Ht-Index for Quantifying the Fractal or Scaling Structure of Geographic Features

Abstract

Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is false, is mainly attributed to the strict definition of fractal dimension as a measure or index for characterizing the complexity of fractals. In this paper, we propose an alternative, the ht-index, to quantify the fractal or scaling structure of geographic features. A geographic feature has ht-index h if the pattern of far more small things than large ones recurs (h-1) times at different scales. The higher the ht-index, the more complex the geographic feature. We conduct three case studies to illustrate how the computed ht-indices capture the complexity of different geographic features. We further discuss how the ht-index is complementary to fractal dimension, and elaborate on a dynamic view behind the ht-index that enables better understanding of geographic forms and processes.

Geospatial big data handling theory and methods: A review and research challenges

“…1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 13 Figure 2 Fractals emerged from big data look very much like the generative fractal -Koch snowflake (Jiang 2015a) What the example illustrated are not only fractals or natural cities generated from big data, but also a new, relaxed definition of fractals. A set or pattern is fractal if there are far more small things than large ones in it, or the scaling pattern of far more small things than large ones recurs multiple times (Jiang 2015b, Jiang & Yin 2014. The new definition is in fact developed from head/tail breaks (Jiang 2013) as a classification scheme for data with a heavy tailed distribution.…”

The fractal nature of maps and mapping

“…However, while this simplistic thinking might be of little use from a physicist point of view (e.g., Bak 1996), it is of great value for mapping or understanding the underlying geographic forms and processes. Geographic forms are fractal rather than Euclidean, and the underlying geographic processes are nonlinear rather than linear (e.g., Batty and Longley 1994, Frankhauser 1994, Salingaros 2005, Benguiui and Blumenfeld-Lieberthal 2007, Chen 2009, Jiang and Yin 2014. This presents the basic perception of geographic forms and processes.…”

A Zipf’s Law-Based Method for Mapping Urban Areas Using NPP-VIIRS Nighttime Light Data

“…Jiang [8] found that the NTL data followed a heavy-tailed distribution, where large values above the average were the minority, termed the head, and small values were the majority, termed the tail. This head/tail breaks method [62] iteratively divides the NTL data around the average into the head and tail parts until the head no longer represents a long-tailed distribution. This method has been verified as a universal and powerful method to delineate urban areas, namely, natural cities by Jiang [8].…”

“…Previous studies [8] have suggested that this sensitive variable is approximately set to 60%. However, Jiang [62] noted that the sensitive variable could be modified in many cases, such as to 50% or even more. Therefore, this sensitive variable in the head/tail breaks method needs more objectivity.…”