Nonparametric Combinatorial Regression for Shape Constrained Modeling

Abstract

We introduce a unified approach for calculating nonparametric shape constrained regression. Enforcement of the shape constraint often accounts for the impact of a physical phenomenon or a specific property. It also improves the model's predicability and facilitates subsequent optimizations. The regression models are built by transforming the problem into the combinatorial domain where the shape constraints are imposed by bounding the combinatorial search space. We start by addressing isotonicity shape constraint using a dynamic programming algorithm and demonstrate how the problem can be mapped to the graph combinatorics domain. Next we show how a number of other important shape constraints including unimodality, convexity, limited level set, and limited slope can be addressed using the same framework. The flexibility of proposed framework enables solving the shape constrained regression problem with an arbitrary user-defined error metric. This flexibility is exploited to add robustness against outliers to the model. The algorithms are described in detail and their computational complexity is established. The performance and effectiveness of the shape constrained regression is evaluated on traces of temperature and humidity measurements from a deployed sensor network where a high degree of accuracy and robustness is demonstrated.