Noise Convolution Models: Fluids in Stochastic Motion, Non-Gaussian Tempo-Spatial Fields, and a Notion of Tilting

Sammanfattning: The primary topic of this thesis is a class of tempo-spatial models which are rather flexible in a distributional sense. They prove quite successful in modeling (temporal) dependence structures and go beyond the limitation of Gaussian models, thus allowing for heavy tails and skewness. By generalizing the construction of the above class of models, it is possible to ‘control’ some random geometric features of the sample path – while keeping the covariance function unaltered. Features such as horizontal and vertical asymmetries (including the question of ‘time- reversibility’ in financial context) and tilting of trajectories. These properties are most prominent in the extremes of the process (but do not exist in e.g. Gaussian models) as shown by means of Rice’s formula for level crossings. Different measures for assessing asymmetries in data records are proposed and model fitting procedures discussed. To combine stochastic and deterministic modeling in the context of numerical weather prediction, we present randomized versions of ‘simple’ physical models based on the shallow water equations. By embedding deterministic shallow water motion into a Gaussian tempo-spatial convolution model, one obtains a velocity field that can be interpreted as stochastically distorted shallow water flow. The methodology is meant to provide prediction, estimation and the handling of uncertainties on various scales.