Radial Basis Functions
Radial basis interpolation is the name given to a large family of exact interpolators. In many ways
the methods applied are similar to those used in geostatistical interpolation (see Section
6.7
), but
without the benefit of prior analysis of variograms. On the other hand they do not make any
assumptions regarding the input data points (other than they are not colinear) and provide
excellent interpolators for a wide range of data. See also Section
8.3.2
for an alternative (neural
network) approach to modeling with radial basis functions.
A great deal of research has been conducted into the quality of these interpolators, across many
disciplines. For terrain modeling and Earth Sciences generally the socalled multiquadric function
has been found particularly effective, as have thin plate splines. The simplest variant of this
method, without smoothing (i.e. as an exact interpolator) can be viewed as a weighted linear
function of distance (or inverse distance) from grid point to data point, plus a “bias” factor,
m
. The
model is of the form:
or the equivalent model, using the untransformed data values and data weights
λ
i
:
In these expressions
z
p
is the estimated value for the surface at grid point
p
;
φ
(
r
i
) is the radial
basis function selected, with
r
i
being the radial distance from point
p
to the
i
th
data point; the
weights
w
i
and
λ
i
and the bias value
m
(or Lagrangian multiplier) are estimated from the data
points. This requires solving a system of
n
linear equations. Using the second of the two models
above the procedure is then essentially the same as for Ordinary Kriging (see further, Section
6.7.2
). For clarity we outline this latter procedure below as a series of steps using matrix notation:
•
Compute the
n
x
n
matrix,
D
, of interpoint distances between all (
x,y
) pairs in the source
dataset (or a selected subset of these)
•
Apply the chosen radial basis function,
φ
(), to each distance in
D
, to produce a new array
Φ
•
Augment
Φ
with a unit column vector and a unit row vector, plus a single entry 0 in position
(
n
+1),(
n
+1). Call this augmented matrix
A
— see below for an illustration
•
Compute the column vector
r
of distances from the grid point,
p
, to each of the source data
points used to create
D
•
Apply the chosen radial basis function to each distance in
r
, to produce a column vector
ϕ
and then create the (
n
+1) column vector
c
as
ϕ
plus a single 1 as the last entry
•
Compute the matrix product
b
=
A
1
c
. This provides the set of
n
weights to be used in the
calculation of the estimated value at
p
, plus the Lagrangian value
m
using the linear equation
cited earlier in this Section. The value
m
is not used in calculating the estimate at
p
in this
formulation
In matrix form the system of linear equations being solved is of the form:
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i.e.
Ab
=
c
hence
b
=
A
1
c
.
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 Summer '08
 Staff
 Regression Analysis, The Land, response variable, GWR

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