A performance comparison of piecewise linear estimation methods.

The response functions in many engineering problems are piecewise smooth functions, although globally they can be highly nonlinear. The linear Shepard algorithm (a moving window weighted least squares method based on linear functions) usually creates reasonable approximations. When used to approximate data obtained from piecewise linear functions, a better approximation near the function creases is obtained when a robust linear Shepard algorithm is used. Due to limitations of the robust linear Shepard algorithm in high dimensions, RIPPLE (residual initiated polynomial-time piecewise linear estimation) was developed for piecewise linear estimation of high dimensional data. RIPPLE selects minimal sets of data based on a minimal residual criterion. Once the best minimal set is obtained, RIPPLE adds other consistent data points using a robust statistical procedure. RIPPLE is resistant to outliers. It produces good approximations even if the data contains significant random errors. The code L2WPMA (least squares weighted piecewise monotonic approximation, ACM TOMS Algorithm 863) calculates a piecewise monotonic approximation to n univariate data points contaminated by random errors. The continuous piecewise linear approximation consists of k (a positive integer provided by the user) monotonic linear splines, alternately monotonically increasing and monotonically decreasing. An optimal approximation is obtained if there are at most k monotonic sections in the exact data. This paper will present a theoretical complexity comparison and an empirical performance comparison of the linear Shepard algorithm, robust linear Shepard algorithm, RIPPLE, and L2WPMA for univariate datasets.