Rank order smoothers with two-dimensional data model

Development of the theory of rank order smoothers is motivated by hardware implementation, as well as statistical, considerations. A recently proposed class of rank order smoothers based on minimization of polyhedral convex functions, involving one-dimensional data models, employs minimization algorithms derived from dynamic programming; computations proceed by storing and manipulating arrays of breakpoints of piecewise linear convex functions. Such an approach has certain advantages over alternatives. Foremost among these is the attainment of a fixed point or "root signal" in a single application of the minimization algorithm. Other advantages are the intelligibility of the connection between the rough data and the root signal smooth derived from it, the absence of any significant problem of how to deal with end values, compatibility with real time applications, and storage and computation time requirements of the order of the number of pieces of rough data. When experimental comparison was made to median smoothers, better smoothing was observed. On the negative side, such smoothers have been largely restricted to types that behave fundamentally like median smoothers, i.e., have the same sorts of fixed points or root signals. It is shown that by letting the data model be two dimensional, and formulating the dynamic programming method so that variables are eliminated in pairs, then the root signal sets of such smoothers are greatly expanded while the various advantages are retained.