First-Order Linear Ordinary Differential Equation for Regression Modelling

Abstract

This paper discusses the data-driven regression modelling using first-order linear ordinary differential equation (ODE). First, we consider a set of actual data and calculate the numerical derivative. Then, a general equation for the first-order linear ODE is introduced. There are two parameters, namely the regression parameters, in the equation, and their value will be determined in regression modelling. After this, a loss function is defined as the sum of squared errors to minimize the differences between estimated and actual data. A set of necessary conditions is derived, and the regression parameters are analytically determined. Based on these optimal parameter estimates, the solution of the first-order linear ODE, which matches the actual data trend, shall be obtained. Finally, two financial examples, the sales volume of Proton cars and the housing index, are illustrated. Simulation results show that an appropriate first-order ODE model for these examples can be suggested. From our study, the practicality of using the first-order linear ODE for regression modelling is significantly demonstrated.