Pitfalls in Parameter Estimation for Delay Differential Equations

• Authors: Christopher T. H. Baker;Christopher A. H. Paul
• Venue: SIAM Journal on Scientific Computing
• Year: 1997

Given a set of data $\{U(\gamma_{i}) \approx u(\gamma_{i};\sfp^{\star})\}$ corresponding to the {\em delay differential equation} \vspace{-3pt} $$ \begin{array}{rcll} u'(t;\sfp) &=& f(t,u(t;\sfp),u(\alpha(t;\sfp);\sfp);\sfp) & \hbox{for } t \ge t_{0}(\sfp),\\ u(t;\sfp) &=& \Psi(t;\sfp) & \hbox{for } t \le t_{0}(\sfp), \end{array} $$ the basic problem addressed here is that of calculating the vector $\sfp^{\star} \in {\blackboardd R}^{n}$. (We also consider neutral differential equations.) Most approaches to parameter estimation calculate $\sfp^{\star}$ by minimizing a suitable objective function that is assumed by the minimization algorithm to be sufficiently smooth. In this paper, we use the derivative discontinuity tracking theory for delay differential equations to analyze how jumps can arise in the derivatives of a natural objective function. These jumps can occur when estimating parameters in lag functions and estimating the position of the initial point, and as such are not expected to occur in parameter estimation problems for ordinary differential equations.