Autonomous Orientation and Geolocation via Celestial Objects

Liu, Cheng Ph.D., Purdue University, December 2013. Non-parametric Spatial Models. Major Professor: Hao Zhang. Covariance functions play a central role in spatial statistics. Parametric covariance functions have been used in most of the existing works on the analysis of spatial data. The primary reason for this is that the classes of parametric covariance functions guarantee that the fitted covariance function is positive definite. In this dissertation, I undertake two non-parametric approaches to modelling the covariance functions. Our approach is motivated by problems that arise in spatial data analysis in recent years. First, it is nontrivial to choose a parametric family among many parametric families of covariance function. A non-parametric covariance function circumvents this problem. Secondly, for a parametric covariance function, the likelihood becomes difficult to compute when the sample size is very large. There are more and more situations where the spatial sample sizes are very large. Although techniques have been developed in recent years that allow for the computation of likelihood for a very large sample size, these techniques can be applied to our non-parametric models as well. Thirdly, the most popular parametric families of covariance function are monotone— that is, the covariance function decreases as the distance increases. Although this monotonicity holds most of the time in applications, there are times it fails to hold such as in the teleconnection in climatology. The dissertation can be divided into two parts. In the first part, we propose a non-parametric low-rank model, which is a non-parametric extension of the parametric low-rank models that have been studied by several authors. A key component in xvi the construction of the non-parametric model is the Lagrange polynomial interpolation. In the second part, we focus on a non-parametric approach that can lead to a covariance function that is appropriate for modelling teleconnection. We will apply this approach to the study of teleconnection of temperature and precipitation across the world.