Statistics and Computer Application

Transcription

1 Statistics and Computer Application GENERATION AND ANALYSIS OF STATISTICAL FRACTALS Bidisha Bandyopadhyay Department of Physics and Astrophysics University of Delhi

2 Outline 1 FRACTALS Types of Fractals Fractal Dimension 2 STATISTICAL FRACTALS Levy Flight Generation Of Levy Flight 3 ANALYSIS The Cumulative Distribution The Methods Asymptotic Method Fourier v/s Asymptotic Determining Fractal Dimension 4 CONCLUSION

3

4 What are FRACTALS?

5 SELF-SIMILAR structures What are FRACTALS?

6 SELF-SIMILAR structures What are FRACTALS? Characteristic FRACTAL DIMENSION

7 What are FRACTALS? SELF-SIMILAR structures Characteristic FRACTAL DIMENSION Not DIFFERENTIABLE anywhere

8 What are FRACTALS? SELF-SIMILAR structures Characteristic FRACTAL DIMENSION Not DIFFERENTIABLE anywhere Reference (Google Images)

9 Types of Fractals

10 Types of Fractals Types of FRACTALS

11 Types of Fractals Types of FRACTALS Deterministic Fractal (e.g. Cantor Set, Mandelbrot Set, Koch Curve)

12 Types of Fractals Types of FRACTALS Deterministic Fractal (e.g. Cantor Set, Mandelbrot Set, Koch Curve)

13 Types of Fractals Types of FRACTALS Deterministic Fractal (e.g. Cantor Set, Mandelbrot Set, Koch Curve) Statistical Fractal (e.g. Levy Flight)

14 Types of Fractals Types of FRACTALS Deterministic Fractal (e.g. Cantor Set, Mandelbrot Set, Koch Curve) Statistical Fractal (e.g. Levy Flight)

15 Fractal Dimension

16 Fractal Dimension How to determine Fractal Dimension?

17 Fractal Dimension Three methods : How to determine Fractal Dimension?

18 Fractal Dimension Three methods : Self-Similarity Method How to determine Fractal Dimension? a = 1 s D

19 Fractal Dimension Three methods : Self-Similarity Method Box Counting Method How to determine Fractal Dimension? a = 1 s D D = log(no. of Boxes containing points) log(scale of Box)

20 Fractal Dimension Three methods : Self-Similarity Method Box Counting Method How to determine Fractal Dimension? a = 1 s D D = log(no. of Boxes containing points) log(scale of Box) General Method D = log(no.ofpoints) log(radius)

21 Levy Flight

22 Levy Flight What are Levy Flights?

23 Levy Flight What are Levy Flights? Markovian Stochastic Processes

24 Levy Flight What are Levy Flights? Markovian Stochastic Processes PDF follow asymptotic Power Law

25 Levy Flight What are Levy Flights? Markovian Stochastic Processes PDF follow asymptotic Power Law Diverging Variance

26 Levy Flight What are Levy Flights? Markovian Stochastic Processes PDF follow asymptotic Power Law Diverging Variance Statistically Self Repeatating (Fractal Dimension=α)

27 Levy Flight

28 Levy Flight Asymptotic Behaviour λ(x) x 1 α

29 Levy Flight Asymptotic Behaviour Behaviour in Fourier Space λ(x) x 1 α ( f(k) = exp [ iµk σ α k α 1 iβ k k (k,α) { tan πα 2 if α 1,0 < α < 2 = 2 πln k if α = 1 f(k) = exp[ k α ] )]

30 Levy Flight

31 Levy Flight f(x) v/s x f(x) x

32 Generation Of Levy Flight

33 Generation Of Levy Flight How to Generate A Levy Flight Distribution?

34 Generation Of Levy Flight How to Generate A Levy Flight Distribution? Asymptotic Method

35 Generation Of Levy Flight How to Generate A Levy Flight Distribution? Asymptotic Method Fourier Method

36 The Cumulative Distribution

37 The Cumulative Distribution F(x) = x 0 f(x)dx

38 The Cumulative Distribution F(x) = x 0 f(x)dx

39 The Methods

40 The Methods Asymptotic Method { 1 0 < x < 1 f(x) = A 1 x > 1 x 1 α A = 1+α α

41 The Methods Asymptotic Method Fourier Method { 1 0 < x < 1 f(x) = A 1 x > 1 x 1 α A = 1+α α f(k) = exp[ k α ]

42 The Methods Figure: Levy Distribution with α = 1.2 at different scales

43 The Methods

44 The Methods Figure: Comparison between Asymptotic Method and Fourier Method

45 Determining Fractal Dimension 11 Log(n) v/s Log (r) "finallog.dat" 1.42x log(n) log(r) Fri May 04 11:48:

46 CONCLUSION

47 CONCLUSION Wide Area of Applications (e.g.structure Formation, Spread of Disease, etc.)

48 CONCLUSION Wide Area of Applications (e.g.structure Formation, Spread of Disease, etc.) Our result gives α = 1.42 with relative error 18.33ì.

49 CONCLUSION Wide Area of Applications (e.g.structure Formation, Spread of Disease, etc.) Our result gives α = 1.42 with relative error 18.33ì. Numerical Methods have their limitation.

50 CONCLUSION Wide Area of Applications (e.g.structure Formation, Spread of Disease, etc.) Our result gives α = 1.42 with relative error 18.33ì. Numerical Methods have their limitation. Further scope for improvement in Numerical Techniques

51 ACKNOWLEDGEMENT I would like to thank Prof.T.R.Seshadri and Dr. Poonam Mehta for their constant support and guidance in this work.

52 THANK YOU