Using polynomial approximations to discover qualitative models

Abstract

The set of ordinary differential equations (ODE) of a physical system determines the precise numerical behavior that will be exhibited in the future. Abstracting an ODE into a qualitative differential equation (QDE) system removes all quantities from the equations and leaves only the structure of the equation system intact. However, a QDE is still very useful in anticipating the future when the qualitative behavior of the system is important rather than the exact numerical solution. Another name for a QDE is a qualitative model. Qualitative relationships make up a qualitative model and it is possible to analyze how a change in one of the variables in the system affects the other variables using these relationships. In other words, qualitative models help us understand the underlying mechanism which determines the behavior that we observe. Writing qualitative models from scratch is a difficult problem which needs to be done by an intelligent expert with domain specific knowledge. Automating the discovery of qualitative models from observations is a difficult problem of machine learning. Various algorithms have been proposed for the solution of this problem in the literature. This thesis presents a new algorithm called LYQUID for the solution of the same problem. The algorithm uses polynomials fitted on observed numerical data as approximations to the underlying real world functions; discovery of qualitative relationships is then performed over those polynomials rather than the original data samples. LYQUID is shown to be a fast and successful learning algorithm which performs very well on benchmark models and tolerates high levels of noise. The algorithm not only relaxes the restrictions over how the data are sampled but it is also capable of working with missing data.