Optimal curve fitting and smoothing using normalized uniform B-splines: a tool for studying complex systems

The basic problem considered in this paper is how to reduce measures of complex systems to easily understood measures. We show that optimal smoothing splines are an ideal means to solve the problem in some cases. Of course, no single method will work for all complex systems and all measures. The problem of constructing optimal curves for given set of data at discrete data points is considered. Both equally-spaced and non equally-spaced data points are treated. The curves are constructed by using B-splines as basis functions, namely as weighted sum of shifted B-splines of degree k. It is then shown that an optimal approximation can be solved without any boundary conditions, wherein explicit solution formulas are presented. A problem of optimal interpolation is also considered in parallel. We apply this technique to the Dow Jones Industrial index for both long and short time periods. We see that long term trends can be easily identified.