L1-Optimal Splines for Outlier Rejection

Abstract: In this article, we consider control theoretic splines with L1 optimization for rejecting outliers in data. Control theoretic splines are either interpolating or smoothing splines, depending on a cost function with a constraint defined by linear differential equations. Control theoretic splines are effective for Gaussian noise in data since the estimation is based on L2 optimization. However, in practice, there may be outliers in data, which may occur with vanishingly small probability under the Gaussian assumption of noise, to which L2-optimized spline regression may be very sensitive. To achieve robustness against outliers, we propose to use L1 optimality, which is also used in support vector regression. A numerical example shows the effectiveness of the proposed method.