A comparison between two OLS-based approaches to estimating urban multifractal parameters

TL;DR

This paper compares two OLS-based methods for estimating urban multifractal parameters, demonstrating that fixing the intercept to zero yields more accurate and normal parameter spectrums within certain scales.

Contribution

It introduces a comparative analysis of OLS approaches for multifractal parameter estimation, recommending the zero-intercept method for better accuracy.

Findings

Zero-intercept regression produces proper multifractal spectra.

Common regression results are often abnormal.

Combining both methods helps identify urban multifractal structures.

Abstract

Multifractal theory provides a new spatial analytical tool to describe urban form and growth, but many basic problems remain to be solved. Among various pending issues, the most significant one is how to obtain proper multifractal dimension spectrums. If an algorithm is improperly used, the parameter values will be abnormal. This paper is devoted to drawing a comparison between two OLS-based approaches for estimating urban multifractal parameters. Using observational data and empirical analysis, we will demonstrate how to utilize the double logarithmic linear regression to evaluate multifractal parameters. The OLS regression analysis has two different approaches. One is to fix the intercept to zero, and the other is not to fix it. The case studies show that the advisable method is to constrain the intercept to zero. The zero-intercept regression yields proper multifractal parameter spectrums within certain scale range of moment order, while the common regression results are not normal. In practice, the zero-intercept regression and the common regression can be combined to calculate multifractal parameters. Comparing the two sets of results, we can judge when and where multifractal urban structure appears. This research will inspire urban scientists to employ a proper technique to estimate multifractal dimensions for urban scientific studies.