Generalized shape constrained spline fitting for qualitative analysis of trends

In this work, we present a generalized method for analysis of data series based on shape constraint spline fitting which constitutes the first step toward a statistically optimal method for qualitative analysis of trends. The presented method is based on a branch-and-bound (B&B) algorithm which is applied for globally optimal fitting of a spline function subject to shape constraints. More specifically, the B&B algorithm searches for optimal argument values in which the sign of the fitted function and/or one or more of its derivatives change. We derive upper and lower bounding procedures for the B&B algorithm to efficiently converge to the global optimum. These bounds are based on existing solutions for shape constraint spline estimation via Second Order Cone Programs (SOCPs). The presented method is demonstrated with three different examples which are indicative of both the strengths and weaknesses of this method.