Univeriate curve-fitting, interpolation, polynomial, spline, Akima
OverviewUnivariate interpolation is an area of curve-fitting which, as opposed to univariate regression analysis, finds the curve that provides an exact fit to a series of two-dimensional data points. It is called univariate as the data points are supposed to be sampled from a one-variable function. Compare this to multivariate interpolation, which aims at fitting data points sampled from a function of several variables. Formally speaking, consider a series of data points and, for the sake of simplicity, consider that , i.e. the points are distinct and are in increasing order with respect to . By interpolating these data points we mean finding a function such that:
Polynomial InterpolationIn the following, let us assume that the interpolation function is polynomial, i.e. is of the type Interpolation/Lagrange. Consider the function
Runge's PhenomenonWhen trying to estimate the error between the original function, from which the series of data points has been sampled, and the polynomial interpolation function , one may notice the following phenomenon. Conside the Runge function:
Spline InterpolationSpline interpolation is somehow a generalization of polynomial interpolation, in that we do not necessarily have to find a single polynomial function to fit the data over the entire interval, but we rather try to find several polynomial functions to fit the data over each subinterval determined by two consecutive data points, while obeying some smoothness conditions. One of the advantages of this generalization is that the resulting interpolation function is less wiggly, as in the case of e.g. Lagrange interpolation. Formally, consider the series of data points in the above paragraphs, which are distinct and ordered increasingly with respect to . A spline interpolation function of degree for the given data points is a function which satisfies the following conditions
- , for all and all , where is some polynomial function with degree less than or equal to
- the derivatives of up to the order are all continuous in the given data points, which basically means that for all we require that:
- George M. Philips, Interpolation and Approximation by Polynomials, Springer-Verlag, New York, 2003.
- Hiroshi Akima, A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures, Journal of the ACM, Vol. 17, No. 4, October 1970, pp. 589-602.