## ← Nonlinear 9.2 Lyapunov Exponents, Kantz, Etc

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Showing Revision 11 created 08/06/2016 by Andrew Medeiros.

1. Like forward Euler, and like backward Euler,
2. the wolff algorithm is a useful thing to talk
3. about in a class because it helps people understand
4. the basic ideas, but no one really uses it anymore.
5. Although, you will see an occasional exception
6. to that statement, usually in the literature of a
7. different field. The primary drawback of the
8. wolff algorithm is its single point nature - the
9. fact that you are picking a starting point and
10. looking for the nearest neighbor at every step.
11. As you can well imagine, any noise in the data
12. might really trash that. The Kantz algorithm
13. and the family of related methods address that
14. by picking a whole bunch of points, tracking them
15. in parallel, averaging their distance to some central
16. point, and watching how that quantity grows.
17. As in the wolff algorithm, you pick a starting point
18. in the trajectory, then you draw an epsilon ball
19. around it, find all the points in the trajectory that
20. are inside that epsilon ball and measure their
21. distances to that central point. Finally, you average
22. all those distances. Now remember, these are all
23. points in a single trajectory. For each one of them, you
24. know where it goes next. If the center point,
25. for example, is the 17th point in the trajectory,
26. you know where the 18th point is, so what you do
27. is you figure out where each of those points
28. goes next...and I'm going to redo this drawing
29. so it's a little less confusing...and this drawing,
30. I'm going to draw in green where each point
31. goes next. This is where the middle one goes next.
32. This one goes here next.
33. This one will go here next.
34. And this one here next, and so on and so forth.
35. This drawing doesn't exactly match the one on the
36. left by the way, it's just a schematic. Then what I do
37. is I compute the distances between the forward image
38. of that center point - which is where the first one went -
39. and the forward images of all those other points.
40. The ratio of the average of all the red distances in
41. this plot to all the average of all the red distances in
42. this plot is a measure of how much the dynamics is
43. stretching out state space around that central point.
44. It's called the stretching factor and that stretching
45. factor is exactly what the Lyapunov exponent is
46. trying to capture. The Kantz algorithm repeats this
47. calculation for a range of different time lengths
48. and a bunch of different initial points.
49. If you plot the results on a log of delta s vs time curve,
50. you'll see a curve like this. Now what's going on here?
51. When time is small, that means that you're not giving
52. things much time to stretch out, so they won't stretch
53. and the ratio will be 1 and the log will be 0.
54. If you let them stretch for a really really long time
55. however, the points in the epsilon ball will spread out all
56. over the attractor and that's as far apart as they can
57. get and that's as far as the volume can grow.
58. That's this upper flat area. In between those
59. asymptotes, there's a scaling region...
60. I apologize for the weakly writing by the way,
62. that I use on the tablet and it's making my writing
63. look awful... Let's think about what a diagonal
64. line on a loglinear plot means. What that means
65. is that the natural log of delta s is proportional
66. to time and the constant of proportionality is a.
67. If we take e to the both sides of this, we'll get...
68. where a is this slope. And state space stretches as
69. e to the lambda t, so the slope of the scaling region, a,
70. is the Lyapunov exponent. You saw a similar thing
71. in the previous segment about calculating the
72. fractal dimension by the way, although the axes
73. were different. There, we were after the
74. relationship between the log of the number of boxes
75. of size epsilon to cover something and the log of
76. one over the size of those boxes, so we plotted
77. log of one over epsilon on the horizontal axis, looked
78. for a scaling region on that curve and took its
79. slope as the value of the capacity dimension.
80. In any calculation like these two things that depends
81. on the existence of a scaling region, it's critically
82. important not to impude the existence of one -
83. that is, to see one there just because you want it there
84. when it's really not. And that's a really subtle problem
85. because scaling regions are very hard to define.
86. You could, for example, fit a line to a chunk of your
87. curve and insist that the r-squared value of that line
88. be above 0.9 or something, but that's an arbitrary
89. threshold. All of this brings up an interesting and
90. important point about nonlinear time series analysis.
91. One that's come up before. This is a very very powerful
92. tool, but it has to be used wisely. And what I mean by
93. wisely, is by a person in the loop. Someone needs
94. to look at those plots and see if they have the right
95. shape before deciding to pick off a fractal dimension
96. or a Lyapunov exponent. If it doesn't have the right
97. shape from your perspective, not mine.
98. Or if that shape doesn't persist as you turn the
99. knobs on the algorithm, like the size of that
100. epsilon ball in the Kantz algorithm, you shouldn't
101. use that curve to pick off those values. And that
102. means, among other things, that these kinds
103. of calculations are all but impossible to automate.
104. By that I mean that if you have 1000 data sets and
105. you want to compute the Lyapunov exponent of
106. all 1000 of them, it's not going to work to
107. simply throw those in the hopper of either of these
108. algorithms and go off and have a beer.
109. The answers won't be right. You need to be involved
110. with the process. That's because the problem is
111. hard and subtle, and you now know how to do that.
112. The reason for this is not that these tools are lousy,
113. but rather that nonlinear time series is a very hard
114. problem. Linear tools are easy to use and they
115. always give you an answer, but if you apply them to
116. a nonlinear system, that answer may well be very
117. wrong. Remember the lamp post. Knowing how these
119. and respect the requirements for what kind of data
120. are needed for them to work properly.
121. The data set needs to be long enough to cover
122. the behavior that you are trying to sample. It needs
123. to be quickly enough sampled to catch all the details
124. in that, and it needs to not have so much noise
125. as to obscure those details. And those are tough
126. requirements sometimes.
127. The TISEAN toolset, by the way, includes the Kantz
128. algorithm for computing Lyapunov exponents from
129. data. All of that was about calculating lambda 1,
130. the largest positive exponent. There are n-1
131. other Lyapunov exponents in an n-dimensional
132. system. How to calculate them? That's much harder.