Discrete piecewise monotonic approximation by a strictly convex distance function

Abstract

Theory and algorithms are presented for the following smoothing problem. We are given n measurements of a real-valued function that have been altered by random errors caused by the deriving process. For a given integer k, some efficient algorithms are developed that approximate the data by minimizing the sum of strictly convex functions of the errors in such a way that the approximated values are made up of at most k monotonic sections. If k = 1 k = 1 , then the problem can be solved by a special strictly convex programming calculation. If k > 1 k > 1 , then there are O ( n k ) O({n^k}) possible choices of the monotonic sections, so that it is impossible to test each one separately. A characterization theorem is derived that allows dynamic programming to be used for dividing the data into optimal disjoint sections of adjacent data, where each section requires a single monotonic calculation. It is remarkable that the theorem reduces the work for a global minimum to O ( n ) O(n) monotonic calculations to subranges of data and O ( k s 2 ) O(k{s^2}) computer operations, where s − 2 s - 2 is the number of sign changes in the sequence of the first divided differences of the data. Further, certain monotonicity properties of the extrema of best approximations with k and k − 1 k - 1 , and with k and k − 2 k - 2 monotonic sections make the calculation quite efficient. A Fortran 77 program has been written and some numerical results illustrate the performance of the smoothing technique in a variety of data sets.