Template-Based Piecewise Affine Regression

We investigate the problem of fitting piecewise affine functions (PWA) to data. Our algorithm divides the input domain into finitely many polyhedral regions whose shapes are specified using a user-defined template such that the data points in each region are fit by an affine function within a desired error bound. We first prove that this problem is NP-hard. Next, we present a top-down algorithm that considers subsets of the overall data set in a systematic manner, trying to fit an affine function for each subset using linear regression. If regression fails on a subset, we extract a minimal set of points that led to a failure in order to split the original index set into smaller subsets. Using a combination of this top-down scheme and a set covering algorithm, we derive an overall approach that is optimal in terms of the number of pieces of the resulting PWA model. We demonstrate our approach on two numerical examples that include PWA approximations of a widely used nonlinear insulin--glucose regulation model and a double inverted pendulum with soft contacts.