Proposal of the “data to equation” algorithm - Inage, Shin-ichi; Yamaguchi, Kazumi (Chaos, Solitons & Fractals, 2018)
A new algorithm to determine the master equation behind time-series data is proposed. For this algorithm, a library of potential terms for the master equation for a finite difference formula was given. Using time-series data, each time-series potential term was estimated. The master equation was written as a linear sum of each potential term. Additionally, each coefficient in front of the potential terms was determined to minimize ∑(linear sum)2 during the given period using the genetic algorithm and Gauss–Seidel method. The genetic algorithm minimized both ∑(linear sum)2 and number of potential terms, because the estimated master equation by the time-series data should be simple. This approach was applied to several functions and so-called Lorenz equations, which demonstrate typical chaotic motion. The time-series data was given through the numerical calculation of sample functions and Lorenz equations. The estimated master equation based on the proposed approach was compared with sample functions and the Lorenz equations. In each case, the coefficients in the equations were in good agreement. There was a small discrepancy between the coefficients of the original Lorenz equations and the estimated master equation. For the Lorenz equations, in the short-term, the attracters were in good agreement. By contrast, the discrepancies influenced the long-term attracters based on chaotic motion.