Model Equations: Parameter Estimation

Abstract

Motions and processes observed in nature are extremely diverse and complex. Therefore, opportunities to model them with explicit functions of time are rather restricted. Much greater potential is expected from difference and differential equations (Sects. 3.3, 3.5 and 3.6). Even a simple one-dimensional map with a quadratic maximum is capable of demonstrating chaotic behaviour (Sect. 6.2). Such model equations in contrast to explicit functions of time describe how a future state of an object depends on its current state or how velocity of the state change depends on the state itself. However, a technology for the construction of these more sophisticated models, including parameter estimation and selection of approximating functions, is basically the same. A simple example: construction of a one-dimensional map $\eta_{n+1}=f(\eta_n,\textbf{c})$ differs from obtaining an explicit temporal dependence $\eta=f(t,\textbf{c})$ only in that one needs to draw a curve through experimental data points on the plane $(\eta_n,\eta_{n+1})$ (Fig. 8.1a–c) rather than on the plane $(t,\eta)$ (Fig. 7.1). To construct model ODEs ${{{\mathrm{d}}{\mathbf{x}}} \mathord{\left/ {\vphantom {{{\mathrm{d}}{\mathbf{x}}} {{\mathrm{d}}t}}} \right. \kern-\nulldelimiterspace} {{\mathrm{d}}t}} = {\mathbf{f}}({\mathbf{x}},{\mathbf{c}})$, one may first get time series of the derivatives ${{\mathrm{d}}x_k}\mathord{\left/{\vphantom {{{\mathrm{d}}x_k}{{\mathrm{d}}t}}} \right. \kern-\nulldelimiterspace} {{\mathrm{d}}t}$ ($k=1,{\ldots},\ D$, where D is a model dimension) via numerical differentiation and then approximate a dependence of ${{\mathrm{d}}x_k}\mathord{\left/{\vphantom {{{\mathrm{d}}x_k}{{\mathrm{d}}t}}} \right. \kern-\nulldelimiterspace} {{\mathrm{d}}t}$ on x in a usual way. Model equations can be multidimensional, which is another difference from the construction of models as explicit functions of time.