Finding Fractal Behavior - Methods & Statistical Analysis

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SUMMARY

This discussion focuses on identifying fractal behavior in datasets using statistical methods. The box method is employed to calculate fractal dimensions, yielding an exponent of approximately 1.85, indicating a power law relationship between box size and the number of boxes. Participants emphasize the need for a statistical approach that explicitly examines self-similarity across different scales, particularly in 1D and 2D patterns. Clarification on the types of measurements and distributions involved is also sought to enhance the analysis.

PREREQUISITES
  • Understanding of fractal dimensions and their calculation methods
  • Familiarity with the box counting method for fractal analysis
  • Knowledge of statistical distributions and their applications
  • Basic concepts of self-similarity in mathematical patterns
NEXT STEPS
  • Research statistical methods for analyzing self-similarity in datasets
  • Explore advanced techniques for calculating fractal dimensions beyond the box method
  • Learn about power law distributions and their implications in fractal analysis
  • Investigate software tools for visualizing and analyzing fractal behavior in 1D and 2D data
USEFUL FOR

Data scientists, statisticians, and researchers interested in fractal analysis and self-similarity in datasets will benefit from this discussion.

Vaal
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Hi, I have been looking for fractal behavior in a data set. I've used the box method to determine fractal dimension by looking at the inverse of box size and the number of boxes needed to enclose the object. These two variables seem to be fairly accurately predicted by a power law (exponent/fractal dimension around 1.85) but I was hoping for a simple statistical method that explicitly looks for self similarity on different scales. Does anyone have any ideas here?

Thanks in advance for any help.
 
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Hey Vaal.

I don't know much about factal dimensions so maybe you could fill in the missing blanks.

If you want to get some kind statistical answer you need to explain the variation involved in terms of a distribution.

What kinds of possible things are you trying to measure (i.e. boxes or areas or lengths with a distribution)? Are you only measuring properties related to boxes? If so do you have a distribution for the sides or the area of the box?
 
The data is a scalar value for all points in a 2D plane. I am most interested in seeing if patterns that are evident on large scales repeat themselves on smaller scales, like is seen in fractals. I'm interested in both 1D and 2D patterns.

Thanks.
 

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