An Example of Nondecomposition in Data Fitting by Piecewise Monotonic Divided Differences of Order Higher Than Two

Abstract

Let n measurements of values of a real function of one variable be given, but the measurements include random errors. For given integers m and σ, we consider the problem of making the least sum of squares change to the data such that the sequence of the divided differences of order m of the fit changes sign at most σ times. The main difficulty in these calculations is that there are about O(n ^σ) combinations of positions of sign changes, so that it is impracticable to test each one separately. Since this minimization calculation has local minima, a general optimization algorithm can stop at a local minimum that need not be a global one. It is an open question whether there is an efficient algorithm that can compute a global solution to this important problem for general m and σ. It has been proved that the calculations when m = 1, which gives a piecewise monotonic fit to the data, and m = 2, which gives a piecewise convex/concave fit to the data, reduce to separating the data into σ + 1 disjoint sets of adjacent data and solving a structured quadratic programming problem for each set. Separation allows the development of some dynamic programming procedures that solve the particular problems in O(n ² + σnlog₂ n) and about O(σn ³) computer operations, respectively. We present an example which shows that the minimization calculation when m ≥ 3 and σ ≥ 1 may not be decomposed into separate calculations on subranges of adjacent data such that the associated divided differences of order m are either non-negative or non-positive. Therefore, the example rules out the possibility of solving the general problem by a similar dynamic programming calculation.