computer sciencefractals
I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most appropriate fractal dimension to look at and what method do you recommend I use to estimate it numerically?
Thanks
Look at any of my published papers! :-)
I've
and then a whole lot of stuff that hasn't yet, or won't ever, make it into print.
My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?
Of course this depends on your definition of fractal. There are nowhere-differentiable functions with graph of Hausdorff dimension 1.
Is the WM function the easiest example of a nowhere differentiable continuous function?
No.
For example, a nowhere-differentiable function due to Kießwetter was designed to be used with high-school students in Germany. English translation in my book: Classics on Fractals
Are these definitions equivalent?
No, the definition with self-similarity is not equivalent to Hausdorff dimension > topological dimension. [Using self-similarity as a definition of fractal should be considered something to use for non-mathematicians who are curious about the subject, but have no hope to understand measures and such for the real definition.]
Is the precise definition still under debate?
Mandelbrot gave the definition: Hausdorff dimension strictly greater than topological dimension. He later wrote that he regretted this, and instead it should be left undefined. Others have provided other definitions. For actual mathematical papers, the authors of course state what they are proving in real mathematical language, not using the word fractal or just using it for the vague explanatory part of the paper.
Added:
Kiesswetter function, two figures from Classics on Fractals
Best Answer
It depends what you want to measure. For real-life data box-counting dimension based on Renyi entropy (of order $q$) is a common choice. For some problems $q=1$ (Shannon entropy) or $q=2$ (collision entropy) may be privileged. You can plot fractal dimension for any $q$ (obtaining rather a function that a single number). Furthermore, you can make a Legendre transform obtaining so-called singularity spectrum (or calculate it directly, see links below).
References, starting from the most accessible: