mega nalysis university of hawaii mathematics department distinguished lecture...

Analysis

Omega

University of Hawaii Mathematics Department

Distinguished Lecture Series

The Geometry of Probability Distributions

February 26 2010

Dr. William F. Shadwick

Omega Analysis Limited

[email protected]

Copyright William F. Shadwick 2010

The Geometry of Probability

Distributions

William F. Shadwick(Joint work with Ana Cascon)

26 February 2010

AnalysisOmega

If F is a cumulative distribution on [A, B] (where eitherof A or B may be infinite) and F has a finite mean µ

then the Omega function of F is defined as

!(x) =I1(x)

I2(x)(1)

where

I1F (x) =! x

AF (z)dz (2)

and

I2F (x) =! B

x1! F (z)dz (3)

Geometry of Distributions 26 Feb 2010 2

AnalysisOmega

The functions I1F and I2F are analogous to the valuesof a put and a call with strike x respectively.

I1F (x) = EF (Max(z ! x,0)) (4)

and

I2F (x) = EF (Max(x! z,0)) (5)

They satisfy a virtual ‘put-call parity’ relation

I2F (x)! I1F (x) = µ! x (6)


AnalysisOmega

!F may be interpreted as the ratio of the value of thedownside to the upside.

The smaller this is, the better o" you are.

The upside and downside balance at the mean.


AnalysisOmega

0

0.9

0.7

0.3

0.1

0.5

1 42 3

1.0

6

0.0

y

0.8

5

0.4

0.2

0.6

µ = 2

F(x)

I1F (µ) = I2F (µ)


AnalysisOmega

Properties of !F :

!(µ) = 1 (7)

d!F

dx=

1

I2F> 0 (8)

limx"A

!F = 0 (9)

limx"B

!F =# (10)


AnalysisOmega

It also follows from the definition and the put-call parityrelation for I1 and I2 that

!F (x) = 1 +x! µF

I2F (x). (11)

We can now see that

!F = !G $% F = G, (12)

since !F=!G implies µF=µG and, from equation 11,

I2F = I2G (13)

Di"erentiating equation 13 gives F=G.


AnalysisOmega

We can also use equation 11 to produce an inverse for-mula for recovering F from !F .

F = 1 +1

!F ! 1+

µ! x

(!F ! 1)2d!F

dx(14)


AnalysisOmega

The Standard Dispersion (a.k.a.‘The om’ ).

We define the om !F by

1

!F=

d!F (µ)

dx. (15)

We have

!F = I1F (µ) = I2F (µ). (16)


AnalysisOmega

0.1

0.7

0.3

0.8

0.6

0.5

0.4

0.2

0.0

x

2.75

x

-1

3.5

2

2.5

-3

3.25

310

3.75

3.0

-2

The om measures spread about the mean. The higherthe concentration the higher the value of !&(µ).


AnalysisOmega

This is a general property. The standard dispersion al-lows us to refine the Markov inequality:For any non-negative random variable x with mean µ,

probability(x > a) 'µ

a. (17)

Our version:

probability(x! µ > b) <!

b. (18)

This is always sharper than (17) for a su#ciently large(and there are distributions where ‘su#ciently large’ isarbitrarily close to the mean.)


AnalysisOmega

Chebychev inequality: For any random variable x withmean µ and variance "2,

probability(|x! µ| > a) '"2

a2 . (19)

Our version:

probability(x > µ + b) <!

b. (20)

and

probability(x < µ! b) <!

b. (21)


AnalysisOmega

1

0.25

y

2

1

0.75

1.25

0.5

43

µ = 2

b=5

(1! F (b))b < ! so probability(x! µ > b) < !b


AnalysisOmega

Properties of the Standard Dispersion

It is easy to check that if

# : x " ax + b (22)

is a proper a#ne transformation and

F = #(G (23)

then

!F = #(!G (24)

It follows from this that ! is translation invariant andscales like the mean.


AnalysisOmega

If two distributions F and G defined on [A, B] have thesame mean µ then

! B

A(F !G)dx = 0. (25)

If, in additon they have the same om then! µ

A(F !G)dx = 0 (26)

and! B

µ(F !G)dx = 0. (27)

By contrast if they have the same standard deviationthen

! B

Ax(F !G)dx = 0. (28)


AnalysisOmega

I 4 I 2 0 2 4

0 .1

0 .2

0 .3

0 .4

Laplace and Normal Distributions with the same

standard deviation (red) and same om (blue).


AnalysisOmega

I 4 I 2 0 2 4

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5

0 .0 6

Absolute Errors in Cumulative Distributions

same sigma(red) and same om (blue).


AnalysisOmega

Standard deviation gives much more weight to outliersthan the om does.

Example: Returns on the S&P 500 for the last 10,361days and the same period excluding 19 October 1987.

$" = 2.23% but $! = 0.29%.


AnalysisOmega

It follows from the scaling property of ! that, when thestandard deviation is defined, the ratio of standard de-viation to standard dispersion is an a#ne invariant. Wehave called this the first C-S Character.

Examples:Uniform distribution: CS1 = 4)

3Normal distribution: CS1 =

)2$

Laplace distribution: CS1 = 2)

2


AnalysisOmega

When distributions have di"erent C-S characters, stan-dard deviation is not a common unit of measurement.

In financial data or instrument measurements subject tonoise, a common number of oms as a threshold for out-liers or noise can be much more appropriate than usingthresholds in standard deviations.


AnalysisOmega

C-S Characters are remarkably robust statistics. Evensmall data samples give results close to population val-ues. They provide a much more practical measure of ‘fattails’ than kurtosis. For the S&P example already cited,$ kurtosis = 240% but $CS1 = 2.8%. (Incidentally theCS1 value of 2.79 proves the returns aren’t normal.)

These statistics are currently being used in investmentrisk management and to remove noise from measure-ments of atmospheric data.(see e.g. Cascon and Shadwick IMCA Journal of Invest-ment Consulting Summer 2006 and Summer 2007)


AnalysisOmega

Properties of the Standard Dispersion Continued

The om is half the mean absolute deviation:

!F =1

2EF (|x! µF |). (29)

Addition formula: If F ="n

i=1 aiFi where the ai arepositive and sum to 1, then µF =

"ni=1 aiµi and

!F =n#

i=1ai!i +

n#

i=1ai

! µF

µi

Fidx (30)


AnalysisOmega

The Geometry Determined by the A#ne Group

The proper a#ne group on the line is the largest sub-group of the di"eomorphism group that preserves theproperty of having a finite first moment.

This group is responsible for the two parameters whichappear in almost all the standard textbook distributions.


AnalysisOmega

The assignment F " log(!F ) commutes with the actionof the a#ne group.

Thus the geometry defined by Omega functions underthis group induces geometry on the space of distribu-tions with finite first moment.

This turns out to have some remarkable structure.


AnalysisOmega

The Equivalence Problem

Coframe on the a#ne group

%1 = ydx (31)

%2 =dy

y(32)

Coframe adapted to the (invariant) I = log(!(x))

!1 = dI =!x

!dx (33)

!2 =dy

y(34)


AnalysisOmega

We have

!1 = J%1 (35)

where

J =!x

y!(36)

Any di"eomorphism that preserves both co-frames alsopreserves J so we have a second functionally independentinvariant.


AnalysisOmega

Now

dI = !1 (37)

anddJ

J= H!1 ! !2, (38)

where

H =!xx!

!2x! 1. (39)


AnalysisOmega

But H depends only on x so it follows that the remaininginformation is in the functional dependence of H on I.

The exceptional equivalence classes are the ones forwhich H is a constant function c.


AnalysisOmega

So far we have not used the information that there isa distribution F for which I = log(!F ). This conditionlimits the possible constant values of H.

When I = log(!F ), we can evaluate H at µ:

H(µ) = 2F (µ)! 1 (40)

from which it follows that !1 < c < 1.


AnalysisOmega

The normal forms corresponding to the exceptional casesc = 0, !1 < c < 0 and 0 < c < 1 are

! = ex (41)

! = x!1c (42)

and

! =( !x)!1c (43)

respectively.


AnalysisOmega

We denote the corresponding exceptional distributionsby CSS0 and CSS& respectively, where & = !1

c . (Thesefirst appeared in a paper with B.A.Shadwick, before wesolved the equivalence problem.)

CSS0 has µ = 0 and ! = 1. It has finite moments of allorders. Its standard deviation is $

$23, which provides a

probabilistic definition of $.


AnalysisOmega

The‘Cascon-Shadwickian’ CSS0 is a global attractor un-der a natural action on the space of Omega functions.This results in a new Central Limit Theorem.

For any integer N > 1, the assignment !F " (!F )N

commutes with the natural action of the a#ne group.

The inversion formula (15) shows (after a long calcula-tion) that !F is the Omega function of a distributionFN . As N tends to #, FN tends to CSS0(µF , !F

N ).


AnalysisOmega

The proof of this result depends only on the behaviourof the invariant H for !&.

H(!&) =1

&H(!) (44)

It is an exercise to show that H is a bounded function,so as & "#, H(!&)" 0.

The convergence is remarkably quick as the followingmovie shows.


AnalysisOmega

Because everything commutes with the a#ne group ac-tion, we may as well pass to the quotient in which:

Any distribution defined on R is replaced by its normalform with µ = 0 and ! = 1

Any distribution defined on a half line bounded belowis replaced by a normal form on [0,#) with µ = 1 (Wehave to use both a#ne parameters to translate the lowerbound to 0 and re-scale the mean to 1 so we cannotchoose the om.)


AnalysisOmega

Any distribution defined on a half line bounded above isreplaced by a normal form on (!#,0] with µ = !1 (Wehave to use both a#ne parameters to translate the lowerbound to 0 and re-scale the mean to !1 so we cannotchoose the om.)

Any distribution defined on a compact interval is replacedby a normal form on [!1,1]. (This uses both a#neparameters so we cannot choose the mean or the om.)


AnalysisOmega

In this quotient, the Cascon-Shadwickian may be re-placed by a '-function with µ = 0 and ! = 0.

(Note that this is quite distinct from a Gaussian '-functionas it is the limit of distributions whose CS character is$

$23 and not

)2$.)


AnalysisOmega

There is nothing (aside from some details in the proof!)to restrict the powers of Omega functions to integers(as the movie illustrated.)

In particular one may take fractional powers, which leadsto a new a#ne invariant: the age of a distribution.


AnalysisOmega

The age is defined as the reciprocal of the smallest powerof !F that is still the Omega function of a distribution.This is clearly an a#ne invariant and only the exceptionaldistribution CSS is ‘ageless’.

Examples:Normal distribution: Age = $)

$($!3)Uniform distribution: Age = 2Exceptional distributions:CSS&: Age = 1

&CSS0: Age =#.


AnalysisOmega

Distributions age in di"erent ways.

Some unimodal distributions (like the normal distribu-tion) start o" as bimodal and age to unimodal.

Some originally tri-modal age to unimodal (like the Laplacedistribution).

Others (like the exceptional distributions) remain uni-modal throughout.


AnalysisOmega

Now we turn to the ‘3 Types Theorem’ of Fisher andTippett. If X is the maximum value of a sample of size n

i.i.d draws of a random variable with distribution G thanProb(X < r) = Gn(r).

The Statistician’s Stability Postulate is that if there is alimiting distribution G as n "# (up to an action of theproper a#ne group) then the limiting distribution mustbe its own attractor so

Gn(x) = G(anx + bn) (45)


AnalysisOmega

The Geometer’s version of the Stability postulate is thatfor all ( * 1

G((x) = G(g(x) (46)

where g( is in the (proper) A#ne group and g1 = Identity.

We have already solved the Equivalence Problem foran invariant function I, which we will now take to beI = log(G) under the action of this group.


AnalysisOmega

We know that the solution depends on the functionalrelation H = H(I) where (Equation 39)

H =GxxG

G2x! 1 =

GxxG!G2x

G2x

(47)

If we write this in terms of derivatives of I = log(G) wehave

H(I) =Ixx

I2x

(48)


AnalysisOmega

We know that I(G() = log(G() = (I(G) and the Stabil-ity Postulate ensures that raising G to the power ( doesnot change the equivalence class.

But we also have (Equation 44)

H((I) =1

(H(I). (49)

Di"erentiating this with respect to ( and evaluating atthe Identity gives

H(I) + IdH

dI= 0. (50)

So H(I) = cI for some constant c.


AnalysisOmega

Combining this with Equation (48) we have

Ixx

Ix= c

Ix

I(51)

ord[log(Ix)]

dx=

d[log(Ic)]

dx(52)

which we solve to find the normal forms correspondingto the Stability Postulate. Integrating once gives

Ix

Ic= c1 (53)

for another constant c1.


AnalysisOmega

So far we have not used the fact that I = log(G) whereG is a probability distribution. This means 0 ' G(x) ' 1so log(G) is negative as is log(!log(G)) = log(!I).

Using these restrictions, in the case that c = 1 Equa-tion 53 gives I = !exp(!c1x + c2) and finally

G(x) = exp(!e!c1x+c2) (54)

This is the Gumbel distribution and its Equivalence Classis given by

H(I) =1

I(55)


AnalysisOmega

The remaining cases which produce the Frechet andWeibull distributions are left as exercises. Their Equiva-lence Classes are given by

H(I) = (1 +1

()I (56)

and

H(I) = (1!1

()I (57)

respectively.


AnalysisOmega

As ( " # both the Frechet and Weibull EquivalenceClasses approach the Gumbel Equivalence Class.

These three cases include all of the relations of the form

H(I) =c

I(58)

so these are all the distribution equivalence classes thatsatisfy the Stability Postulate.


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