Some Applications of Functional Networks in Statistics and Engineering

Abstract

Functional networks are a general framework useful for solving a wide range of problems in probability, statistics, and engineering applications. In this article, we demonstrate that functional networks can be used for many general purposes including (a) solving nonlinear regression problems without the rather strong assumption of a known functional form, (b) modeling chaotic time series data, (c) finding conjugate families of distribution functions needed for the applications of Bayesian statistical techniques, (d) analyzing the problem of stability with respect to maxima operations, which are useful in the theory and applications of extreme values, and (e) modeling the reproductivity and associativity laws that have many applications in applied probability. We also give two specific engineering applications—analyzing the Ikeda map with parameters leading to chaotic behavior and modeling beam stress subject to a given load. The main purpose of this article is to introduce functional networks and to show their power and usefulness in engineering and statistical applications. We describe the steps involved in working with functional networks including structural learning (specification and simplification of the initial topology), parametric learning, and model-selection procedures. The concepts and methodologies are illustrated using several examples of applications.