The Geometry of GeneralizedHyperbolic Random Field

Hanadi M. Mansour

Yarmouk UniversityFaculty of Science

Supervisor:

Dr. Mohammad AL-Odat

Abstract

Random Field Theory

The Generalized Hyperbolic Random Field

Simulation Study

Conclusions and Future Work

Abstract

In this thesis, we introduce a new non-Gaussian random field called the generalized hyperbolic random field. We show that the generalized hyperbolic random field generates a family of random fields. We study the properties of this field as well as the geometry of its excursion set above high thresholds.We derive the expected Euler characteristic of its excursion set in a close form.

Abstract –Cont.

Also we find an approximation to the expected number of its local maxima above high thresholds.

We derive an approximation to size of one connected component (cluster) of its excursion set above high threshold.

We use simulation to test the validity of this approximation. Finally we propose some future work.

BACK

In this chapter, we introduce to the random field theory and give a brief review of literature.

Most of the material covered in this chapter is based on Adler (1981), Worsely (1994) and Alodat (2004).

Random Field Theory

Random fields

We may define the random field as a collection of random variables

together with a collection of measures or distribution functions.

Random fields –Cont.

A Gaussian random field (GRF) with covariance function R( s, t ) is stationary or homogenous if its covariance function depends only on the difference between two points t, s as follows:

R ( s , t ) = R ( s – t )

And is isotropic if its covariance function depends only on distance between two points t, s as follows:

R ( s , t ) = R ( ║t – s║ )

Excursion set

Let be a random field. For any fixed real number u and any subset we may define the excursion set of the field X (t) above the level u to be the set of all points for t Є C which X (t) ≥ u

i.e.; the excursion setAu (X) = Au (X , C) = {t Є C : X (t) ≥ u}

Excursion set – Cont.

If X (t) is a homogeneous and smooth Gaussian random field, then with probability approaching one as , the excursion set is a union of disjoint connected components or clusters such that each cluster contains only one local maximum of X (t) at its center.

u

Expectation of Euler characteristic

The Euler characteristic simply counts ( the number of connected components) - (number of holes) in Au (Y)

As u gets large, these holes disappear, and as a result the Euler characteristic counts only the number of connected components.

According to Hasofer (1978), the following approximation is accurate.

uasYAEutYP u

Ctsup

Expectation of Euler characteristic – Cont.

Adler (1981) derived a close form of the Expectation of Euler characteristic when the random field is a Gaussian as the following:

Where:

0YVar

21d

1d212

d

u

2π

uΗΛ2

uΧμYΑχΕ

detexp

2

1

0

21

1 2211

d

jj

jdj

d !jd!judΓuH

.

Euler characteristic intensity

Let be an isotropic random field. Cao and Worsley (1999) define , the jth Euler characteristic intensity of the field by

uPYj

dRCttY ,

jRtY

1,,0,0det

0,0

11

.

1

..juuYYYYE

juYP

uPj

j

jj

Yj

Euler characteristic intensity –Cont.

Cao and Worsley (1999) are give the values of for j = 0, 1, 2, 3 when the random field is a Gaussian.

Also, they give the following approximation

uPYj

upCutYP Yj

d

jj

Ct

0

sup

Expectation of the number of local maxima

For a random field Y (t) above the level u.

Let denote the number of local maxima.

Adler (1981) gives the following formula if the random field is a Gaussian

As it follows that u

uO

uuCYAME d

dd

u11

2

2exp

21

212

1

CtYAM u ,

XAMEXAE uu

Expected volume of one cluster using the PCH

YAxE

uFCVE

u

Yd

1

VE

Poisson clumping heuristic (PCH) technique can be employed to find an approximation to the mean value of the volume of one cluster to get the following approximation for

Distribution of the maximum cluster volume

In this section, we will describe how to approximate of the maximum volume of the clusters of the excursion set of a stationary random field Y (t) using the Poisson clumping heuristic approach given by Aldous(1989).

The same procedure was adopted by Friston et al. (1994) to find the distribution of the maximum volume of the excursion set of a single Gaussian random field.

Distribution of the maximum cluster volume –Cont.

Then we have the following formula for the distribution of the maximum cluster

vVPCNVP ud 1max exp1

BACK

The Generalized Hyperbolic Random Field (GHRF)

Let be a Gaussian random field with zero mean and variance equal to one, also let W be a generalized inverse Gaussian random variable independent of .

We define the Generalized Hyperbolic Random Field (GHRF) by:

Where:

tY

tX

CttX ,

tXWWtY

R,

Generalized hyperbolic distribution (GHD)

A random vector Y is said to have a d- dimensional generalized hyperbolic distribution with parameters if and only if it has the joint density

Where

xK

xc

d

dt

21

2

21

2

11 tt yyxq

1

221

2 exp

td

d

Y y

q

qcKyf

,,,,,

Generalized hyperbolic distribution (GHD) – Cont.

We note that the generalized hyperbolic distribution is closed under marginal and conditioning distributions, also it is easy to see that it is closed under affine transformation.

Some special cases

We derive from the generalized hyperbolic distribution the following distributions:1. The one dimensional normal inverse

Gaussian (NIG) distribution.2. The one - dimensional Cauchy distribution3. The variance Gamma distribution.4. The d-dimensional skewed t distribution.5. The d-dimensional student t distribution.

Properties of GHRF tY

tY

tY tY1. The isotropy of .

2. The is also continuous in mean square sense.

3. The is almost surely continuous at t*.

4. The GHRF has the mean square partial derivatives in the ith direction at t.

5. The GHRF is ergodic.

Properties of GHRF -Cont.

6. For every k and every set of points t1,…,tk

C the vector has a multivariate generalized hyperbolic distribution.

7. Differentiability of implies the differentiability of

8. The mean and covariance functions of the GHRF are:

tX

WEtm stRWEWstR XY ,var, 2

tY

ktYtY ,...,1

Expectation of Euler characteristicof (GHRF)

In this section we derive the Expectation of Euler characteristic when the random field generalized hyperbolic random field.

Theorem:

The Expected Euler characteristic of is given by:

CXAEECYAE

WWuWu ,,

CYAu ,

Expectation of Euler characteristicof (GHRF) – Cont.

Then we obtain the following formula:

21

0

21

0

213 1

21

2!1!

1

,

d

j

jd

i

iijdi

j

j

u

ui

jd

jdj

KC

CYAE

ijij

ij

Expectation of Euler characteristicof (GHRF) – Cont.

Where

ijd

RK

C

CCddC

uCC

u

ij

w

wd

21

,0,,2

2

det

exp2

21

21

2

23

2

2

Euler characteristic intensity of Y(t)

Theorem For the GHRF the jth Euler characteristic intensity of is given by:

Based on the previous theorem we have found the values of for j = 0, 1, 2 and 3 in our work.

tY CttY ,

WWuPEuP X

jYj

uP Xj

Expected number of local maximaof Y(t)

Since W varies from 0 to ∞ then we cannot obtain a close form for the expectation of the number of local maxima, but we will obtain the expected number of local maxima of by separating into two parts as follows:

utYP

Ctsup

tY

dwwfw

wutXP

dwwfw

wutXPutYP

Ct

a

CtCt

0

0

sup

supsup

Expected number of local maximaof Y (t) –Cont.

We ignore the second term from the above integral if a is large enough, then we approximate

And we get the following approximation

XAMEbyw

wutXPW

WuCt

sup

dwwfXAMEutYPa

WWu

Ct

0

sup

Size distribution of one component

In this section, we derive an approximation to the distribution of the size of one connected component of .

When To do this, we approximate the field near a local maximum at t = 0 by the quadratic form

YAu

tYu

tYttYtYtY tt 02100

...*

Size distribution of one component -Cont.

The cluster size (the size of one connected component of ) is approximated by V the volume of the d-dimensional ellipsoid

Where:

WEuQ

12

2

dw

d

d

YAu

2

2

det

2dd

d

Q

wEV

uYE

Mean volume of one cluster using PCH

In this section ,we will derive approximation to the mean value of the volume of one cluster of the excursion set of using Poisson clumping heuristic.

dRCttY ,

Mean volume of one cluster using PCH -Cont

For d = 2 we get the approximation formula

21

21

21

21

21

21

2

0

1

KKuC

dwwfw

wuCVE

d

BACK

Comparing the exact and the approximate distributions

The following figures show the simulation results for different values of , FWHM, grid, and λ.

,,,u

Empirical distributions F and G of V at different thresholds for:

72,15,2,1,0 gridfwhm

Fig: 4.1

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

3.50.0378

4.50.0312

5.50.0314

72,15,2,1,0 gridfwhm

Table: 1

Empirical distributions F and G of V at different thresholds for:

72,10,2,1,0 gridfwhm

Fig: 4.3

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1324

2.50.0666

3.50.0556

72,10,2,1,0 gridfwhm

Table: 3

Empirical distributions F and G of V at different thresholds for:

72,10,5.0,1,2,0 gridfwhm

Fig: 4.4

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1086

2.50.1568

3.50.1514

72,10,5.0,1,2,0 gridfwhm

Table: 4

Empirical distributions F and G of V at different thresholds for:

72,10,5.0,0 gridfwhm

Fig: 4.7

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.2354

2.50.2222

3.50.2148

72,10,5.0,0 gridfwhm

Table: 7

Empirical distributions F and G of V at different thresholds for:

82,15,5.0,1,25.0,0 gridfwhm

Fig: 4.8

Empirical distributions F and G of V at different thresholds for:

82,15,5.0,1,25.0,0 gridfwhm

ud ( F, G)

1.50.0198

2.50.0608

3.50.0782

Table: 8

Empirical distributions F and G of V at different thresholds for:

72,10,5.0,10 gridfwhm

Fig: 4.10

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

4.50.1820

5.50.1700

6.50.1360

72,10,5.0,10 gridfwhm

Table: 10

Empirical distributions F and G of V at different thresholds for:

72,10,5.0,1,21,0 gridfwhm

Fig: 4.11

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1052

2.50.0550

3.50.0564

72,10,5.0,1,21,0 gridfwhm

Table: 11

Empirical distributions F and G of V at different thresholds for:

72,15,5.0,1,21,0 gridfwhm

Fig: 4.13

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1154

2.50.0564

3.50.0590

72,15,5.0,1,21,0 gridfwhm

Table: 13

Empirical distributions F and G of V at different thresholds for:

Fig: 4.15

72,15,1,1,21,0 gridfwhm

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1026

2.50.0338

3.50.0322

Table: 15

72,15,1,1,21,0 gridfwhm

Empirical distributions F and G of V at different thresholds for:

82,20,1,21,0 gridfwhm

Fig: 4.16

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

1.50.1084

2.50.0244

3.50.0510

Table: 16

82,20,1,21,0 gridfwhm

Empirical distributions F and G of V at different thresholds for:

72,10,2,15.0,0 gridfwhm

Fig: 4.17

Empirical distributions F and G of V at different thresholds for:

ud ( F, G)

100.0872

150.0364

200.0704

Table: 17

72,10,2,15.0,0 gridfwhm

Discussion of simulation results

From the above Figures we note the following:

1. The CDF G(x) is very close to the CDF of F(x) for different values of .

2. As the level u increases, the CDF G(x) becomes closer to the CDF F (x) in most of the cases.

FWHMu ,,,,,

BACK

Conclusion

In this thesis, we introduced a new random field called the generalized hyperbolic random field.

This field generates a family of random fields, this makes the generalized hyperbolic random field flexible to use in modeling many random responses.

We studied the geometry of the excursion set of the generalized hyperbolic random field.

Conclusion –Cont.

If the random field is homogeneous and smooth, then above high threshold, the excursion set is a disjoint union of connected components or clusters.

Moreover, we derived the expectation of the Euler characteristic in a closed form.

On the other hand, we tried to derive the expectation of the number of local maxima, but it was unfeasible to get this in a closed form because the threshold varies from 0 to ∞.

Conclusion –Cont.

Then, we approximated the expectation of the number of local maxima by the tail distribution of the supremum of the generalized hyperbolic random field.

We also approximated the tail distribution of the supremum of the generalized hyperbolic random field by the expectation of the Euler characteristic.

Conclusion –Cont.

As another part of the thesis, we also derived a closed form approximation to the distribution of the size of one connected component as well as a closed form approximation to the distribution of the excess height of the GHRF above high thresholds.

We discussed the properties of the generalized hyperbolic random field and showed that the Gaussian random field admits mean square differentiability, isotropy, moduli of continuity.

Conclusion –Cont.

Finally we conduct a comparison between the approximate cluster size distribution and the exact cluster size distribution using simulation study.

The results shows that our approximation is very good and valid for large thresholds.

Future work

1. Conjunction of GHRF’s.

2. Predicting the GHRF.

3. Volume and surface area of the body above

the excursion set.

4. Estimation of the parameters . ,,,,,

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