The fractal dimension logistic curve model offers a promising approach to understanding the dynamic patterns of urban development. By deriving formulas for growth rate and acceleration of fractal dimension, we can pinpoint the inflection points in urban growth and delineate distinct stages of development. Previous studies have focused on analyzing fractal dimension values over long time spans, which means determining inflection points within a sufficiently long time series and studying the growth patterns and stages within that fixed span of time. However, the applicability of this method to studying and predicting urban growth stages in different short-term periods remains uncertain and warrants further investigation. Furthermore, the commonly used calculation method for fractal dimension, the box-counting method, often yields results influenced by the time span or scale and may introduce errors in the results obtained through least squares regression. To address this issue, a new simplified fractal index value obtained through the head/tail breaks method was applied as a supplement to the box-counting method. This new method better captures the complexity of urban development and improves the effectiveness of dividing urban growth stages. To validate our approach, we conducted a case study using continuous short-span time series data from Changchun City. By applying the fractal dimension logistic curve model, we identified inflection points of past years and divide growth stages based on urban development background across different research periods, this achieves a new application of using the logistic model based on short-span time series to predict the growth stages of different periods in the past. Our findings aim to provide valuable insights and decision support for urban planning and development initiatives.

中文翻译：

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使用分形维度 Logistic 曲线建模和头尾断裂法进行城市增长阶段分析


分形维数 Logistic 曲线模型为理解城市发展的动态模式提供了一种很有前途的方法。通过推导出分形维数的增长速度和加速公式，我们可以确定城市增长的拐点并描绘不同的发展阶段。以前的研究侧重于分析长期跨度内的分形维数值，这意味着确定足够长的时间序列内的拐点，并研究该固定时间跨度内的生长模式和阶段。然而，这种方法在研究和预测不同短期城市增长阶段的适用性仍不确定，值得进一步研究。此外，分形维数的常用计算方法（盒计数法）通常会产生受时间跨度或比例影响的结果，并且可能会在通过最小二乘回归获得的结果中引入误差。为了解决这个问题，通过头/尾断裂方法获得的新简化分形指数值被应用于盒计数方法的补充。这种新方法更好地捕捉了城市发展的复杂性，并提高了划分城市增长阶段的有效性。为了验证我们的方法，我们使用来自长春市的连续短跨度时间序列数据进行了案例研究。通过应用分形维数 Logistic 曲线模型，识别过去几年的拐点，并根据不同研究时期的城市发展背景划分增长阶段，实现了使用基于短跨度时间序列的 Logistic 模型预测过去不同时期增长阶段的新应用。 我们的研究结果旨在为城市规划和发展计划提供有价值的见解和决策支持。