spap2
Least-squares spline approximation
Syntax
Description
spline = spap2(knots,k,x,y) k with the given knot sequence knots
for which
(*) y(:,j) = f(x(j)), all j
in the weighted mean-square sense, meaning that the sum
is minimized, with default weights equal to 1. The data
values y(:,j) can be scalars, vectors, matrices, or
ND-arrays, and |z|2 is the sum of
the squares of all the entries of z. Data points with the
same site are replaced by their average.
If the sites x satisfy the Schoenberg-Whitney
conditions
then there is a unique spline of the given order and knot sequence satisfying
(*) exactly. No spline is returned unless (**) is satisfied for some subsequence
of x.
spap2(,
with l,k,x,y) l a positive integer, returns the B-form of a
least-squares spline approximant, but with the knot sequence chosen for you. The
knot sequence is obtained by applying aptknt to an appropriate
subsequence of x. The resulting piecewise-polynomial consists
of l polynomial pieces and has k-2
continuous derivatives. If you feel that a different distribution of the
interior knots might do a better job, follow this up
with
sp1 = spap2(newknt(spline),k,x,y));
spap2({knorl1,...,knorlm},k,{x1,...,xm},y)
provides a least-squares spline approximation to gridded
data. Here, each knorli is either a knot sequence or a
positive integer. Further, k must be an
m-vector, and y must be an
(r+m)-dimensional array, with
y(:,i1,...,im) the datum to be fitted at the
site
[x{1}(i1),...,x{m}(im)], all i1, ...,
im. However, if the spline is to be scalar-valued, then,
in contrast to the univariate case, y is permitted to be an
m-dimensional array, in which case
y(i1,...,im) is the datum to be fitted at the
site
[x{1}(i1),...,x{m}(im)], all i1, ...,
im.
spap2({knorl1,...,knorlm},k,{x1,...,xm},y,w)
also lets you specify the weights. In this m-variate case,
w must be a cell array with m entries,
with w{i} a nonnegative vector of the same size as
xi, or else w{i} must be empty, in
which case the default weights are used in the ith
variable.
Examples
Calculate the Least-Squares Approximation of a Cubic Spline
This example shows how to compute the least-squares
approximation to the data x, y, by
cubic splines with two continuous derivatives, basic interval
[a..b], and interior breaks
xi, provided xi has all its
entries in (a..b) and the conditions (**) are satisfied.
sp = spap2(augknt([a,xi,b],4),4,x,y)
In that case, the approximation consists of
length(xi)+1 polynomial pieces. If you only want
to get a cubic spline approximation consisting of l
polynomial pieces, use instead
sp = spap2(l,4,x,y);
If the resulting approximation is not satisfactory, try using a larger
l. Else use
sp = spap2(newknt(sp),4,x,y);
for a better distribution of the knot sequence. Repeat this process multiple times to increase the fidelity of the distribution.
As another example, spap2(1,2,x,y); provides the
least-squares straight-line fit to data
x,y, while
w = ones(size(x)); w([1 end]) = 100; spap2(1,2,x,y,w);
forces that fit to come very close to the first data point and to the last.
Compute and Plot the Least Square Approximation of a Bivariate Function
This example shows how to create a bivariate function, and compute and plot its least-square approximation.
Generate the data for the approximation and the bivariate function.
x = -2:.2:2; y=-1:.25:1; [xx, yy] = ndgrid(x,y); z = exp(-(xx.^2+yy.^2));
Compute the least-square approximation and plot it.
sp = spap2({augknt([-2:2],3),2},[3 4],{x,y},z);
fnplt(sp)
Input Arguments
Output Arguments
Algorithms
spcol is called on to provide the
almost block-diagonal collocation matrix
(Bj,k(xi)),
and slvblk solves the linear system
(*) in the (weighted) least-squares sense, using a block QR factorization.
Gridded data are fitted, in tensor-product fashion, one variable at a time, taking advantage of the fact that a univariate weighted least-squares fit depends linearly on the values being fitted.