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Re: "Elastic string" (type of traveling-salesman) paradigm for sampling
Hi Curtis,
if you can solve a problem in different ways, you should pick the
simplest one. Further, a good measure of dimension should not depend
(strongly) on how you pick the points. Therefore, I would replace
separated points by a continuous line by linear interpolation (assuming
the point are ordered). Then you can simple put your measure-stick along
this line.
hope this helps, Daniel
Curtis Osterhoudt wrote:
> Dear MathGroup,
>
> This is a question which might open up far more cans of worms than I'd
> like, but it has got me thinking, and I can always use help with that.
> Forgive the length of the post!
>
> I'm writing some code (in Mathematica) to calculate the fractal
> dimension(s) of nominally 1D datasets; an example would be discretely-sampled
> {time, single_coordinate} sets. This brings me to my first question: other
> than the simple approach described in "The Fractal Dimension of the Blues"
> notebook[1], are there relatively accessible Mathematica codes out there
> already?
>
> Regardless of the answer to the first question, I've another. One way in which
> to sample a discrete data set (or even a continuous function) is to imagine
> an elastic string stretched between the first and last data points. Then,
> specify how many (straight) sections the string is allowed to be deformed in.
> For just one section, the string just goes straight from the first to the
> last point. For two sections, the string goes from the first point, to
> somewhere on the dataset, and then from there to the last point.
> The "elastic" part of the wording is to imply that the string's length is
> required to be the shortest possible, passing through the first, last, and
> any intermediate sampling points. (The possibility of allowing for
> interpolation between data points causes massive problems---at least in my
> mind---as to how sample things correctly, so I'll leave that concept aside
> for the moment.) This is a sort of "free" traveling-salesman problem, with
> the salesman picking and choosing a set number of points to visit between the
> first and last.
> I'm imagining a rather icky optimization problem (especially for large
> datasets), and wonder if anyone has some hints or suggestions as to how to
> make relatively quick code. I'm totally open to the spewing-forth of ideas,
> whether or not they're very carefully considered.
> As a first guess at the algorithm, I can fix the first and last points of the
> string, then run a "bead" affixed to the string along each of the remaining
> points, and see which one ends up minimizing the string length (obviously a
> point between the first and last points in both dimensions, _if_ the sampled
> data allows for such a thing). As more points are allowed to be sampled, it's
> almost certain that previously-affixed points will have to become unglued,
> and attach themselves elsewhere.
>
> Any suggestions? This may be a common problem with some famous and
> efficient solutions, but I'm not familiar with them. Even some good websites
> or journal article suggestions would be very welcome.
>
> Best wishes,
> Curtis O.
>
>
>
>
> [1] Available as
> http://calcand.math.uiuc.edu/courseware/Old%20Stuff/Pictures_and_Math_Fun/Fowler's%20Neat%20Graphics/13fracBluWRI.nb
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