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# Attractors distribution

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### Attractors distribution

1. 1. Attractors of DistributionGeneralized Central limit theorem and Stable distribution Xiong Wang 王雄 Centre for Chaos and Complex Networks City University of Hong Kong 1
2. 2. Outline Normal distribution most prominent probability distribution in simple system. Central limit theorem Why normal distribution is so normal Power Law most prominent probability distribution in complex system. Generalized central limit theorem Stable distribution: Attractor family of 2 distributions
3. 3. Part 1NORMAL DISTRIBUTION 3
4. 4. Probability density function 4
5. 5. Moment and variance 5
6. 6. Normal distribution  where parameter μ is the mean or expectation (location of the peak)  and σ 2 is the variance, the mean of the squared deviation, 6
7. 7. 3-sigma rule about 99.7% are within three standard deviations 7
8. 8. Part 2CENTRAL LIMIT THEOREM 8
9. 9. Central limit theorem 9
10. 10. Central limit theorem  The central limit theorem states that the sum of a number of independent and identically distributed (i.i.d.) random variables with finite variances will tend to a normal distribution as the number of variables grows. C:chaosTalklevyIllustratingTheCentralLimitThe http://demonstrations.wolfram.com/IllustratingTheCentralLimitTheoremWith 10 SumsOfUniformAndExpone/
11. 11. Other distributions can beapproximated by the normal The binomial distribution B(n, p) is approximately normal N(np, np(1 − p)) for large n and for p not too close to zero or one. The Poisson(λ) distribution is approximately normal N(λ, λ) for large values of λ. The chi-squared distribution χ2(k) is approximately normal N(k, 2k) for large ks. The Students t-distribution t(ν) is approximately normal N(0, 1) when ν is large. 11
12. 12. Galton Board If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals According to the central limit theorem the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large. http://www.youtube.com/watch?v=xDIyAOBa_yU C:chaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOf BernoulliRandomV.cdf 12
13. 13. Principle of maximum entropy According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the default. 13
14. 14. Another viewpoint For a given mean and variance, the corresponding normal distribution is the continuous distribution with the maximum entropy. Therefore, the assumption of normality imposes the minimal prior structural constraint beyond these moments 14
15. 15. Summary of normaldistribution First, the normal distribution is very tractable analytically, that is, a large number of results involving this distribution can be derived in explicit form. 15
16. 16. Summary of normaldistribution Second, the normal distribution arises as the outcome of the central limit theorem, which states that under mild conditions the sum of a large number of random variables is distributed approximately normally. Finally, the "bell" shape of the normal distribution makes it a convenient choice for modelling a large variety of random variables encountered in practice. 16
17. 17. 17
18. 18. Part 3POWER LAW 18
19. 19. Income distribution 19
20. 20. 20
21. 21. Normal vs Power law You can hardly find a person twice as tall as you Fair enough… This is normal distribution But you can easily find a person 10000 times richer than you… Extremely unfair… This is power law distribution 21
22. 22. Examples of power lawsa. Word frequency: Estoup.b. Citations of scientific papers: Price.c. Web hits: Adamic and Hubermand. Copies of books sold.e. Diameter of moon craters: Neukum & Ivanov.f. Intensity of solar flares: Lu and Hamilton.g. Intensity of wars: Small and Singer.h. Wealth of the richest people.i. Frequencies of family names: e.g. US & Japan not Korea.j. Populations of cities.
23. 23. 23
24. 24. The Power Law Phenomenon Power Law Bell Curve Distribution Many nodes with Most nodes few links have the sameNo. of nodes No. of nodeswith k links with k links number of links # of links (k) # of links No highly (k) A few nodes connected nodes with many links
25. 25. Part 4GENERALIZED CENTRAL LIMIT THEOREM 25
26. 26. Attraction basin of Gaussian The central limit theorem states that the sum of a number of independent and identically distributed (i.i.d.) random variables with finite variances will tend to a normal distribution as the number of variables grows. All distribution with finite variance form the attraction basin of Gaunssian.what about the distribution having infinite variance? 26
27. 27. Characteristic function 27
28. 28. Gaussian pdf and itscharacteristic function C:chaosTalklevy01FourierTransformPairs. cdf 28
29. 29. Characteristic function as amoment generating function 29
30. 30. Cauchy–Lorentz distribution 30
31. 31. Cauchy–Lorentz distribution PDF Characteristic function Observe that the characteristic function is not differentiable at the origin: So the Cauchy distribution does not have an expected value 31 or Variance.
32. 32. Generalized central limittheorem A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution f(x;α,0,c,0) as the number of variables grows. 32
33. 33. Stable distribution In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha- stable distribution. 33
34. 34.  Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters β and α, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures). C:chaosTalklevyStableDensityFunction.cdf
35. 35. Characteristic function ofStable distribution A random variable X is called stable if its characteristic function is given by 35
36. 36. Symmetric α-stable distributionswith unit scale factor 36
37. 37. Skewed centered stabledistributions with different β 37
38. 38. Unified normal and power law For α = 2 the distribution reduces to a Gaussian distribution with variance σ2 = 2c2 and mean μ; the skewness parameter β has no effect The asymptotic behavior is described, for α < 2 38
39. 39. Log-log plot of skewed centered stable distribution PDFs showing thepower law behavior for large x. Again the slope of the linear portionsis equal to -(α+1)
40. 40. Concluding RemarksThe importance of stable probabilitydistributions is that they are "attractors" forproperly normed sums of independent andidentically-distributed (iid) random variables.The normal distribution is one family of stabledistributions.Without the finite variance assumption the limitmay be a stable distribution, which has thepower law behavior for large x. 40
41. 41. AnalogyChen’s attractor family Stable distribution family At first, Lorenz attractor  At first, normal was found distribution was found For a long time, this was thought as the only story… Then a question raised naturally… Could there be any extension? Then a family of  Then a family of attractor attractors were found of distribution were found which unified Lorenz and which unified normal Chen attractor distribution and power law
42. 42. Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com 42