Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty

Ivan Contreras,Josep Vehi,Remei Calm,Miguel A Sainz,Pau Herrero

doi:10.3390/math9060631

Abstract

Complex systems are usually affected by various sources of uncertainty, and it is essential to account for mechanisms that ensure the proper management of such disturbances. This paper introduces a novel approach to solve symbolic regression problems, which combines the potential of Grammatical Evolution to obtain solutions by describing the search space with context-free grammars, and the ability of Modal Interval Analysis (MIA) to handle quantified uncertainty. The presented methodology uses an MIA solver to evaluate the fitness function, which represents a novel method to manage uncertainty by means of interval-based prediction models. This paper first introduces the theory that establishes the basis of the proposed methodology, and follows with a description of the system architecture and implementation details. Then, we present an illustrative application example which consists of determining the outer and inner approximations of the mean velocity of the water current of a river stretch. Finally, the interpretation of the obtained results and the limitations of the proposed methodology are discussed.

Highlights

Continuous improvements in algorithmic problem-solving, together with an increase in the availability of high-performance computing, have resulted in a new generation of precise and highly detailed mathematical models [1]
The purpose of this paper is to present a computational framework for solving symbolic regression problems, which is able to cope with the uncertainty encountered in real-life problems, and to make predictions accounting for such uncertainty
The presented approach acknowledges the existence of uncertainty and uses intervals to power a Grammatical Evolution (GE) system architecture aiming to resolve regression problems

Summary

Introduction

Continuous improvements in algorithmic problem-solving, together with an increase in the availability of high-performance computing, have resulted in a new generation of precise and highly detailed mathematical models [1]. The identification and management of uncertainty is essential to manage the complexity that occurs in numerous engineering applications [2] Methods that model these uncertainties seek to develop robust mechanisms designed to remain flexible and resilient to appropriately react to new situations. The uncertainty of a system can arise from many sources, such as disturbances from the physical environment (e.g., variability in the flow controlling a water level), noise from devices that collect data (e.g., blood glucose readings from a continuous blood glucose monitor [3]), the representation of errors in physical quantities (e.g., the error in the gravitational acceleration constant g = 9.807 ± 0.027 m/s2), truncation errors in floating point operations, etc These uncertainties will lead to undesired system behaviors if not addressed properly. Methodologies able to manage such uncertainty in a robust manner are required

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Discussion

Conclusion