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# Fractal

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### Fractal

1. 1. Fractal: Correlation Dimension 1
2. 2. 2 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 2
3. 3. 3 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 3
4. 4. Phase Space Assuming that if scientists are willing to study the motion of a rocket, it is not enough for them to only know the height of the rocket at anytime. They also want to know the rocket’s velocity. And then, they can draw the v-h figure. The space that v-h is in is phase space. 4
5. 5. 5 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 5
6. 6. Attractor Attractor is an area in phase space. It draws the system’s phase trail to itself. Or the system moves along its attractor in phase space. Phase space of the pendulum in the earth. 6
7. 7. Attractor How about the attractor of the pendulum in the vacuum? What is the attractor going to be if the suspension point moves like a pendulum? 7
8. 8. 8 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 8
9. 9. Strange Attractor When these attractors (or the motions within them) cannot be easily described as simple combinations of fundamental geometric objects, these attractors are called strange attractors. Chen’s system. 𝑥 = 𝜎 𝑦 − 𝑥 𝑦 = 𝜌 − 𝜎 𝑥 − 𝑥𝑧 + 𝜌𝑦 𝑧 = 𝑥𝑦 − 𝛽𝑧 9
10. 10. Strange Attractor 𝜎 = 35, 𝛽 = 3, 𝜌 = 28, 𝑎 = 1, 𝑏 = 1 𝑎𝑛𝑑 𝑐 = 1, then the phase space is shown following, 10
11. 11. Strange Attractor Two properties It is sensitive to minor variations of initial values. 𝒂 = 𝟏, 𝒃 = 𝟏 𝒂𝒏𝒅 𝒄 = 𝟏 𝒂 = 𝟏, 𝒃 = 𝟏 𝒂𝒏𝒅 𝒄 = 𝟎. 𝟗 The attractor has a fractal structure. 11
12. 12. 12 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 12
13. 13. Fractal What is the Fractal? A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. Self-similarity What kind of fractal characteristic should we focus on? 13
14. 14. Fractal Self-similarity, which may be manifested as: Exact self-similarity: identical at all scales; e.g. Koch snowflake 14
15. 15. Fractal Quasi self-similarity: approximates the same pattern at different scales; e.g., the Mandelbrot set Qualitative self-similarity: as in a time series 15
16. 16. Fractal Statistical self-similarity: repeats a pattern stochastically, so numerical or statistical measures are preserved across scales; e.g., the well- known example ---- the coastline of Britain. What characteristic in a fractal should we focus on? 16
17. 17. Fractal How long is the coastline of Britain. 17 Unit = 200 km, length = 2400 km (approx.) Unit = 50 km, length = 3400 km
18. 18. Fractal How long is the coastline of Britain? However, the founder of fractal, Mandelbrot found that, as the scale of measurement becomes smaller, the measured length of the coastline rises without limit, bays and peninsulas revealing ever- smaller subbays and subpeninsulas —— at least down to atomic scale, where the process does finally come to an end perhaps. 18
19. 19. Fractal How long is the coastline of Britain? In fact, Mandelbrot said, any coastline is—in a sense—infinitely long. In other sense, the answer depends on the length of our ruler. What characteristic in a fractal should we focus on? Fractal dimension 19
20. 20. 20 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 20
21. 21. Fractal Dimension What is the fractal dimension? Fractal dimension is a way of measuring qualities :the degree of irregularity in an object or the efficiency of the object in taking up space. E.g. a twisting coastline, despite its immeasurability in terms of length, nevertheless has a certain characteristic degree of irregularity. The degree remains constant over different scales. More specifically, the fractal dimension of Britain’s coastline is 1.26. 21
22. 22. Fractal Dimension Several formal mathematical definitions of different types of fractal dimension are listed below: Box counting dimension Information dimension Correlation dimension Hausdorff dimension … Although for some classic fractals all these dimensions coincide, in general they are not equivalent. 22
23. 23. Fractal Dimension Strange Attractor and Fractal Dimension Strange Attractor: reflects the system motion trend – It’s sensitive to minor variations of initial values – It has a fractal structure For a dynamic system, if it has some different working statuses, its attractors in these statuses are different. That means, their fractal dimensions are also different. We can analyze fractal dimension of its strange attractor and then know the system’s working status. 23
24. 24. Fractal dimension What information can we use to draw the strange attractor of a system? Exact self-similarity Quasi self-similarity Statistical self-similarity Qualitative self-similarity: as in a time series And use Correlation Dimension 24
25. 25. 25 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 25
26. 26. Correlation Dimension Correlation Dimension changes sensitively with the change of attractor. Therefore, if a dynamic system works in different statuses, in other words, it has different movement trends, their attractors are different and their correlation dimensions are also different. How to calculate the correlation dimension? Obtain the output of the system—time series Calculate the delay time 𝝉 Reconstruct the phase space Calculate the correlation dimension Determine fractal structure and embedded dimension 𝒎 26
27. 27. Correlation Dimension Obtain the output of the system—time series The time series is anything that you can record when the system is working. E.g. the engine speed, amplitude, velocity, acceleration and etc.. We denote it as following: Calculate the delay time 𝜏 In order to determine the delaytime 𝜏, we useAutocorrelationMethod. Use Autocorrelation function: . 27
28. 28. Correlation Dimension Calculate the delay time 𝜏 And 𝜏 meets this condition: , where R(0) is the initial value of Autocorrelation Function. 28
29. 29. Correlation Dimension Reconstruct the phase space We notice that, for a non-linear dynamic system, 𝑥𝑖 is one- dimension, which couldn’t reflect entirely phase space. Therefore, we need to reconstruct the phase space and transform the low dimension phase space 𝒙 intoa high dimension phase space 𝑿: Where 𝑚 is embedding dimension, N is the number of vectors in the new phase space and 𝑁 = 𝑛 − (𝑚 − 1)𝜏. 29
30. 30. Correlation Dimension Calculate the correlation dimension We use 𝜀𝑖𝑗 𝑚 , 𝑖 ≠ 𝑗, to denote the Euclidean Distance of all the points in 𝑋: Then the correlation integral can be calculated as follow: where 𝑯 is Heaviside Function and: 30
31. 31. Correlation Dimension Calculate the correlation dimension Then the correlation dimension, denoted as D, is calculated by: 𝐷 = 𝑑ln𝐶 𝑚(𝑟) 𝑑ln𝑟 . Thus far, we just need to draw the diagram between ln𝐶 𝑚(𝑟) and ln𝑟 and find the scale-free zone, in which the line of ln𝐶 𝑚(𝑟)—ln𝑟 is approximately a straight line, in other words, the slope in this zone is relatively constant. The D is that slope. 31
32. 32. Correlation Dimension Determine fractal structure and embedded dimension 𝑚 we use Saturation Correlation Dimension Method to determine fractal structure and embedded dimension: As the growth in m, when D is relatively stable, the attractor in phase space is a fractal, m is embedded dimension and now, the correlation dimension D is correct. 32
33. 33. 33 Outline Phase Space Attractor Strange Attractor Fractal Fractal Dimension Correlation Dimension Application 33
34. 34. Application In our work, we mainly focus on predicting the direction that the car will turn on. There are three directions that the green vehicle can go. That means these directions represent three trends and three attractors, as well as three parts of correlation dimensions. Time series—vehicle’s acceleration 34
35. 35. Application 35 The acceleration data are shown in (a). Using the Saturation Correlation Dimension Method and drawing the diagram of m- D (b), we are able to find that, the correlation dimension is stable.
36. 36. Application Result From this figure, the correlation dimensions for turning right, turning left and going straight are 10, 12 and 5. Therefore, it’s easy for us to distinguish these three motions. 36
37. 37. Thank You