A binary search algorithm for univariate data approximation and estimation of extrema by piecewise monotonic constraints

Abstract

The piecewise monotonic approximation problem makes the least changes to n univariate noisy data so that the piecewise linear interpolant to the new values is composed of at most k monotonic sections. The term “least changes” is defined in the sense of a global sum of strictly convex functions of changes. The main difficulty in this calculation is that the extrema of the interpolant have to be found automatically, but the number of all possible combinations of extrema can be ${\mathcal {O}}(n^{k-1})$, which makes not practicable to test each one separately. It is known that the case $k=1$ is straightforward, and that the case $k>1$ reduces to partitioning the data into at most k disjoint sets of adjacent data and solving a $k=1$ problem for each set. Some ordering relations of the extrema are studied that establish three quite efficient algorithms by using a binary search method for partitioning the data. In the least squares case the total work is only ${\mathcal {O}}(n \sigma +k\sigma \log _2\sigma )$ computer operations when $k \ge 3$ and is only ${\mathcal {O}}(n)$ when $k=1$ or 2, where $\sigma -2$ is the number of sign changes in the sequence of the first differences of the data. Fortran software has been written for this case and the numerical results indicate superior performance to existing algorithms. Some examples with real data illustrate the method. Many applications of the method arise from bioinformatics, energy, geophysics, medical imaging, and peak finding in spectroscopy, for instance.