Partially linear models and their applications to change point detection of chemical process data

In many chemical data sets, the amount of radiation absorbed (absorbance) is related to the concentration of the element in the sample by Lambert–Beer's law. However, this relation changes abruptly when the variable concentration reaches an unknown threshold level, the so-called change point. In the context of analytical chemistry, there are many methods that describe the relationship between absorbance and concentration, but none of them provide inferential procedures to detect change points. In this paper, we propose partially linear models with a change point separating the parametric and nonparametric components. The Schwarz information criterion is used to locate a change point. A back-fitting algorithm is presented to obtain parameter estimates and the penalized Fisher information matrix is obtained to calculate the standard errors of the parameter estimates. To examine the proposed method, we present a simulation study. Finally, we apply the method to data sets from the chemistry area. The partially linear models with a change point developed in this paper are useful supplements to other methods of absorbance–concentration analysis in chemical studies, for example, and in many other practical applications.