Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays

Ahmed A Mahmoud,Vijanth S Asirvadam,Mohana S Muthuvalu,Sarat C Dass

doi:10.1155/2017/6148934

Ahmed A Mahmoud, Vijanth S Asirvadam + Show 2 more

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https://doi.org/10.1155/2017/6148934

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Journal: Complexity	Publication Date: Jan 1, 2017
Citations: 1	License type: CC BY 4.0

Affiliation: Universiti Teknologi Petronas

Abstract

This article presents statistical inference methodology based on maximum likelihoods for delay differential equation models in the univariate setting. Maximum likelihood inference is obtained for single and multiple unknown delay parameters as well as other parameters of interest that govern the trajectories of the delay differential equation models. The maximum likelihood estimator is obtained based on adaptive grid and Newton-Raphson algorithms. Our methodology estimates correctly the delay parameters as well as other unknown parameters (such as the initial starting values) of the dynamical system based on simulation data. We also develop methodology to compute the information matrix and confidence intervals for all unknown parameters based on the likelihood inferential framework. We present three illustrative examples related to biological systems. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2014b.

Highlights

Delay differential equations (DDEs) are widely used to model many real life phenomena, especially in science and engineering
To find the Maximum likelihood estimators (MLEs) numerically, we develop a two-stage numerical procedure consisting of an adaptive grid procedure, followed by a gradient descent algorithm
We presented the method of maximum likelihood for estimating parameters in delayed differential equations

Summary

Introduction

Delay differential equations (DDEs) are widely used to model many real life phenomena, especially in science and engineering. A univariate delay differential equation model (DDEM) with multiple delays equates the real valued observations, yi, as noisy realizations from an underlying DDE: yi = x (ti) + εi, i = 0, 1, 2, . M, is the jth delay term with delay parameter τj > 0, and θ = Θp) is a vector of other parameters of interest that govern the trajectories of the underlying DDE in (2). Equations (1) and (2) constitute a univariate DDEM in the most general form. In a DDEM, the parameters θr and τj are often unknown and have to be estimated based on observations yi, i = 0, 1, .

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