Qualitative trend analysis based on a mixed-integer representation

Abstract

Shape constrained spline fitting is a useful method to impose prior knowledge onto flexible semi-parametric models during parameter estimation. Most typically, the function shape is imposed through order restrictions on the regression coefficients. The intended shape is considered known or selected based on heuristic rules. In this study, we present a method to estimate the optimal set of order restrictions to segment a univariate data series into episodes with distinct shapes. This is also known as the qualitative trend analysis (QTA) problem. The obtained solution uses a trade-off between lack-of-fit and model complexity. Further, our practical implementation takes inspiration from the generalized order restricted information criterion (GORIC) for inequality-constrained model selection. From this, one learns (a) that QTA can be formulated as a mixed-integer quadratic program (MIQP) and (b) that the newly proposed mixed order restricted information criterion (MORIC) enables optimal segmentation. This is illustrated through didactic case studies.