
- outdata - values of permutation entropy for ordinal patterns with tied ranks
EXAMPLE OF USE (with a plot):
indata = rand( 1, 7777 ); % generate random data points
for i = 4000:7000 % generate change of data complexity
indata( i ) = 4*indata( i - 1 )*( 1 - indata( i - 1 ) );
delay = 1; % delay 1 between points in ordinal patterns (successive points)
order = 3; % order 3 of ordinal patterns (4-points ordinal patterns)
windowSize = 512; % 512 ordinal patterns in one sliding window
outdata = PEeq( indata, delay, order, windowSize );
ax1 = subplot( 2, 1, 1 ); plot( indata, 'k', 'LineWidth', 0.2 );
grid on; title( 'Original time series' );
ax2 = subplot( 2, 1, 2 );
plot( length(indata) - length(outdata)+1:length(indata), outdata, 'k', 'LineWidth', 0.2 );
grid on; title( 'Values of permutation entropy for ordinal patterns with tied ranks' );
linkaxes( [ ax1, ax2 ], 'x' );
The method is based on precomputing values of successive ordinal patterns of order d, using the fact that they are "overlapped" in d points, and on precomputing successive values of the permutation entropy related to "overlapping" successive time-windows .
CHOICE OF ORDER OF ORDINAL PATTERNS
The larger order of ordinal patterns is, the better permutation entropy estimates complexity of the underlying dynamical system [KUU14]. But for time series of finite length too large order of ordinal patterns leads to an underestimation of the complexity because not all ordinal patterns representing the system can occur [KUU14]. Therefore for practical applications, orders = 3...7 are often used [BP02,RMW13,ZZR12].
In [AZS08] the following rule for choice of order is recommended:
5*(order + 1)! < windowSize.
CHOICE OF SLIDING WINDOW LENGTH
Window size should be chosen in such way that time series is stationary within the window (for example, for EEG analysis 2 seconds sliding windows are often used) so that distribution of ordinal patterns would not change within the window [BKP02,KUU14(Section 2.2),U15(Section 5.1.2)].
CHOICE OF DELAY BETWEEN POINTS IN ORDINAL PATTERNS
I would recommend choosing different delays and comparing results (see, for example, [KUU14, Section 2.2-2.4] and [U15, Chapter 5] for more details) though delay = 1 is often used for practical applications.
Choice of delay depends on particular data analysis you perform [RWM13,KUU14] and on sampling rate of the data. For example, if you are interested in low-frequency part of signals it makes sense to use larger delays.
[BKP02] Bandt C., Pompe B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, APS
[BQM2012] Bian, C., Qin, C., Ma, Q.D. and Shen, Q., 2012. Modified permutation-entropy analysis of heartbeat dynamics. Physical Review E, 85(2), p.021906.
[UK13] Unakafova, V.A., Keller, K., 2013. Efficiently Measuring Complexity on the Basis of Real-World Data. Entropy, 15(10), 4392-4415.
[U15] Unakafova, V.A., 2015. Investigating measures of complexity for dynamical systems and for time systems (Doctoral dissertation, Lübeck, Univ., Diss., 2015).
Valentina Unakafova (2020). Permutaton entropy with tied ranks (fast algorithm) (https://www.mathworks.com/matlabcentral/fileexchange/63794-permutaton-entropy-with-tied-ranks-fast-algorithm), MATLAB Central File Exchange. Retrieved .
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