Discovery of Equations: Experimental Evaluation of Convergence

Robert Zembowicz, Jan M. Zytkow

Systems that discover empirical equations from data require large scale testing to become a reliable research tool. In the central part of this paper we discuss two convergence tests for large scale evaluation of equation finders and we demonstrate that our system, which we introduce earlier, has the desired convergence properties. Our system can detect a broad range of equations useful in different sciences, and can be easily expanded by addition of new variable transformations. Previous systems, such as BACON or ABACUS, disregarded or oversimplified the problems of error analysis and error propagation, leading to paradoxical results and impeding the true world applications. Our system treats experimental error in a systematic and statistically sound manner. It propagates error to the transformed variables and assigns error to parameters in equations. It uses errors in weighted least squares fitting, in the evaluation of equations, including their acceptance, rejection and ranking, and uses parameter error to eliminate spurious parameters. The system detects equivalent terms (variables) and equations, and it removes the repetitions. This is important for convergence tests and system efficiency. Thanks to the modular structure, our system can be easily expanded, modified, and used to simulate other equation finders.

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