"Elastic string" (type of traveling-salesman) paradigm for sampling discrete data sets

*To*: mathgroup at smc.vnet.net*Subject*: [mg78198] "Elastic string" (type of traveling-salesman) paradigm for sampling discrete data sets*From*: Curtis Osterhoudt <cfo at lanl.gov>*Date*: Mon, 25 Jun 2007 07:08:17 -0400 (EDT)*Organization*: LANL*References*: <200706131143.HAA07211@smc.vnet.net> <200706210953.FAA27469@smc.vnet.net>*Reply-to*: cfo at lanl.gov

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 -- ========================================================== Curtis Osterhoudt cfo at remove_this.lanl.and_this.gov PGP Key ID: 0x4DCA2A10 Please avoid sending me Word or PowerPoint attachments See http://www.gnu.org/philosophy/no-word-attachments.html ==========================================================

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