Box-Counting Dimension Calculation for Fractal Dimension Analysis

This article presents a method for calculating the fractal dimension of one-dimensional curves through program implementation based on the box-counting dimension principle. The developed program enables comprehensive analysis of fractal characteristics in 1D curves and demonstrates practical applications of box-counting methodology. During development, we addressed critical computational considerations including efficient data structures (such as dynamic arrays for storing box counts) and optimized algorithms (implementing logarithmic scaling analysis for dimension calculation). The implementation utilizes C++ programming language, leveraging its object-oriented capabilities to create a BoxCountingDimension class with key methods like calculateBoxCount() and computeFractalDimension(). This program facilitates deeper investigation into fractal properties of 1D curves and finds practical applications in data analysis domains, particularly in financial time-series analysis such as stock price volatility modeling where fractal dimensions help characterize market complexity. The core algorithm involves systematically covering the curve with varying box sizes ε, counting occupied boxes N(ε), and determining the fractal dimension from the slope of the log(N(ε)) versus log(1/ε) plot through linear regression analysis.