A multivariate function form quite different from the tensor-product construct is the scattered translates form, or stform for short. As the name suggests, it uses arbitrary or scattered translates ψ(· –c_{j}) of one fixed function ψ, in addition to some polynomial terms. Explicitly, such a form describes a function
$$f\left(x\right)={\displaystyle \sum _{j=1}^{n-k}\psi \left(x-{c}_{j}\right){a}_{j}+p\left(x\right)}$$
in terms of the basis function ψ, a sequence (c_{j}) of sites called centers and a corresponding sequence (a_{j}) of n coefficients, with the final k coefficients, a_{n-k+1},...,a_{n}, involved in the polynomial part, p.
When the basis function is radially symmetric, meaning that ψ(x) depends only on the Euclidean length |x| of its argument, x, then ψ is called a radial basis function, and, correspondingly, f is then often called an RBF.
At present, the toolbox works with just one kind of stform, namely a bivariate thin-plate spline and its first partial derivatives. For the thin-plate spline, the basis function is ψ(x) = φ(|x|^{2}), with φ(t) = tlogt, i.e., a radial basis function. Its polynomial part is a linear polynomial, i.e., p(x)=x(1)a_{n – 2}+x(2)a_{n – 1}+a_{n}. The first partial derivative with respect to its first argument uses, correspondingly, the basis function ψ(x)=φ(|x|^{2}), with φ(t) = (D_{1}t)·(logt+1) and D_{1}t = D_{1}t(x) = 2x(1), and p(x) = a_{n}.
A function in stform can be put together from its center sequence centers
and
its coefficient sequence coefs
by the command
f = stmak(centers, coefs, type);
with the string type
one of 'tp00'
, 'tp10'
, 'tp01'
,
to indicate, respectively, a thin-plate spline, a first partial of
a thin-plate spline with respect to the first argument, and a first
partial of a thin-plate spline with respect to the second argument.
There is one other choice, 'tp'
; it denotes a thin-plate
spline without any polynomial part and is likely to be used only during
the construction of a thin-plate spline, as in tpaps
.
A function f in stform depends linearly on its coefficients, meaning that
$$f\left(x\right)={\displaystyle \sum _{j=1}^{n}{\psi}_{j}\left(x\right){a}_{j}}$$
with ψ_{j} either a translate of the
basis function Ψ or else some polynomial. Suppose you wanted
to determine these coefficients a_{j} so
that the function f matches prescribed values at
prescribed sites x_{i}.
Then you would need the collocation matrix (ψ_{j}(x_{i})).
You can obtain this matrix by the command stcol
(centers,x,type)
.
In fact, because the stform has a_{j} as
the jth column, coefs(:,j)
,
of its coefficient array, it is worth noting that stcol
can also supply the transpose of
the collocation matrix. Thus, the command
values = coefs*stcol(centers,x,type,'tr');
would provide the values at the entries of x
of
the st function specified by centers
and type
.
The stform is attractive because, in contrast to piecewise polynomial forms, its complexity is the same in any number of variables. It is quite simple, yet, because of the complete freedom in the choice of centers, very flexible and adaptable.
On the negative side, the most attractive choices for a radial
basis function share with the thin-plate spline that the evaluation
at any site involves all coefficients. For example, plotting a scalar-valued
thin-plate spline via fnplt
involves
evaluation at a 51-by-51 grid of sites, a nontrivial task when there
are 1000 coefficients or more. The situation is worse when you want
to determine these 1000 coefficients so as to obtain the stform of
a function that matches function values at 1000 data sites, as this
calls for solving a full linear system of order 1000, a task requiring
O(10^9) flops if done by a direct method. Just the construction of
the collocation matrix for this linear system (by stcol
)
takes O(10^6) flops.
The command tpaps
, which
constructs thin-plate spline interpolants and approximants, uses iterative
methods when there are more than 728 data points, but convergence
of such iteration may be slow.
After you have constructed an approximating or interpolating
thin-plate spline st
with the aid of tpaps
(or directly via stmak
), you can use the following commands:
fnbrk
to obtain its parts or change
its basic interval,
fnval
to evaluate
it
fnplt
to
plot it
fnder
to construct its two first
partial derivatives, but no higher order derivatives as they become
infinite at the centers.
This is just one indication that the stform is quite different
in nature from the other forms in this toolbox, hence other fn...
commands
by and large don't work with stforms. For example, it makes no sense
to use fnjmp
, and fnmin
or fnzeros
only
work for univariate functions. It also makes no sense to use fnint
on a function in stform because
such functions cannot be integrated in closed form.
The command Ast =
fncmb
(st,A)
can be
used on st
, provided A
is something
that can be applied to the values of the function described by st
.
For example, A
might be 'sin'
,
in which case Ast
is the stform of the function
whose coefficients are the sine of the coefficients of st
.
In effect, Ast
describes the function obtained
by composing A
with st
. But,
because of the singularities in the higher-order derivatives of a
thin-plate spline, there seems little point to make fndir
or fntlr
applicable
to such a st
.