Parameter Estimation for Time Varying Dynamical Systems using Least Squares Support Vector Machines

Abstract This paper develops a new approach based on Least Squares Support Vector Machines (LS-SVMs) for parameter estimation of time invariant as well as time varying dynamical SISO systems. Closed-form approximate models for the state and its derivative are first derived from the observed data by means of LS-SVMs. The time-derivative information is then substituted into the system of ODEs, converting the parameter estimation problem into an algebraic optimization problem. In the case of time invariant systems one can use least-squares to solve the obtained system of algebraic equations. The estimation of time-varying coefficients in SISO models, is obtained by assuming an LS-SVM model for it.

[1]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[2]  Robert E. Kalaba,et al.  Quasilinearization and the estimation of chemical rate constants from raw kinetic data [by] R. Bellman [and others] , 1967 .

[3]  J. Varah.  A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations , 1982 .

[4]  Lorenz T. Biegler,et al.  Nonlinear parameter estimation: A case study comparison , 1986 .

[5]  Gunilla Wikström,et al.  Toolbox for Parameter Estimation and Simulation in Dynamic Systems with Application to Chemical Kinetics , 1995 .

[6]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[7]  Johan A. K. Suykens,et al.  Optimal control by least squares support vector machines , 2001, Neural Networks.

[8]  Carmen G. Moles,et al.  Parameter estimation in biochemical pathways: a comparison of global optimization methods. , 2003, Genome research.

[9]  Fernando Pérez-Cruz,et al.  Support Vector Regression for the simultaneous learning of a multivariate function and its derivatives , 2005, Neurocomputing.

[10]  Keh-Dong Shiang,et al.  A perturbation-based estimate algorithm for parameters of coupled ordinary differential equations, applications from chemical reactions to metabolic dynamics , 2009, Comput. Methods Programs Biomed..

[11]  Vivek Dua,et al.  An Artificial Neural Network approximation based decomposition approach for parameter estimation of system of ordinary differential equations , 2011, Comput. Chem. Eng..

[12]  Paul Hofmarcher.  Primal and dual model representations in kernel-based learning , 2011 .