| Popis: |
A brief survey is presented for the main methods that are used in least squares data smoothing by adjusting the signs of divided differences of the smoothed values. The most distinctive feature of the smoothing approach is that it provides automatically a piecewise monotonic or a piecewise convex/concave fit to the data. The data are measured values of a function of one variable that contain random errors. As a consequence of the errors, the number of sign alterations in the sequence of mth divided differences is usually unacceptably large, where m is a prescribed positive integer. Therefore, we make the least sum of squares change to the measurements by requiring the sequence of the divided differences of order m to have at most k−1 sign changes, for some positive integer k. Although, it is a combinatorial problem, whose solution can require about O(nk) quadratic programming calculations in n variables and n−m constraints, where n is the number of data, very efficient algorithms have been developed for t... |
