Approximating, modeling, and testing Gaussian processes on stream networks

Geostatistical analysis for spatial or spatial-temporal data plays an important role not only in the field of statistics, but also in a broad scope of disciplines including geography, hydrology, agriculture, etc. Traditionally, spatial data are collected and analyzed on planar or spherical domains. More recently, researchers have started to analyze spatial data on networks, including a specific type of spatial network called a stream network. On a stream network, a feature of interest is measured and it is assumed to be influenced by other factors. Such feature is referred to as the response variable. Typical response variables are stream chemistry quantities such as the density of environmental pollutants and measurements of fish population such as counts of fish. Statistical models can be built to make predictions of the response variable as new data come in. Even though some current modeling approaches can predict the response accurately, a strong assumption, known as quasi-stationarity, is required to make these models valid. In this thesis, we propose some new models by generalizing current models to relax the assumption. The performance of the existing models and the newly developed models are compared using data on trout density in a stream network in Wyoming, USA.