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The Geometry of Generalized Hyperbolic Random Field

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## The Geometry of Generalized Hyperbolic Random Field

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**Yarmouk University**Faculty of Science The Geometry of GeneralizedHyperbolic 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**Chapter Two**Random Field Theory**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, sas follows: R ( s , t ) = R ( s– t ) • And is isotropic if its covariance function depends only on distance between two points t, sas 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 , 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.**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: .**Euler characteristic intensity**• Let be an isotropic random field. Cao and Worsley (1999) define , the jth Euler characteristic intensity of the field by**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**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**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**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 BACK**Chapter Three**The Generalized Hyperbolic Random Field (GHRF)**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:**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**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: • The one dimensional normal inverse Gaussian (NIG) distribution. • The one - dimensional Cauchy distribution • The variance Gamma distribution. • The d-dimensional skewed t distribution. • The d-dimensional student t distribution.**Properties of GHRF**• The isotropy of . • The is also continuous in mean square sense. • The is almost surely continuous at t*. • The GHRF has the mean square partial derivatives in the ith direction at t. • The GHRF is ergodic.**Properties of GHRF -Cont.**• For every k and every set of points t1,…,tk C the vector has a multivariate generalized hyperbolic distribution. • Differentiability of implies the differentiability of • The mean and covariance functions of the GHRF are:**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:**Expectation of Euler characteristicof (GHRF) – Cont.**• Then we obtain the following formula:**Expectation of Euler characteristicof (GHRF) – Cont.**• Where**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.**Expected number of local maximaof Y(t)**• Since W varies from 0to ∞ 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:**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**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**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:**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.**Mean volume of one cluster using PCH -Cont**• For d = 2 we get the approximation formula BACK**Chapter Four**Simulation Study**Comparing the exact and the approximate distributions**• The following figuresshow the simulation results for different values of , FWHM, grid, and λ.**Empirical distributions F and G of V at different thresholds**for: Fig: 4.1**Empirical distributions F and G of V at different thresholds**for: Table: 1**Empirical distributions F and G of V at different thresholds**for: Fig: 4.3**Empirical distributions F and G of V at different thresholds**for: Table: 3**Empirical distributions F and G of V at different thresholds**for: Fig: 4.4**Empirical distributions F and G of V at different thresholds**for: Table: 4**Empirical distributions F and G of V at different thresholds**for: Fig: 4.7**Empirical distributions F and G of V at different thresholds**for: Table: 7**Empirical distributions F and G of V at different thresholds**for: Fig: 4.8**Empirical distributions F and G of V at different thresholds**for: Table: 8**Empirical distributions F and G of V at different thresholds**for: Fig: 4.10**Empirical distributions F and G of V at different thresholds**for: Table: 10**Empirical distributions F and G of V at different thresholds**for: Fig: 4.11