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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