Model Equations: Parameter Estimation

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