Abstract

Citation

Researchers should cite this work as follows:

• Ramsay, J. O. (2023). 2007-Ramsay-Hooker-Campbell-Cao-Parameter estimation for differential equations-a generalized smoothing approach. SIMIODE, QUBES Educational Resources. doi:10.25334/CN0T-JT79

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Ramsay, J. O., G. Hooker, D. Campbell and J. Cao. 2007. Parameter estimation for differential equations: a generalized smoothing approach. J. R. Statist. Soc. B.  69(5): 741–796.

See https://ideas.repec.org/a/bla/jorssb/v69y2007i5p741-796.html .  Accessed 27 March 2023.

Summary: We propose a new method for estimating parameters in models that are defined by a system of non-linear differential equations. Such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation. The paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non-linear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive estimates and confidence intervals, and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology. The performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus.

Keywords: differential equation; dynamic system; estimating equation; functional data analysis; Gauss–Newton method; parameter cascade; profiled estimation, parameter estimation