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How complex is a fractal?: Head/tail breaks and fractional hierarchy
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Industrial Development, IT and Land Management, Land management, GIS.ORCID iD: 0000-0002-2337-2486
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Industrial Development, IT and Land Management, Land management, GIS.ORCID iD: 0000-0001-9328-9584
2018 (English)In: Journal of Geovisualization and Spatial Analysis, ISSN 2509-8810Article in journal (Refereed) Epub ahead of print
Abstract [en]

A fractal bears a complex structure that is reflected in a scaling hierarchy, indicating that there are far more small things than large ones. This scaling hierarchy can be effectively derived using head/tail breaks—a clustering and visualization tool for data with a heavy-tailed distribution—and quantified by a head/tail breaks-induced integer, called ht-index, indicating the number of clusters or hierarchical levels. However, this integral ht-index has been found to be less precise for many fractals at their different phrases of development. This paper refines the ht-index as a fraction to measure the scaling hierarchy of a fractal more precisely within a coherent whole and further assigns a fractional ht-index—the fht-index—to an individual data value of a data series that represents the fractal. We developed two case studies to demonstrate the advantages of the fht-index, in comparison with the ht-index. We found that the fractional ht-index or fractional hierarchy in general can help characterize a fractal set or pattern in a much more precise manner. The index may help create intermediate map scales between two consecutive map scales.

Place, publisher, year, edition, pages
2018.
Keywords [en]
Ht-index;Fractal;Scaling;Complexity;Fht-index
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:hig:diva-26168DOI: 10.1007/s41651-017-0009-zOAI: oai:DiVA.org:hig-26168DiVA, id: diva2:1183548
Available from: 2018-02-18 Created: 2018-02-18 Last updated: 2018-03-22Bibliographically approved
In thesis
1. Topological and Scaling Analysis of Geospatial Big Data
Open this publication in new window or tab >>Topological and Scaling Analysis of Geospatial Big Data
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Geographic information science and systems face challenges related to understanding the instinctive heterogeneity of geographic space, since conventional geospatial analysis is mainly founded on Euclidean geometry and Gaussian statistics. This thesis adopts a new paradigm, based on fractal geometry and Paretian statistics for geospatial analysis. The thesis relies on the third definition of fractal geometry: A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. Therefore, the terms fractal and scaling are used interchangeably in this thesis. The new definition of fractal is well-described by Paretian statistics, which is mathematically defined as heavy-tailed distributions. The topology of geographic features is the key prerequisite that enables us to see the fractal or scaling structure of the geographic space. In this thesis, topology refers to the relationship among meaningful geographic features (such as natural streets and natural cities).

The thesis conducts topological and scaling analyses of geographic space and its involved human activities in the context of geospatial big data. The thesis utilizes the massive, volunteered, geographic information coming from LBSM platforms, which are the global OpenStreetMap database and countrywide, geo-referenced tweets and check-in locations. The thesis develops geospatial big-data processing and modeling techniques, and employs complexity science methods, including heavy-tailed distribution detection and head/tail breaks, along with some complex network analysis. Head/tail breaks and the induced ht-index are a powerful tool for geospatial big-data analytics and visualization. The derived scaling hierarchies, power-law metrics, and network measures provide quantitative insights into the heterogeneity of geographic space and help us understand how it shapes human activities at city, country, and world scales. 

Place, publisher, year, edition, pages
Gävle: Gävle University Press, 2018. p. 73
Series
Studies in the Research Profile Built Environment. Doctoral thesis ; 7
Keywords
Third definition of fractal, scaling, topology, power law, head/tail breaks, ht-index, complex network, geospatial big data, natural cities, natural streets
National Category
Computer and Information Sciences Earth and Related Environmental Sciences
Identifiers
urn:nbn:se:hig:diva-26197 (URN)978-91-88145-24-6 (ISBN)978-91-88145-25-3 (ISBN)
Public defence
2018-05-16, Lilla Jadwiga-salen, Kungsbäcksvägen 47, Gävle, 10:00 (English)
Opponent
Supervisors
Available from: 2018-04-24 Created: 2018-03-04 Last updated: 2018-04-25

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Jiang, BinMa, Ding

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