# On the Mathematical Validity of the Higuchi Method

## TL;DR

This paper rigorously analyzes the Higuchi method for estimating fractal dimensions, confirming its accuracy for certain functions and highlighting its potential instability under perturbations.

## Contribution

It provides a precise mathematical formulation of the Higuchi method and evaluates its validity and robustness for different classes of functions.

## Key findings

- Accurately estimates the dimension for non-fractal functions
- Provides a geometrical interpretation of the algorithm
- Shows the method can be unstable under perturbations

## Abstract

In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show that it yields the correct dimension for a class of non-fractal functions. Furthermore, it will be shown that the algorithm follows a geometrical approach and therefore gives a reasonable estimate of the fractal dimension of a fractal function. We conclude the paper by discussing the robustness of the method and show that it can be highly unstable under perturbations.