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# lsq_splin

weighted least squares cubic spline fitting

### Calling Sequence

[y, d] = lsq_splin(xd, yd [, wd], x)

### Arguments

- xd, yd
vectors of the same size, data to be fitted by a cubic spline

- wd
(optional) a vector of same format than

`xd`

and`yd`

, weights of the least square fit.- x
a strictly increasing (row or column) vector, breakpoints of the cubic spline

- y, d
vectors of same format than

`x`

, the triplet (x,y,d) defines the approximated cubic spline.

### Description

This function computes an approximated cubic spline
*s* for the data `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 chosen 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:

for all *fin S*, i.e. realizes the minimum of
the sum of the squared errors over all functions of
*S*.

The spline *s* is completely 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.

### Remarks

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 occurred.

### Examples

// this is an artificial example where the data 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); clf() plot2d(xd,[ye yd ys],style=[2 -2 3], ... leg="exact function@experimental measures (gaussian perturbation)@fitted spline") xtitle("a least square spline") show_window()

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