I think I have a spline problem, and I would like to
implement the solution in S. There are a lot of spline algorithms and I am
looking for some direction on what is most appropriate. I need a spline that can
be made to extrapolate certain data points as described below. Of course it
needs to happen algorithmically as I have lots of this data. Can a Bezier be
fit to data?
Consider a material volume sampled in discrete
non-uniform intervals. There is a continuous trend in properties that is
integrated over each interval. We are aware of the specific form of the
continuous trend, but it is concealed by the interval nature of the data. The
specific need is to identify the ‘true’ location and value of the
minima and maxima of the function representing the data. Perhaps an example…
In the following figure, the numbers represent the sampled
layers: the height of the bars is the value of the sampled property. The x axis
is depth into the material volume. The asterisk represents the true value and
location of the minima and maxima. We know that they are lower and higher than
the interval sampled data, and (critically) that the value of the maxima or
maxima is offset from the center of the sampled volume, depending on the
trajectory of the change between sampled layers. Perhaps you can imagine a
smooth line connecting the asterisks which preserves the area of the
corresponding bars.
*
33333
33333
* 33333 *
1111 33333 444
1111 222 33333 444 5*5
1111 2*2 33333 444 555 6*6
1111 222 33333 444 555 666