lsq_splin — weighted least squares cubic spline fitting
[y, d] = lsq_splin(xd, yd [, wd], x)
vectors of the same size, datas to be fitted by a cubic spline
(optional) a vector of same format than xd
and yd
,
weights of the least square fit.
a strictly increasing (row or column) vector, breakpoints of the cubic spline
vectors of same format than x
, the triplet (x,y,d) defines the approximated
cubic spline.
This function computes an approximated cubic spline s for the datas
xd, yd, wd
(in the following m is supposed to be the
length of these vectors) and from a choice of the spline breakpoints x
(for instance if you want n breakpoints uniformly choosen you may use
x=linspace(min(xd),max(xd),n))
). If S is the space of all
cubic splines functions with breakpoints x1 < x2 < ... < xn then
the resulting spline s is such that:
_m_ _m_ \ 2 \ 2 / wd(k) (s(xd(k)) - yd(k)) <= / wd(k) (f(xd(k)) - yd(k)) --- --- k=1 k=1
for all f in S, i.e. realizes the minimum of the sum of the squared errors over all functions of S.
The spline s is completly defined by the triplet (x,y,d)
(y
and d
are the vectors of the spline ordinates and
first derivatives at the xi 's : yi=s(xi) and di=s'(xi)) and
its evaluation at some points must be done by the interp function.
When wd
is not given, all the points have the same weight 1.
A point (xd(k),yd(k)) is considered in the fit if xd(k) in [x1,xn] and wd(k) > 0. In particular you can put a null (or even negative) weight to all data points you want to ignore in the fitting. When the total number of points taken into account in the fit procedure is (strictly) less than 4 an error is issued.
The vector xd
do not need to be in increasing order.
Depending on the number and on the positions of the xd(k) 's and on the choice of the x(i) 's there may be several solutions but only one is selected. When this occurs a warning message is displayed in the Scilab command window. This function is intended to be used when m is much larger than n and in this case no such problem may occured.
// this is an artifical example where the datas xd and yd // are build from a perturbed sin function a = 0; b = 2*%pi; sigma = 0.1; // standard deviation of the gaussian noise m = 200; // number of experimental points xd = linspace(a,b,m)'; yd = sin(xd) + grand(xd,"nor",0,sigma); n = 6; // number of breakpoints x = linspace(a,b,n)'; // compute the spline [y, d] = lsq_splin(xd, yd, x); // use equal weights // plotting ye = sin(xd); ys = interp(xd, x, y, d); xbasc() plot2d(xd,[ye yd ys],style=[2 -2 3], ... leg="exact function@experimental measures (gaussian perturbation)@fitted spline") xtitle("a least square spline") xselect()