GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 38.1 Overview

B-splines are commonly used as basis functions to fit smoothing
curves to large data sets. To do this, the abscissa axis is
broken up into some number of intervals, where the endpoints
of each interval are called *breakpoints*. These breakpoints
are then converted to *knots* by imposing various continuity
and smoothness conditions at each interface. Given a nondecreasing
knot vector
t = {t_0, t_1, ..., t_{n+k-1}},
the n basis splines of order k are defined by

B_(i,1)(x) = (1, t_i <= x < t_(i+1) (0, else B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x) + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)

for i = 0, ..., n-1. The common case of cubic B-splines is given by k = 4. The above recurrence relation can be evaluated in a numerically stable way by the de Boor algorithm.

If we define appropriate knots on an interval [a,b] then the B-spline basis functions form a complete set on that interval. Therefore we can expand a smoothing function as

f(x) = \sum_i c_i B_(i,k)(x)

given enough (x_j, f(x_j)) data pairs. The coefficients c_i can be readily obtained from a least-squares fit.

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |