LOCALLY MONOTONIC REGRESSION

The concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of a signal. The median filter is a suboptimal smoother under this measure of smoothness, since a filter pass does necessarily yield a locally monotonic output; even if a locally monotonic output does result, there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal. Locally monotonic regression is a technique for the optimal smoothing of finite-length discrete real signals under such a criterion. A theoretical framework where the existence of locally monotonic regressions is proven and algorithms for their computation are given. Regression is considered as an approximation problem in R(n), the criterion of approximation is derived from a semimetric and the approximating set is the collection of signals sharing the property of being locally monotonic.