IV JIAAIS, Osvaldo Rosso Presentation

2. Chaos & Noise
OBJETIVE:
Given a time series,
can we said if it originated by
a chaotic low dimensional dynamics or
it is originated by a stochatics
dynamics ?

3. Noise & Chaotic time series sheare some
characteristic which make them almost
indistiguishable
A wide-band power spectrum
Power spectrum of type f^(-k), with k>0
A delta like auto-correlation function
An irregular behavior of the measured signals
Chaos & Noise

4. Chaos & Information

5. Information Theory Quantifiers
We can use Information Theory based
quantifiers to characterize chaotic systems !!!
Entropic Measures
 Shannon,
 Tsallis,
 Renyi
Fisher’s Information Measure
Generalized Statistical Complexity

6. Information Theory Quantifiers

7. Information Theory Quantifiers

8. • Complexity ?
Information Theory Quantifiers

9. • simple pat
H = 0
Mondrian
The Simple & The Complex

10. • simple pattern
• order
H = 0
Mondrian
The Simple & The Complex

11. H ≠ 0
Pollock
The Simple & The Complex

12. • no pattern
• disorder
H = 1
Pollock
The Simple & The Complex

13. H ≠ 0
Bosch
The Simple & The Complex

14. •
H ≠ 0
Bosch
The Simple & The Complex

15. The Simple & The Complex

16. • some orden
• some patterns
H ≠ 0
Bosch
The Simple & The Complex

17. The Simple & The Complex
H = 0
C = 0
H ≠ 0
C ≠ 0
H = 1
C = 0

18. Complexity

19. The Simple & The Complex
H = 0
C = 0
H ≠ 0
C ≠ 0
H = 1
C = 0

20. Crystal & Ideal Gas
CRYSTAL IDEAL GAS
 High ordered system  Completely disordered
system
 Minimal information stored
in the system
 Maximal information stored
in the system
 Probability Distribution
Function P in phase space:
pj = 1 for j = k
pj = 0 for j ≠ k
 Probability Distribution
Function P in phase space:
pj = 1/N for j = 1, ... , N
(Pequiprobability distribution)
 Maximum D( P, Pe)  Minimum D( P, Pe)

21. Crystal & Ideal Gas
CRYSTAL IDEAL GAS
 High ordered system  Completely disordered
system
 Minimal information stored
in the system
 Maximal information stored
in the system
 Probability Distribution
Function P in phase space:
pj = 1 for j = k
pj = 0 for j ≠ k
 Probability Distribution
Function P in phase space:
pj = 1/N for j = 1, ... , N
(Pequiprobability distribution)
 Maximum D( P, Pe)  Minimum D( P, Pe)

22. Crystal & Ideal Gas
CRYSTAL IDEAL GAS
 High ordered system  Completely disordered
system
 Minimal information stored
in the system
 Maximal information stored
in the system
 Probability Distribution
Function P in phase space:
pj = 1 for j = k
pj = 0 for j ≠ k
 Probability Distribution
Function P in phase space:
pj = 1/N for j = 1, ... , N
(Pequiprobability distribution)
 Maximum D( P, Pe)  Minimum D( P, Pe)

23. Crystal & Ideal Gas
CRYSTAL IDEAL GAS
 High ordered system  Completely disordered
system
 Minimal information stored
in the system
 Maximal information stored
in the system
 Probability Distribution
Function P in phase space:
pj = 1 for j = k
pj = 0 for j ≠ k
 Probability Distribution
Function P in phase space:
pj = 1/N for j = 1, ... , N
(equiprobability distribution)
 Maximum D( P, Pe)  Minimum D( P, Pe)

24. Crystal & Ideal Gas
CRYSTAL IDEAL GAS
 High ordered system  Completely disordered
system
 Minimal information stored
in the system
 Maximal information stored
in the system
 Probability Distribution
Function P in phase space:
pj = 1 for j = k
pj = 0 for j ≠ k
 Probability Distribution
Function P in phase space:
pj = 1/N for j = 1, ... , N
(equiprobability distribution)
 Maximum D( P, Pe)  Minimum D( P, Pe)

25. Statistical Complexity
R. López-Ruiz, H. L. Mancini, and X. Calbet.
A statistical measure of complexity.
Physics Letter A 209 (1995) 321-326.

26. Disorder H

27. Disequilibrium Q

28. Selection of the information measure J

29. Selection of Distance D

30. Generalized
Statistical Complexity Measures
O. A. Rosso, M. T. Martín, A. Figliola, K. Keller, and A. Plastino.
EEG analysis using wavelet-based information tools.
Journal Neuroscience Methods 153 (2006) 163-182.

31. Maximum and Minimum of
Generalized Statistical Complexity Measures
M. T. Martín, A. Plastino, and O. A. Rosso,
Generalized statistical complexity measures: Geometrical and analytical
properties. Physica A 369 (2006) 439-462.

32. Maximum and Minimum of
Generalized Statistical Complexity Measures

33. N = 6
Maximum and Minimum of
Generalized Statistical Complexity Measures

34. Given a time series
we can define the associate probability distribution
function based on
• Frequency counting
• Histogram of amplitudes
• Binary distribution
• Frequency representation (Fourier Transform)
• Frequency bands representation (Wavelet Transform)
• Ordinal patterns (Bandt-Pompe methodology)
• Horizontal Visibility Graph
Probability distribution P

35. • PD
Chaos & Noise
White noise
K = 0
Logistic
Map
R = 4
PDF-Histogram PDF-Histogram

36. Chaos & Noise
White noise
K = 0
Logistic
Map
R = 4
B

37. Chaos & Noise
White noise
K = 0
Logistic
Map
R = 4
B
But Chaotic Dynamics is NOT Stochastic Dynamics !!!

38. Probability Distribution Function
 The PDF must to include the time causality
 Bandt and Pompe Methodology
Bandt C, Pompe B
Permutation entropy: a natural complexity measure for time
series. Phys. Rev. Lett. 88 (2002) 174102.
 Horizontal Visibility Graph
Lacasa L, Luque B, Ballesteros F, Luque J, Nuno JC (2008)
From time series to complex networks: The visibility graph.
Proc. Natl. Acad. Sci. USA 105: 4972–4975.

39. Probability Distribution Function
 The PDF must to include the time causality
 Bandt and Pompe Methodology
Bandt C, Pompe B
Permutation entropy: a natural complexity measure for time
series. Phys. Rev. Lett. 88 (2002) 174102.
 Horizontal Visibility Graph
Lacasa L, Luque B, Ballesteros F, Luque J, Nuno JC
From time series to complex networks: The visibility graph.
Proc. Natl. Acad. Sci. USA 105 (2008) 4972–4975.

40. Bandt-Pompe PDF

41. Bandt-Pompe PDF

42. Bandt-Pompe PDF

43. Bandt-Pompe PDF
Ordinal patterns in a simple time series. (A) Ordinal patterns at the dimension d = 3. (B) Illustration of the
ordinal procedure for d = 3 for sine and white noise time series. (C) Probability distribution of ordinal
patterns π.

44. Logistic Map & White Noise

45. Chaos & Noise

46. Chaos & Noise

47. Chaos & Noise

48. Chaos + Noise

49. N = 10**5 ;
D = 6
O. A. Rosso, L. C. Carpi, P. M. Saco, M. Gómez Ravetti, H. A. Larrondo, A. Plastino
The Amigó paradigm of forbidden/missing patterns: a detailed analysis, The European Physics Journal B 85 (2012) 419 – 430
Chaos + Noise

50. O. A. Rosso, L. C. Carpi, P. M. Saco, M. Gómez Ravetti, H. A. Larrondo, A. Plastino
The Amigó paradigm of forbidden/missing patterns: a detailed analysis, The European Physics Journal B 85
(2012) 419 – 430
Chaos + Noise

51. Some Applications

52. • Thank a lot !!!
• Questions ?
• Gracias !!!
• Preguntas ?
Information Theory Quantifiers