An Approach to Estimating Quadratic Logistic Model Parameters of Fractal Dimension Curves

TL;DR

This paper presents a novel nonlinear regression approach to estimate parameters of quadratic logistic models for fractal dimension curves, demonstrated on urban growth data from Beijing.

Contribution

It introduces a new method combining differentiation, discretization, and regression to estimate quadratic logistic model parameters effectively.

Findings

Method successfully applied to Beijing's urban fractal data.

Estimates quadratic logistic parameters with high accuracy.

Applicable to various fields beyond urban science.

Abstract

The fractal dimension curves of urban form and growth fall into two categories: One can be described by common logistic function, and the other can be described with quadratic logistic function. The approach to estimating the parameter of the ordinary logistic model has been developed. However, how to estimate the parameter of quadratic logistic model is still a problem. This paper is devoted to finding a nonlinear regressive approach for estimating parameter values of quadratic logistic model of fractal dimension curves. The process can be summarized as below. First, differentiating quadratic logistic function in theory with respect to time yields a growth rate equation of fractal dimension. Second, discretizing the growth rate equation yields a nonlinear regressive model of fractal dimension curve. Third, applying the least squares method to the nonlinear regressive equation yields the capacity parameter value of the quadratic logistic model. Fourth, substituting the capacity parameter value into the quadratic logistic model and changing it into a quasilinear form, we can estimate the other parameter values by ordinary linear regression analysis. In this way, a practical quadratic logistic model of fractal dimension curves can be gained. The approach is applied to multifractal dimension curves of Beijing city to show its effectiveness. The method can be extended to estimate the parameter values of quadratic logistic models in many fields besides urban science.