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

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.