Re: "Elastic string" (type of traveling-salesman) paradigm for sampling

*To*: mathgroup at smc.vnet.net*Subject*: [mg78236] Re: "Elastic string" (type of traveling-salesman) paradigm for sampling*From*: dh <dh at metrohm.ch>*Date*: Wed, 27 Jun 2007 05:11:47 -0400 (EDT)*References*: <200706131143.HAA07211@smc.vnet.net> <200706210953.FAA27469@smc.vnet.net> <f5o803$53s$1@smc.vnet.net>

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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