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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 tC 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: Εχ 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 . VarY 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 PY 0 u Y Pj u .. E Y j det Y ,j0 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 Psup Y t u j C p j u tC 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 EM 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 EM 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 Ex 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 PVmax N 1 exp d C u PV1 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: mt E W RY t , s 2 var W EW RX 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 1i u d 1 2 j i i i j!d 1 j !2 j 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 ji 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 tC follows: u w P sup Y t u P sup X t f wdw tC 0 tC w a u w X t f wdw 0 Psup tC 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 Psup X t by E M Au W X w tC W And we get the following approximation a Psup Y t u E M Au W X f w dw tC 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 0t t Y 0t 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 EW 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 wdw w 0 EV 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 , , , , , . Bibliography [1] Adler, R. J. (1981). The Geometry of Random Fields. John Wiley and Sons, New York. [2] Adler, R. J. (1999). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10:1, 1-74. [3] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag, New York. [4] Alodat, M. T. (2004). Detecing conjunctions using clusters volumes. Ph.D, McGill University, Montreal, Quebec, Canada. [5] Alodat, M. T. and Aludaat K. M. (2007). The generalized hyperbolic process. Brazilian Journal probability and Statistics. (accepted) [6] Alodat, M. T. and AL-Rawwash, M. Y. (2007). Skew Gauusian Random Field. [7] Barndorff- Nielsen, Ole Stelzer and Robert (2004). 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