version 1.5.3 (815 KB) by
Valentina Unakafova

Efficiently computing values of permutation entropy from 1D time series in sliding windows

function outdata = PE( indata, delay, order, windowSize )

computes efficiently [1] values of permutation entropy [2] for orders=1...8 of ordinal patterns from 1D time series in sliding windows. See more ordinal-patterns based measures at www.mathworks.com/matlabcentral/fileexchange/63782-ordinal-patterns-based-analysis--beta-version-

NOTES

1 Order of ordinal patterns is defined as in [1,3,7,8], i.e. order = n-1 for n defined as in [2]

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 [2].

INPUT

- indata - 1D time series (1 x N points)

- delay - delay between points in ordinal patterns (delay = 1 means successive points)

- order - order of the ordinal patterns (order + 1 is the number of points in ordinal patterns)

- windowSize - size of sliding window ( = number of ordinal patterns within sliding window)

OUTPUT

- outdata - (1 x (N - windowSize - order*delay) values of permutation entropy within [0,1] since each sliding window contains windowSize ordinal patterns but uses in fact (windowSize + order*delay + 1) points).

INTERPRETATION

The larger the values of permutation entropy (in the range from 0 to 1) are, the higher diversity of ordinal patterns is and the more complex input data are.

CITING THE CODE

[a] Unakafova, V.A., Keller, K., 2013. Efficiently measuring complexity on the basis of real-world data. Entropy, 15(10), 4392-4415.

[b] Unakafova, Valentina (2015). Fast permutation entropy (www.mathworks.com/matlabcentral/fileexchange/44161-permutation-entropy--fast-algorithm-), MATLAB Central File Exchange. Retrieved Month Day, Year.

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 ) );

end

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 = PE( indata, delay, order, windowSize );

figure;

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' );

linkaxes( [ ax1, ax2 ], 'x' );

CHOICE OF ORDER OF ORDINAL PATTERNS

The larger order of ordinal patterns is, the better permutation entropy estimates complexity of the underlying dynamical system [3]. 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 [3]. Therefore, for practical applications, orders = 3...7 are often used [2,4,5,8].

In [6] 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 [2,8], [3,Section 2.2], [7,Section 5.1.2].

CHOICE OF DELAY BETWEEN POINTS IN ORDINAL PATTERNS

I would recommend choosing different delays and comparing results (see, for example, [3, Section 2.2-2.4] and [7, Chapter 5] for more details) though delay = 1 is often used for practical applications.

Choice of delay depends on particular data analysis you perform [3,4], on type of pre-processing 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.

REFERENCES

[1] Unakafova, V.A., Keller, K., 2013. Efficiently measuring complexity on the basis of real-world Data. Entropy, 15(10), 4392-4415.

[2] Bandt, C. and Pompe, B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17), p.174102.

[3] Keller, K., Unakafov, A.M. and Unakafova, V.A., 2014. Ordinal patterns, entropy, and EEG. Entropy, 16(12), pp.6212-6239.

[4] Riedl, M., Muller, A. and Wessel, N., 2013. Practical considerations of permutation entropy. The European Physical Journal Special Topics, 222(2), pp.249-262.

[5] Zanin, M., Zunino, L., Rosso, O.A. and Papo, D., 2012. Permutation entropy and its main biomedical and econophysics applications: a review. Entropy, 14(8), pp.1553-1577.

[6] Amigo, J.M., Zambrano, S. and Sanjuan, M.A., 2008. Combinatorial detection of determinism in noisy time series. EPL (Europhysics Letters), 83(6), p.60005.

[7] Unakafova, V.A., 2015. Investigating measures of complexity for dynamical systems and for time series (Doctoral dissertation, University of Luebeck).

[8] Keller, K., and M. Sinn. Ordinal analysis of time series. Physica A: Statistical Mechanics and its Applications 356.1 (2005): 114—120

Valentina Unakafova (2021). Permutation entropy (fast algorithm) (https://www.mathworks.com/matlabcentral/fileexchange/44161-permutation-entropy-fast-algorithm), MATLAB Central File Exchange. Retrieved .

Created with
R2012b

Compatible with any release

**Inspired:**
Change-point detection using the conditional entropy of ordinal patterns

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!Create scripts with code, output, and formatted text in a single executable document.

Fulin CaicityskyMM xuervin