Kernel - based Copula Processes by Eddie K . H . Ng A thesis submitted in conformity

Kernel-based Copula Processes Eddie K. H. Ng Doctor of Philosophy Graduate Department of Electrical & Computer Engineering University of Toronto 2010 The field of time-series analysis has made important contributions to a wide spectrum of applications such as tide-level studies in hydrology, natural resource prospecting in geo-statistics, speech recognition, weather forecasting, financial trading, and economic forecasts and analysis. Nevertheless, the analysis of the non-Gaussian and non-stationary features of time-series remains challenging for the current state-of-art models. This thesis proposes an innovative framework that leverages the theory of copula, combined with a probabilistic framework from the machine learning community, to produce a versatile tool for multiple time-series analysis. I coined this new model Kernel-based Copula Processes (KCPs). Under the new proposed framework, various idiosyncracies can be modeled compactly via a kernel function for each individual time-series, and long-range dependency can be captured by a copula function. The copula function separates the marginal behavior and serial dependency structures, thus allowing them to be modeled separately and with much greater flexibility. Moreover, the codependent structure of a large number of time-series with potentially vastly different characteristics can be captured in a compact and elegant fashion through the notion of a binding copula. This feature allows a highly heterogeneous model to be built, breaking free from the homogeneous limitation of most conventional models. The KCPs have demonstrated superior predictive power when used to forecast a multitude of data sets from meteorological and financial areas. Finally, the versatility of the KCP model is exemplified when it was successfully applied to non-trivial classification problems unaltered.