Transcript The Geometry of Generalized Hyperbolic Random Field

Yarmouk University
Faculty of Science
The Geometry of Generalized
Hyperbolic Random Field
Hanadi M. Mansour
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
Random Field Theory
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 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 set
Au (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 u   , 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.
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.
  E  A Y  as u  


P
sup
Y
t

u


u
 tC

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:
Εχ  Αu Y  
Where:
 d 1 
 2 


 u2
μ d  Χ  exp  
 2
  21 
 det Λ  Η d 1 u 
 
  
2π 
j

 1 u d 1 2 j
H d 1 u   Γ d  
j
j  0 j ! d  1  2 j ! 2
.
  VarY 0
d 1
2
Euler characteristic intensity
Let Y t , t  C  R
be an isotropic
random field. Cao and Worsley (1999)
Y
P
define j u  , the jth Euler characteristic
j
intensity of the field Y t   R by
d
 PY 0  u 

 
Y


Pj u    
 ..
 E Y j det   Y
 
,j0

Y

.
j 1
j 1


 0, Y  u  j 1 0, u  , j  1

Euler characteristic intensity –Cont.
Cao and Worsley (1999) are give the
values of PjY u  for j = 0, 1, 2, 3 when
the random field is a Gaussian.
Also,
they
approximation
give
the
following


Y
Psup Y t   u     j C p j u 
 tC
 j 0
d
Expectation of the number of local
maxima
For a random field Y (t) above the level u.
Let M  Au Y , t  C denote the number of local
maxima.
Adler (1981) gives the following formula if the
random field is a Gaussian
1
2
EM  Au Y  
 d C   u
d 1
2
As u   it follows that
 u2
exp  
 2
d 1
2


 1  O 1  
 

 u 

E  Au  X   EM  Au  X 
Expected volume of one cluster
using the PCH
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 E V 
E V  
 d C 1  FY u 
Ex Au Y 
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
PVmax   N  1  exp   d C  u PV1  v
BACK
The Generalized Hyperbolic
Random Field (GHRF)
Let X t , t  C 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 X t  .
We define Y t  the Generalized Hyperbolic
Random Field (GHRF) by:
Y t     W  W X t 
Where:
,  R
Generalized hyperbolic distribution
(GHD)
A random vector Y is said to have a ddimensional generalized hyperbolic distribution
with parameters  ,  , ,  ,  ,  if and only if it
has the joint density
cK
fY  y 
Where
q
 q
d

2
1 d

  
2 2


exp  y   
t
q  x   y      y        
 x        
c
2   K  x 
1
t

t

d
2
t
1
2

1
d

  
2

1

1


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 distribution
3. The variance Gamma distribution.
4. The d-dimensional skewed t distribution.
5. The d-dimensional student t distribution.
Y t 
Properties of GHRF
1. The isotropy of Y t .
2. The Y t  is also continuous in mean
square sense.
3. The Y t  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 Y t1 ,..., Y t k  vector
has a multivariate generalized hyperbolic
distribution.
7. Differentiability of X t  implies the
differentiability of Y t 
8. The mean and covariance functions of the
GHRF are:
mt      E W 
RY t , s    2 var W   EW R X t , s 
Expectation of Euler characteristic
of (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 Au Y , C 
is given by:
  

E  Au Y , C   EW  E    Au   W  X , C 



  
W

Expectation of Euler characteristic
of (GHRF) – Cont.
Then we obtain the following formula:
E  Au Y , C  
 d 1 
 2  d 1 2 j


C3
 
j 0
i 0
K 
  

  ij
ij


    1   j
  

 ij
d 1 2 j 
i
d 1 2 j i i
 1 u   
 

j
i

j!d  1  j !2 
Expectation of Euler characteristic
of (GHRF) – Cont.
Where      u   2
     2
C 3  2C 2 exp  u   
C2 
d d C  det  



2 
d 1
2

 

Cw 
2 K  
Cw

,  ,  0,   R
 d 1
   
 j i
 2 

ij
1
2
Euler characteristic intensity of Y(t)
Theorem
For the GHRF Y t , t  C the jth Euler
characteristic intensity of Y t is given by:

 X  u    W 
P u   E Pj 

W

 
Y
j
Based on the previous theorem we have
found the values of PjX u  for j = 0, 1, 2
and 3 in our work.
Expected number of local maxima
of 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 Y t  by
 into two parts as
separating P


sup
Y
t

u


 tC

follows:

u    w 


P sup Y t   u    P sup X t  
 f wdw 
 tC
 0  tC
w

a


u    w 
X t  
 f wdw
0 P sup
tC
w

Expected number of local maxima
of Y (t) –Cont.
We ignore the second term from the above
integral if a is large enough, then we
approximate



u    w 
 


P sup X t  
by E M  Au   W  X 
w 


W

 

And we get the following approximation
t C
a




 


Psup Y t   u    E M Au   W  X   f w dw


 tC
 0 
W

 
Size distribution of one component
In this section, we derive an approximation to
the distribution of the size of one connected
component of Au Y  .
When u   To do this, we approximate the
field Y t  near a local maximum at t = 0 by the
quadratic form
Y t 
*
1 t ..
Y 0   t Y 0 t  t Y 0 t
2
t
.
Size distribution of one component Cont.
The cluster size (the size of one connected
component of Au Y  ) is approximated by V the
volume of the d-dimensional ellipsoid
V 
Where:
d
2
2 E wd
det Q 
d
2
E Y u
Q   u     EW 
wd 

d
2
 d 


 2  1
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 Y t , t  C  R d using
Poisson clumping heuristic.
Mean volume of one cluster using
PCH -Cont
For d = 2 we get the approximation formula
   u    w 


 d C 1   
 f wdw 
w


0


EV  


K 1   
K 1   




2
2
C 2 u   



1
1
1







 2





2
2








 




 





1
2
BACK
Comparing the exact and the
approximate distributions
The following figures show the simulation
results for different values of u ,  ,  ,  ,
FWHM, grid, and λ.
Empirical distributions F and G of V
at different thresholds for:
    0,     1,   2, fwhm  15, grid  2 7
Fig: 4.1
Empirical distributions F and G of V
at different thresholds for:
    0,     1,   2, fwhm  15, grid  2 7
u
d ( F, G)
3.5
0.0378
4.5
0.0312
5.5
0.0314
Table: 1
Empirical distributions F and G of V
at different thresholds for:
    0,     1,   2, fwhm  10, grid  2 7
Fig: 4.3
Empirical distributions F and G of V
at different thresholds for:
    0,     1,   2, fwhm  10, grid  2 7
u
d ( F, G)
1.5
0.1324
2.5
0.0666
3.5
0.0556
Table: 3
Empirical distributions F and G of V
at different thresholds for:
    0,   2,   1,   0.5, fwhm  10, grid  2 7
Fig: 4.4
Empirical distributions F and G of V
at different thresholds for:
    0,   2,   1,   0.5, fwhm  10, grid  2 7
u
d ( F, G)
1.5
0.1086
2.5
0.1568
3.5
0.1514
Table: 4
Empirical distributions F and G of V
at different thresholds for:
    0,       0.5, fwhm  10, grid  2 7
Fig: 4.7
Empirical distributions F and G of V
at different thresholds for:
    0,       0.5, fwhm  10, grid  2 7
u
d ( F, G)
1.5
0.2354
2.5
0.2222
3.5
0.2148
Table: 7
Empirical distributions F and G of V
at different thresholds for:
  0,   0.5   2,   1,   0.5, fwhm  15, grid  28
Fig: 4.8
Empirical distributions F and G of V
at different thresholds for:
  0,   0.5   2,   1,   0.5, fwhm  15, grid  28
u
d ( F, G)
1.5
0.0198
2.5
0.0608
3.5
0.0782
Table: 8
Empirical distributions F and G of V
at different thresholds for:
    0   1,     0.5, fwhm  10, grid  2 7
Fig: 4.10
Empirical distributions F and G of V
at different thresholds for:
    0   1,     0.5, fwhm  10, grid  2 7
u
d ( F, G)
4.5
0.1820
5.5
0.1700
6.5
0.1360
Table: 10
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   0.5, fwhm  10, grid  2 7
Fig: 4.11
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   0.5, fwhm  10, grid  2 7
u
d ( F, G)
1.5
0.1052
2.5
0.0550
3.5
0.0564
Table: 11
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   0.5, fwhm  15, grid  2 7
Fig: 4.13
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   0.5, fwhm  15, grid  2 7
u
d ( F, G)
1.5
0.1154
2.5
0.0564
3.5
0.0590
Table: 13
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   1, fwhm  15, grid  2 7
Fig: 4.15
Empirical distributions F and G of V
at different thresholds for:
  0,   1   2,   1,   1, fwhm  15, grid  2 7
u
d ( F, G)
1.5
0.1026
2.5
0.0338
3.5
0.0322
Table: 15
Empirical distributions F and G of V
at different thresholds for:
  0,   1     2,   1, fwhm  20, grid  28
Fig: 4.16
Empirical distributions F and G of V
at different thresholds for:
  0,   1     2,   1, fwhm  20, grid  28
u
d ( F, G)
1.5
0.1084
2.5
0.0244
3.5
0.0510
Table: 16
Empirical distributions F and G of V
at different thresholds for:
  0,   0.5     1,   2, fwhm  10, grid  2 7
Fig: 4.17
Empirical distributions F and G of V
at different thresholds for:
  0,   0.5     1,   2, fwhm  10, grid  2 7
u
d ( F, G)
10
0.0872
15
0.0364
20
0.0704
Table: 17
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 u,  , ,  ,  , FWHM .
2. As the level u increases, the CDF G(x) becomes
closer to the CDF F (x) in most of the cases.
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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