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⛄ 内容介绍
OPA（序数模式分析）工具箱用于多元时间序列的非线性分析，基于序数模式的度量变得越来越流行 [1-5]，这些度量可以高效计算 [6,7] 并可视化：-
排列entropy (cfg.method = 'PE') [2]
- 具有并列等级的序数模式的排列熵 (cfg.method = 'eqPE') [4,8]
- 排列熵和序数模式分布 (cfg.method = 'opdPE ') [3]
- 序数模式的条件熵 (cfg.method = 'cePE') [6]
- 稳健的排列熵 (cfg.method = 'rePE') [4,7]
⛄ 部分代码
%% compute permutation entropy in sliding windows
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'PE'; % compute permutation entropy
cfg.order = 3; % ordinal pattens of order 3 (4-points ordinal patterns)
cfg.delay = 2; % delay 2 between points in ordinal patterns
% (one point between successive points in ordinal patterns)
cfg.windowSize = 512; % window size = 512 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute permutation entropy and ordinal distributions in sliding windows
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'opdPE'; % compute permutation entropy
cfg.order = 3; % ordinal pattens of order 3 (4-points ordinal patterns)
cfg.orderSeq = 6; % ordinal pattens of order 6 for plotting their sequence (7-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns (successive points)
cfg.windowSize = 1024; % window size = 1024 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute all the implemented measures simultaneously for comparison
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'all'; % compute all implemented ordinal-patterns-based measures
cfg.order = 4; % ordinal patterns of order 4 (5-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns
cfg.windowSize = 512; % window size = 512 time steps
cfg.lowerThreshold = 0.2; % the distance considered negligible between points
cfg.upperThreshold = 200; % the distance between points most probably related to artifact
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute conditional entropy of ordinal patterns in sliding windows
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'CE'; % we compute conditional entropy of ordinal patterns
cfg.order = 3; % ordinal pattens of order 3 (4-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns (successive points)
cfg.windowSize = 512; % window size = 512 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute robust permutation entropy
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'rePE'; % compute robust permutation entropy
cfg.order = 6; % ordinal patterns of order 6 (7-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns
cfg.windowSize = 2048; % window size = 2048 time steps
cfg.lowerThreshold = 0.2; % the distance that is considered negligible between points
cfg.upperThreshold = 100; % the distance between points most probably related to artifact
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute permutation entropy for ordinal patterns with tied ranks in sliding windows
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'PEeq'; % compute permutation entropy for ordinal patterns with tied ranks
cfg.order = 3; % ordinal pattens of order 3 (4-points ordinal patterns)
cfg.delay = 3; % delay 3 between points in ordinal patterns
% (2 points between successive points in ordinal patterns)
cfg.windowSize = 1024; % window size = 1024 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute permutation entropy for several channels
load( 'tonicClonic.mat' );
indata( 2, : ) = rand( 1, length( indata ) );
cfg = [];
cfg.method = 'PE'; % compute robust permutation entropy
cfg.order = 3; % ordinal patterns of order 3 (4-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns
cfg.windowSize = 1024; % window size = 1024 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
%% compute permutation entropy and conditional entropy of ordinal patterns
% for different parameters of logistic map (we use low-level functions for the example)
orbitLength = 10^4;
% take different r values
order = 7; % for ordinal pattens of order 7 (8-points ordinal patterns)
delay = 1; % for delay 1 (successive points in ordinal patterns)
windowSize = orbitLength - order*delay;
r = 3.5:5*10^(-4):4;
peValues = zeros( 1, length( r ) );
ceValues = zeros( 1, length( r ) );
leValues = LEofLogisticMap( 3.5, 4, 5*10^(-4) );
indata = zeros( 1, orbitLength );
for i = 1:length( r )
if ( rem( i, 10 ) == 0 )
disp( [ 'Calculating entropies for r = ' num2str( r( i ) ) ' from 4' ] );
end
indata( 1, 1 ) = rand( 1, 1 );
for j = 2:orbitLength
indata( j ) = r( i )*indata( j - 1 )*( 1 - indata( j - 1 ) );
end
peValues( i ) = PE( indata, delay, order, windowSize );
ceValues( i ) = CondEn( indata, delay, order, windowSize - delay );
end
figure;
linewidth = 0.5;
markerSize = 2;
plot( r, leValues, 'k', 'LineWidth', linewidth ); grid on; hold on;
plot( r, peValues, 'go', 'markerSize', markerSize ); grid on; hold on;
plot( r, ceValues, 'bo', 'markerSize', markerSize ); grid on; hold on;
legend( 'LE', 'PE', 'CE' );
xlabel( 'Values of parameter r for logistic map x(t)=r*x(t-1)*(1-x(t-1))' );
%% INEFFICIENT METHOD: compute permutation entropy in sliding windows with an old method
% just for comparison in terms of speed with fast (PE.m) method
load( 'tonicClonic.mat' );
cfg = [];
cfg.method = 'oldPE'; % compute permutation entropy
cfg.order = 6; % ordinal pattens of order 6 (7-points ordinal patterns)
cfg.delay = 1; % delay 1 between points in ordinal patterns (successive points)
cfg.windowSize = 512; % window size = 512 time steps
cfg.time = 0:1/102.4:179.999; % OPTIONAL time axis for plotting
cfg.units = 'seconds'; % OPTIONAL units of time for plotting
outdata = OPanalysis( cfg, indata );
⛄ 运行结果
⛄ 参考文献
REFERENCES:
[1] Amigo, J.M., Keller, K. and Unakafova, V.A., 2015. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems-Series B, 20(10).
[2] Bandt C., Pompe B., Permutation entropy: a natural complexity measure for time series. Physical review letters, 2002, APS
[3] Keller, K., and M. Sinn. Ordinal analysis of time series. Physica A: Statistical Mechanics and its Applications 356.1 (2005): 114--120
[4] Keller, K., Unakafov, A.M. and Unakafova, V.A., 2014. Ordinal patterns, entropy, and EEG. Entropy, 16(12), pp.6212-6239.
[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] Unakafova, V.A., Keller, K., 2013. Efficiently measuring complexity on the basis of real-world Data. Entropy, 15(10), 4392-4415.
[7] Unakafova, V.A., 2015. Investigating measures of complexity for dynamical systems and for time series (Doctoral dissertation, University of Luebeck).
[8] 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.
[9] 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.
[10] Cao, Y., Tung, W.W., Gao, J.B. et al., 2004. Detecting dynamical changes in time series using the permutation entropy. Physical Review E, 70(4), p.046217.
[11] Riedl, M., Muller, A. and Wessel, N., 2013. Practical considerations of permutation entropy. The European Physical Journal Special Topics, 222(2), pp.249-262
