Discovering New Concepts in Interfacial Reaction and Engineering for
Template Synthesis of Nanomaterials, Encapsulation, and Dynamic Control of Cell-Biomaterial Interfaces

Interpolating and Smoothing Cubic Splines

Starting with SplineExample97.xls, study the functions contained within the cells and how they relate to the data carefully.  Note that many of the functions in the cells have braces {...} around them.  These braces means that the functions have been entered as array- formulas, read about these in the following section. Note that the custom spline functions can also be inserted using the Excel menu item: Insert - Function - User Defined.  The latter allows the use of these functions without having to remember the syntax, albeit they still have to be entered as array formulas.  The smoothing spline routine is designed to produce a cubic spline approximation to a data set in which the function values are noisy.  The resulting natural cubic spline has knots at all the data abscissas (XDATA), but does not merely interpolate the data.  The smoothing spline function S is chosen to minimize the total curvature subject to the constraint where fi are the data points, wi are the weights or error estimates for the data, and s is the smoothing parameter which should have a value of ~1 if the error estimates are correct. The alternative cross-validation smoothing spline routine available in this package automatically chooses a value for s so that the smoothing spline best approximates the value at individual data points fi using all data points except the i-th. This approach assumes that all data points have equal weights.