Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Least-squares spline approximation

`spap2(knots,k,x,y) `

spap2(l,k,x,y)

sp = spap2(...,x,y,w)

spap2({knorl1,...,knorlm},k,{x1,...,xm},y)

spap2({knorl1,...,knorlm},k,{x1,...,xm},y,w)

`spap2(knots,k,x,y) `

returns
the B-form of the spline * f* of order

`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

$$\sum _{j}w(j)|y(:,j)-f\left(x(j)\right){|}^{2}$$

is minimized, with default weights equal to 1.
The data values `y(:,j)`

may be scalars, vectors,
matrices, even ND-arrays, and |* z*|

If the sites `x`

satisfy the (Schoenberg-Whitney)
conditions

$$\begin{array}{l}\text{knots}(j)x(j)\text{knots}(j+k)\\ (**)\text{}j=1,\mathrm{...},\text{length}(x)=\text{length(knots)}-k\end{array}$$

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(l,k,x,y) `

, with `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`

. `T`

he 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(sp),k,x,y));

`sp = spap2(...,x,y,w) `

lets you specify the weights `w`

in the error measure
(given above). `w`

must be a vector of the same size
as `x`

, with nonnegative entries. All the weights
corresponding to data points with the same site are summed when those
data points are replaced by their average.

`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 `i`

th
variable.

sp = spap2(augknt([a,xi,b],4),4,x,y)

is the least-squares approximant 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 in some fashion. In that case, the
approximant consists of `length(xi)+1`

polynomial
pieces. If you do not want to worry about the conditions (**) but
merely want to get a cubic spline approximant 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 possibly better distribution of the knot sequence. In fact, if that helps, repeating it may help even more.

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.

`spcol`

is called
on to provide the almost block-diagonal collocation matrix (* B_{j}*,

`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.

Was this topic helpful?