Unified Metrics for Characterizing the Fractal Nature of Geographic Features

Abstract

The term fractal is used to describe an object displaying self-similarity at different scales. This self-similarity can be measured by either the power-law exponent or the ht-index, which is a recently proposed method for characterizing the fractal nature of geographic features. Although increasingly popular in geography, the ht-index is not sensitive to changes or evolutions of fractals, limiting its usefulness as an alternative “fractal dimension” to the power-law exponent. Two improvements to the ht-index were suggested in the literature, namely, the cumulative rate of growth (the CRG index) and the ratio of areas in a rank-size plot (the RA index). The CRG index is sensitive but not monotonic, however, with respect to the evolution of fractals. The RA index is both sensitive and monotonic but not interpretable in fractal terms. In this article, two novel metrics, referred to as unified metrics, are proposed by combining advantages of the ht-index and all of its improvements, being simultaneously easy to interpret, monotonic with respect to the evolution of fractals, and sensitive to changes in the evolution. The usefulness of unified metrics was demonstrated by both numerical experiments and case studies. Given that the idea behind the ht-index has led to a relaxed, emerging popular definition of fractals, the proposed unified metrics have great potential to be used as the standard fractal dimension along with such a definition.

碎形的概念, 用来描绘在不同尺度展现出自我相似性的物体。此般自我相似性, 能够以幂律指数或是 ht 指标进行衡量, 且为晚近提出分类地理特徵的碎形本质之方法。尽管 ht 指标在地理学中越来越盛行, 但它对于碎形的改变或演化并不敏感, 使其作为幂律指数之外的另类 “碎形维度” 的效用受到限制。本文献指出两个改进 ht 指标的方式, 亦即成长的累积率 (CRG 指标), 以及在等级规模测定中的面积比率 (RA 指标) 。CRG 指标对于碎形的演化是敏感的, 但却不单调。RA 指标同时是敏感且单调的, 但却无法以碎形的概念进行诠释。本文透过结合 ht 指标的优势及其所有改进之处, 提出两个被指称为统一数值统计的崭新数值统计, 同时容易诠释、 在碎形的演化上是单调的, 且对演化的改变具敏感性。本文同时透过数值实验和案例研究, 证实统一数值统计的效用。有鉴于 ht 指标背后的概念导致鬆散且逐渐盛行的碎形定义, 本文提出的统一数值统计, 有广大的潜力运用作为随着此一定义的标准碎形维度。

El término fractal se usa para describir un objeto que despliega auto-similitud a diferentes escalas. Esta auto-similitud puede medirse, bien por el exponente de la ley de potencia, o por el ht-índice, el cual es un método propuesto recientemente para caracterizar la naturaleza fractal de los rasgos geográficos. Aunque cada vez es más popular en geografía, el ht-índice no es sensible a los cambios o evoluciones de los fractales, limitando su utilidad como una “dimensión fractal” alternativa al exponente de la ley de potencia. Se sugirieron dos mejoras al ht-índice en la literatura, o sea la tasa acumulativa de crecimiento (índice CRG) y la ratio de áreas en un rango de tamaño de lotes (el índice RA). El índice CRG, sin embargo, es sensible pero no monótono, con respecto de la evolución de los fractales. El índice RA es tanto sensible como monótono, pero no es interpretable en términos fractales. En este artículo se proponen dos medidas novedosas, denominadas métricas unificadas, combinando las ventajas del ht-índice y todas sus mejoras, lo que lo hace simultáneamente fácil de interpretar, monótono con respecto a la evolución de los fractales y sensible a los cambios en la evolución. La utilidad de las métricas unificadas se demostró tanto mediante experimentos numéricos como en estudios de casos. Considerando que la idea detrás del ht-índice ha llevado a una nueva y distendida definición popular de los fractales, la métrica unificada que se propone tiene un gran potencial de usarse como la dimensión fractal estándar junto con tal definición.

Key Words:

• fractal
• head/tail breaks
• heavy-tailed distribution
• ht-index
• scaling structure

关键词：:

• 碎形
• 头／尾分割
• 重尾分佈
• ht指标
• 尺度化结构。

Palabras clave:

• fractal
• rupturas cabeza/cola
• distribución de cola pesada
• ht-índice
• estructura de escala

Acknowledgments

We thank the editor, Mei-Po Kwan, and anonymous reviewers for their constructive comments, which greatly improved this article. The first author also thanks Hong Zhang at Southwest Jiaotong University for helpful discussions. Thanks also due to Meihui Xie, Kun Tian, and Xiaofei Zhao at Tsinghua University for their early support of this work. Zhao Liu served as the corresponding author for this article.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 41471383, 41571414, and 41401434).

Additional information

Notes on contributors

Peichao Gao

PEICHAO GAO was a graduate student in the Department of Civil Engineering at Tsinghua University, Beijing, China, when this article was written and is now a PhD Candidate in the Department of Land Surveying and Geo-Informatics at The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. E-mail: [email protected]. His research interests include fractals, entropies, and their applications in geography.

Zhao Liu

ZHAO LIU is an Associate Professor in the Department of Civil Engineering at Tsinghua University, Beijing, China. E-mail: [email protected]. His research interests include geospatial analysis, geovisualization, and remote sensing.

Gang Liu

GANG LIU is an Associate Professor in the College of Earth Sciences at Chengdu University of Technology, Sichuan, China. E-mail: [email protected]. His research interests include transportation, geospatial analysis, and cartography.

Hongrui Zhao

HONGRUI ZHAO is a Professor in the Department of Civil Engineering at Tsinghua University, Beijing, China. E-mail: [email protected]. Her research interests include geospatial analysis, quantitative remote sensing, and landscape ecology.

Xiaoxiao Xie

XIAOXIAO XIE is a graduate student in the Department of Civil Engineering at Tsinghua University, Beijing, China. E-mail: [email protected]. Her research interests include time geography.