## Quasi-likelihood functions on the sphere

Student thesis: Master Thesis and HD Thesis

- Emil Færk

4. term, Mathematics, Master (Master Programme)

This thesis treats the theory of point processes on the sphere with a focus on quasi-likelihood estimation. The purpose is to introduce theory for point processes on the sphere and investigate a new numeric approximation method for estimating functions and compare it to the Nyström approximation method and implement both in R.

To treat the general theory of point processes we define the surface measure, which is used on the sphere instead of the Lebesgue measure. We introduce some general properties and formulas, which are the foundation of the thesis. Furthermore summary statistics for point processes on the sphere and there non-parametric estimates are presented. These are deduced by applying the theory of the Palm distribution, hence the Palm distribution, and its properties are introduced for point processes on the sphere.

In addition to that some clustered point processes are presented, where the log Gaussian Cox process is used to test the implementation of the quasi-likelihood function.

The quasi-likelihood function is a estimation function, hence we present the theory of estimation functions, where the Godambe information criteria is used to obtain a sufficient condition for a optimal estimation function. This condition is used to obtain a Fredholm integral equation of the second kind, where the solution is a function φ depending on the points of the point process and a parameter vector. The solution φ is the function solving the estimation function optimally. The exact solution of φ is generally not explicit. Therefore two numeric approximation methods are introduced. The first method is based on the theory of Mercer's representation for complex covariance functions. This method does only apply, when the intensity function is constant, i.e. the point process is assumed isotropic, and the observation window is the sphere. When applying Mercer's representation to numerically solve φ, we discover that the numeric solution of φ is a constant, hence the optimal first order estimating function is constant. This implies that the optimal estimate for the intensity is the intuitive estimate.

The second approximation method, we introduce, is the Nyström approximation method. This method applies in a more general context and can be written as a quasi-likelihood function, i.e. a optimal estimating function. The quasi-likelihood function obtain by using Nyström approximation is implemented in R as a function s2quasi which is available in appendix.

In hindsight it is discussed, whether a constant function is a exact solution to φ, when the point process is assumed isotrpoic, and the observation window is the sphere. Therefore the optimal estimate of the intensity is the intuitive estimate.

The implemented R function s2quasi is used to fit a log Gaussian Cox process to the dataset "galaxies", when assuming constant intensity and isotropic covariance function with observation window W=S

Many of the properties and formulas for point process theory on R

Estimation of the intensity function when assumed a isotropic point process with the sphere as observation window gives that the optimal estimate of the intensity function is the intuitive estimate. Further work would involve simulation studies of the quasi-likelihood function compared to a composite likelihood function, examples with quasi-likelihood estimation for inhomogeneous intensity function and estimation of the pair correlation function in practice using either the K-function or the non-parametric estimate of the pair correlation function.

To treat the general theory of point processes we define the surface measure, which is used on the sphere instead of the Lebesgue measure. We introduce some general properties and formulas, which are the foundation of the thesis. Furthermore summary statistics for point processes on the sphere and there non-parametric estimates are presented. These are deduced by applying the theory of the Palm distribution, hence the Palm distribution, and its properties are introduced for point processes on the sphere.

In addition to that some clustered point processes are presented, where the log Gaussian Cox process is used to test the implementation of the quasi-likelihood function.

The quasi-likelihood function is a estimation function, hence we present the theory of estimation functions, where the Godambe information criteria is used to obtain a sufficient condition for a optimal estimation function. This condition is used to obtain a Fredholm integral equation of the second kind, where the solution is a function φ depending on the points of the point process and a parameter vector. The solution φ is the function solving the estimation function optimally. The exact solution of φ is generally not explicit. Therefore two numeric approximation methods are introduced. The first method is based on the theory of Mercer's representation for complex covariance functions. This method does only apply, when the intensity function is constant, i.e. the point process is assumed isotropic, and the observation window is the sphere. When applying Mercer's representation to numerically solve φ, we discover that the numeric solution of φ is a constant, hence the optimal first order estimating function is constant. This implies that the optimal estimate for the intensity is the intuitive estimate.

The second approximation method, we introduce, is the Nyström approximation method. This method applies in a more general context and can be written as a quasi-likelihood function, i.e. a optimal estimating function. The quasi-likelihood function obtain by using Nyström approximation is implemented in R as a function s2quasi which is available in appendix.

In hindsight it is discussed, whether a constant function is a exact solution to φ, when the point process is assumed isotrpoic, and the observation window is the sphere. Therefore the optimal estimate of the intensity is the intuitive estimate.

The implemented R function s2quasi is used to fit a log Gaussian Cox process to the dataset "galaxies", when assuming constant intensity and isotropic covariance function with observation window W=S

^{2}. The estimate of the intensity function using s2quasi corresponds in this setting to the intuitive estimate of the intensity.Many of the properties and formulas for point process theory on R

^{d}can be adapted to point processes on the sphere with clear differences in the non-parametric estimate of the pair correlation function and the relation between the pair correlation function and the K-function.Estimation of the intensity function when assumed a isotropic point process with the sphere as observation window gives that the optimal estimate of the intensity function is the intuitive estimate. Further work would involve simulation studies of the quasi-likelihood function compared to a composite likelihood function, examples with quasi-likelihood estimation for inhomogeneous intensity function and estimation of the pair correlation function in practice using either the K-function or the non-parametric estimate of the pair correlation function.

Language | Danish |
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Publication date | 9 Jun 2017 |

Number of pages | 56 |