Parameter range reduction for ODE models using monotonic discretizations

Abstract

This paper analyzes the effectiveness of various monotonic discretizations of an ODE in a parameter range reduction algorithm. Several properties of discretizations are given, and five classes of discretizations are defined for various step numbers s. The range reduction algorithm that employs these discretizations is described. Using both analytical results based on the prototypical model x ′ = λx, and empirical results based on two more complicated models, it is shown that one particular class of discretizations (the A1OUT class) results in the tightest bounds on the parameters. This result is shown to be attributed to a certain characteristic value, A 0 , of the discretization. Accumulation of these discretizations is also defined, and its usefulness in the range reduction algorithm is described.

An algorithm for continuous piecewise linear bounding of discrete time series data

“…We were motivated in particular by ODE parameter identification schemes based on monotonic discretizations [20,19]. Such schemes make use of the fact that data points lie in some known interval.…”

“…In some applications, it is necessary to obtain an upper and lower estimate at any given time, requiring two continuous representations: one bounding the data above and the other below. We are motivated in particular by the need for continuous upper and lower bounds of discrete data for use in a parameter range reduction algorithm for ordinary differential equation models [20,19]. Parameter estimation in this context is typically accomplished by selecting a set of initial parameter values, numerically integrating the model equations and comparing the result to the time series data.…”

Parameter Range Reduction in Ordinary Differential Equation Models

“…The algorithm will make use of a specific family of linear multistep discretizations. In [14], it was shown that the best discretization formulae for parameter range reduction were called A1OUT discretizations, whose form is…”

“…This paper presents an improvement on a parameter range reduction method first introduced in [13,14]. The algorithm discretizes the model equations and uses these discretizations to quickly prune regions of parameter space that are deemed to be inconsistent with the data.…”

“…The final result is a collection of boxes from which a better initial parameter selection can be made. The method in [13,14] required that the discretization over a given time window be monotonic with respect to each parameter. We relax here this condition and show that it results in a smaller region of parameter space that fails to be inconsistent.…”