Estimation of shape constrained functions in dynamical systems and its application to gene networks

Inspired by estimation and identification of biological and engineering systems subject to constraints, this paper addresses nonparametric estimation of monotone functions contained in a class of dynamical systems. A two-stage estimation procedure is proposed. At the first stage, partial state estimation is performed via trend filtering techniques. At the second stage, a penalized spline (or P-spline for short) estimator is used to estimate monotone functions. The highlight of the paper is asymptotic analysis of the monotone P-spline estimator formulated as a constrained optimization problem. The uniform Lipschitz property is established for optimal spline coefficients. By approximating the estimator by a solution of a differential equation with a constrained right-hand side, the paper develops asymptotic normality at interior points and establishes convergence rates. The proposed estimator is applied to estimation of a monotone regulatory function in a gene regulatory network.