Sigmoid Data Fitting by Least Squares Adjustment of Second and Third Divided Differences

We consider the performance of two data smoothing methods that provide sigmoid fits by adjustment of divided differences on some test problems. Thus we investigate the accuracy and the efficiency of the methods for smoothing a variety of data points, our conclusions being drawn from numerical results. The first method is a least squares data smoothing calculation subject to nonnegative third divided differences. The second method is a non-linear least squares data smoothing calculation subject to one sign change in the second divided differences. Both methods employ structured quadratic programming calculations, which take into account the form of the constraints and make efficient use of the banded matrices that occur in the subproblems during the iterations of the quadratic programming calculations. The total work of each method, in practice, is of quadratic complexity with respect to the number of data. Our results expose some weaknesses of the methods. Therefore they may be helpful to the development of new algorithms that are particularly suitable for sigmoid data fitting calculations. Our results expose also some strengths of the methods, which they may be useful to particular scientific analyses, e.g. sigmoid phenomena, and to strategic management practices, i.e. economic substitution.