Modelling temporal biomarkers with semiparametric nonlinear dynamical systems

SummaryDynamical systems based on differential equations are useful for modelling the temporal evolution of biomarkers. Such systems can characterize the temporal patterns of biomarkers and inform the detection of interactions between biomarkers. Existing statistical methods for dynamical systems deal mostly with single time-course data based on a linear model or generalized additive model. Hence, they cannot adequately capture the complex interactions between biomarkers; nor can they take into account the heterogeneity between systems or subjects. In this article, we propose a semiparametric dynamical system based on multi-index models for multiple-subjects time-course data. Our model accounts for between-subject heterogeneity by incorporating system-level or subject-level covariates into the dynamical systems, and it allows for nonlinear relationships and interactions between the combined biomarkers and the temporal rate of each biomarker. For estimation and inference, we consider a two-step procedure based on integral equations from the proposed model. We propose an algorithm that iterates between estimation of the link function through splines and estimation of the index parameters, and which allows for regularization to achieve sparsity. We prove model identifiability and derive the asymptotic properties of the estimated model parameters. A benefit of our approach is the ability to pool information from multiple subjects to identify the interactions between biomarkers. We apply the method to analyse electroencephalogram data for patients affected by alcohol dependence. The results provide new insights into patients’ brain activities and demonstrate differential interaction patterns in patients compared to control subjects.