Permutation Entropies and Chaos

Transcription

1 Permutation Entropies and Chaos Russell Hawkins Complexity Sciences Center Physics Department University of California, Davis June 2nd, 2015 Russell Hawkins (UC Davis) Group Meeting 1 / 30

2 Nanoelectromechanical Systems 500 nm Roukes group Nano Letters 2011 Sub-micron scale resonators Mechanical behavior electrically measure/controlled Subject to thermal fluctuations We want to make Maxwell Demons with them! Russell Hawkins (UC Davis) Group Meeting 2 / 30

3 Nanobeams = Duffing Oscillators Compress a beam, eventually buckles up/down Can model bistable beam dynamics with Duffing equation Russell Hawkins (UC Davis) Group Meeting 3 / 30

4 Duffing Equation ẍ + γẋ αx + βx 3 = G cos ωt Interested in mapping behavior over parameter space Also want to characterize behavior of experimental system Exploring utility of Permutation Entropy Russell Hawkins (UC Davis) Group Meeting 4 / 30

5 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Russell Hawkins (UC Davis) Group Meeting 5 / 30

6 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Russell Hawkins (UC Davis) Group Meeting 5 / 30

7 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Simple algorithm Russell Hawkins (UC Davis) Group Meeting 5 / 30

8 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Simple algorithm Fast Russell Hawkins (UC Davis) Group Meeting 5 / 30

9 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Simple algorithm Fast Robust to noise Russell Hawkins (UC Davis) Group Meeting 5 / 30

10 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Simple algorithm Fast Robust to noise Can get Shannon entropy rate of process in infinite limit Russell Hawkins (UC Davis) Group Meeting 5 / 30

11 Permutation Entropy Measure chaos directly from data, without equations of motion? Permutation Entropy: Doesn t need description of process Simple algorithm Fast Robust to noise Can get Shannon entropy rate of process in infinite limit Data requirement grows quickly Russell Hawkins (UC Davis) Group Meeting 5 / 30

12 Algorithm Look at length n window Russell Hawkins (UC Davis) Group Meeting 6 / 30

13 Algorithm Look at length n window Rearrange elements in increasing order Russell Hawkins (UC Davis) Group Meeting 6 / 30

14 Algorithm Look at length n window Rearrange elements in increasing order Record permutation of elements Russell Hawkins (UC Davis) Group Meeting 6 / 30

15 Algorithm Look at length n window Rearrange elements in increasing order Record permutation of elements Repeat for all length n windows in time series Russell Hawkins (UC Davis) Group Meeting 6 / 30

16 Algorithm Look at length n window Rearrange elements in increasing order Record permutation of elements Repeat for all length n windows in time series PE = Shannon entropy of distribution over permutations: H n = π p(π) log 2 p(π) where π = set of all n! permutations Russell Hawkins (UC Davis) Group Meeting 6 / 30

17 A Simple Example x = (5, 8, 10, 11, 7) Windows of length n = 3: Russell Hawkins (UC Davis) Group Meeting 7 / 30

18 A Simple Example Windows of length n = 3: x = (5, 8, 10, 11, 7) (5, 8, 10): x 1 < x 2 < x 3, permutation is 123 Russell Hawkins (UC Davis) Group Meeting 7 / 30

19 A Simple Example x = (5, 8, 10, 11, 7) Windows of length n = 3: (5, 8, 10): x 1 < x 2 < x 3, permutation is 123 (8, 10, 11): x 1 < x 2 < x 3, permutation is 123 Russell Hawkins (UC Davis) Group Meeting 7 / 30

20 A Simple Example x = (5, 8, 10, 11, 7) Windows of length n = 3: (5, 8, 10): x 1 < x 2 < x 3, permutation is 123 (8, 10, 11): x 1 < x 2 < x 3, permutation is 123 (10, 11, 7): x 3 < x 1 < x 2, permutation is 312 Russell Hawkins (UC Davis) Group Meeting 7 / 30

21 A Simple Example x = (5, 8, 10, 11, 7) Windows of length n = 3: (5, 8, 10): x 1 < x 2 < x 3, permutation is 123 (8, 10, 11): x 1 < x 2 < x 3, permutation is 123 (10, 11, 7): x 3 < x 1 < x 2, permutation is 312 H 3 = 2/3 log 2 2/3 1/3 log 2 1/3 H bits Russell Hawkins (UC Davis) Group Meeting 7 / 30

22 Permutations Entropy Rate Define For a broad class of systems, h Hn = lim n n 1 h = h µ Permutations partition time series by relative values of neighbors Derives partitions from the data itself Side steps issue of finding generating partition Russell Hawkins (UC Davis) Group Meeting 8 / 30

23 Duffing Oscillator Permutation Entropy Rate Focus on 1 degree of freedom (position) Sample continuous time data with at least twice Nyquist cutoff frequency Compare with Kolmogorov-Sinai Entropy (largest Lyapunov exponent) Russell Hawkins (UC Davis) Group Meeting 9 / 30

24 Duffing Oscillator Entropy Rate 2.5 Entropy Rate Amplitude Frequency 0.00 Russell Hawkins (UC Davis) Group Meeting 10 / 30

25 Duffing Oscillator Permutation Entropy Rate 2.5 Permutation Entropy Amplitude Frequency Russell Hawkins (UC Davis) Group Meeting 11 / 30

26 Permutation Entropy Rate vs KS Entropy Rate 2.5 Permutation Entropy Entropy Rate Amplitude Amplitude Frequency Frequency 0.00 Russell Hawkins (UC Davis) Group Meeting 12 / 30

27 Permutation Entropy for 1-D Maps To get a better grip on convergence properties, look at simpler systems: Tent Map Logistic Map Don t need to worry about sampling frequency issue, h µ much easier to calculate. Russell Hawkins (UC Davis) Group Meeting 13 / 30

28 Tent Map h n Convergence Russell Hawkins (UC Davis) Group Meeting 14 / 30

29 Tent Map h n Convergence Russell Hawkins (UC Davis) Group Meeting 15 / 30

30 Logistic Map h n Convergence Russell Hawkins (UC Davis) Group Meeting 16 / 30

31 Logistic Map h n Convergence Russell Hawkins (UC Davis) Group Meeting 17 / 30

32 Tent Map h n vs h µ Russell Hawkins (UC Davis) Group Meeting 18 / 30

33 Logistic Map h n vs h µ Russell Hawkins (UC Davis) Group Meeting 19 / 30

34 Permutation Excess Entropy Defined analogously to standard Excess Entropy: E lim n (H n h n) Russell Hawkins (UC Davis) Group Meeting 20 / 30

35 Permutation Excess Entropy Defined analogously to standard Excess Entropy: E lim n (H n h n) Converges to E for same class of processes for which h = h µ Russell Hawkins (UC Davis) Group Meeting 20 / 30

36 Permutation Excess Entropy Defined analogously to standard Excess Entropy: E lim n (H n h n) Converges to E for same class of processes for which h = h µ Length n approximate: E n H n h µ n Russell Hawkins (UC Davis) Group Meeting 20 / 30

37 Wrinkle for 1-D Maps 1-D maps don t have a unique Excess Entropy! Different generating partitions give different EE values All generating partitions converge to same h µ, but may differ in how they converge Which Excess Entropy does E converge to for 1-D maps? Russell Hawkins (UC Davis) Group Meeting 21 / 30

38 Tent Map Binary Excess Entropy Russell Hawkins (UC Davis) Group Meeting 22 / 30

39 Tent Map E* Russell Hawkins (UC Davis) Group Meeting 23 / 30

40 Tent Map E* Russell Hawkins (UC Davis) Group Meeting 24 / 30

41 Tent Map E* Russell Hawkins (UC Davis) Group Meeting 25 / 30

42 Logistic Map Binary Excess Entropy Russell Hawkins (UC Davis) Group Meeting 26 / 30

43 Logistic Map E* Russell Hawkins (UC Davis) Group Meeting 27 / 30

44 Logistic Map E Russell Hawkins (UC Davis) Group Meeting 28 / 30

45 Logistic Map E Russell Hawkins (UC Davis) Group Meeting 29 / 30

46 Conclusions Permutation Entropy is cool! Definitely captures qualitatively chaos/order in continuous time systems Wants to converge to h µ for 1-D maps Takes its time in doing so Permutation Excess Entropy is cool! Tantalizing, raises more questions than it answers Which Excess Entropy does it converge too? Reflects what kind of partition permutations are Russell Hawkins (UC Davis) Group Meeting 30 / 30