Globally COnstrained Local Function Approximation via Hierarchical Modelling, a Framework for System Modelling under Partial Information

Local function approximations concern fitting low order models to weighted data in neighbourhoods of the points where the approximations are desired. Despite their generality and convenience of use, local models typically suffer, among others, from difficulties arising in physical interpretation of the parameters and data sparsity in high dimensional situations (or the so called curse of dimensionality). While estimation in parametric global moels, on the other hand, may eliminte the majority of these problems, it generally raises other important issues such as how an appropriate structure should be obtained. This paper presents a new approach for system modelling under partial (global) information (or the so called Gray-box modelling) that seeks to perserve the benefits of the global as well as local methodologies sithin a unified framework. While the proposed technique relies on local approximations, constraints are introduced to ensure the conformity of the estimates to a gien global structure. Hierarchical models are then utilized as a tool to ccomodate global model uncertainties via parametric variabilities within the structure. The global parameters and their associated uncertainties are estimated simultaneously with the (local estimates of) function values. The approach is applied to modelling of a linear time variant dynamic system under prior linear time invariant structure where local regression fails as a result of high dimensionality.