Studies in non-Gaussian analysis

This thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.

The combination of neural estimates in prediction and decision problems

In this dissertation, different ways of combining neural predictive models or neural-based forecasts are discussed. The proposed approaches consider mostly Gaussian radial basis function networks, which can be efficiently identified and estimated through recursive/adaptive methods. Two different ways of combining are explored to get a final estimate – model mixing and model synthesis –, with the aim of obtaining improvements both in terms of efficiency and effectiveness. In the context of model mixing, the usual framework for linearly combining estimates from different models is extended, to deal with the case where the forecast errors from those models are correlated. In the context of model synthesis, and to address the problems raised by heavily nonstationary time series, we propose hybrid dynamic models for more advanced time series forecasting, composed of a dynamic trend regressive model (or, even, a dynamic harmonic regressive model), and a Gaussian radial basis function network. Additionally, using the model mixing procedure, two approaches for decision-making from forecasting models are discussed and compared: either inferring decisions from combined predictive estimates, or combining prescriptive solutions derived from different forecasting models. Finally, the application of some of the models and methods proposed previously is illustrated with two case studies, based on time series from finance and from tourism.