Attractors distribution

1

Attractors of Distribution

Generalized Central limit theorem and Stable distribution

Xiong Wang 王雄

Centre for Chaos and Complex Networks

City University of Hong Kong

1

2

Outline

Normal distributionmost prominent probability distribution in simple system

Central limit theoremWhy normal distribution is so normal

Power Law most prominent probability distribution in complex system

Generalized central limit theoremStable distribution Attractor family of distributions

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NORMAL DISTRIBUTION

Part 1

3

4

Probability density function

5

Moment and variance

Normal distribution

where

parameter μ

is the mean

or

expectation

(location of

the peak)

and σthinsp2 is the

variance the

mean of the

squared

deviation

6

3-sigma rule

about 997 are within three standard

deviations7

CENTRAL LIMIT THEOREM

Part 2

8

9

Central limit theorem

10

Central limit theorem

The central limit

theorem states that the

sum of a number of

independent and

identically distributed

(iid) random

variables with finite

variances will tend to a

normal distribution as

the number of

variables grows

CchaosTalklevyIllustratingTheCentralLimitThe

httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith

SumsOfUniformAndExpone

Other distributions can be

approximated by the normal

The binomial distribution B(nthinspp) is

approximately normal N(npthinspnp(1 minus p)) for

large n and for p not too close to zero or one

The Poisson(λ) distribution is approximately

normal N(λthinspλ) for large values of λ

The chi-squared distribution χ2(k) is

approximately normal N(kthinsp2k) for large ks

The Students t-distribution t(ν) is

approximately normal N(0thinsp1) when ν is large 11

Galton Board

If the probability of bouncing right on a pin is p (which equals 05 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

httpwwwyoutubecomwatchv=xDIyAOBa_yU

CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf

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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

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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 moments14

Summary of normal

distribution

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

Summary of normal

distribution

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

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POWER LAW

Part 3

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Income distribution

19

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Normal vs Power law

You can hardly find a person twice as tall as

you

Fair enoughhellip

This is normal distribution

But you can easily find a person 10000 times

richer than youhellip

Extremely unfairhellip

This is power law distribution

21

Examples of power laws

a Word frequency Estoup

b Citations of scientific papers Price

c Web hits Adamic and Huberman

d Copies of books sold

e Diameter of moon craters Neukum amp 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 eg US amp Japan not Korea

j Populations of cities

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The Power Law Phenomenon

Most nodes have the same number of links

No highly connected nodes

Many nodes with few links

A few nodes with many links

of links (k) of links (k)

No

of

node

s w

ith

k li

nks

No

of

node

s w

ith

k li

nks

Bell CurvePower Law Distribution

GENERALIZED CENTRAL LIMIT THEOREM

Part 4

25

Attraction basin of Gaussian

The central limit theorem states that the sum

of a number of independent and identically

distributed (iid) 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

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what about the distribution having infinite variance

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Characteristic function

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Gaussian pdf and its

characteristic function

CchaosTalklevy01FourierTransformPairs

cdf

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Characteristic function as a

moment generating function

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CauchyndashLorentz distribution

CauchyndashLorentz 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

or Variance31

Generalized central limit

theorem

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 lt α lt 2 (and therefore having infinite

variance) will tend to a stable distribution

f(xα0c0) as the number of variables grows

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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 Leacutevy alpha-

stable distribution

33

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)

CchaosTalklevyStableDensityFunctioncdf

A random variable X is called stable if its

characteristic function is given by

Characteristic function of

Stable distribution

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Symmetric α-stable distributions

with unit scale factor

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Skewed centered stable

distributions with different β

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 lt 2

Unified normal and power law

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Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)

4040

Concluding Remarks

The importance of stable probability

distributions is that they are attractors for

properly normed sums of independent and

identically-distributed (iid) random variables

The normal distribution is one family of stable

distributions

Without the finite variance assumption the limit

may be a stable distribution which has the

power law behavior for large x

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Analogy

Chenrsquos attractor family

At first Lorenz attractor

was found

Then a family of

attractors were found

which unified Lorenz and

Chen attractor

Stable distribution family

At first normal

distribution was found

Then a family of attractor

of distribution were found

which unified normal

distribution and power

law

For a long time this was thought as the only storyhellipThen a question raised naturallyhellip

Could there be any extension

4242

Xiong Wang 王雄

Centre for Chaos and Complex Networks

City University of Hong Kong

Email wangxiong8686gmailcom

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