Annals of Data Science

, Volume 5, Issue 4, pp 549–563 | Cite as

Fractal Dimension Calculation for Big Data Using Box Locality Index

  • Rong Liu
  • Robert Rallo
  • Yoram Cohen


The box-counting approach for fractal dimension calculation is scaled up for big data using a data structure named box locality index (BLI). The BLI is constructed as key-value pairs with the key indexing the location of a “box” (i.e., a grid cell on the multi-dimensional space) and the value counting the number of data points inside the box (i.e., “box occupancy”). Such a key-value pair structure of BLI significantly simplifies the traditionally used hierarchical structure and encodes only necessary information required by the box-counting approach for fractal dimension calculation. Moreover, as the box occupancies (i.e., the values) associated with the same index (i.e., the key) are aggregatable, the BLI grants the box-counting approach the needed scalability for fractal dimension calculation of big data using distributed computing techniques (e.g., MapReduce and Spark). Taking the advantage of the BLI, MapReduce and Spark methods for fractal dimension calculation of big data are developed, which conduct box-counting for each grid level as a cascade of MapReduce/Spark jobs in a bottom-up fashion. In an empirical validation, the MapReduce and Spark methods demonstrated good effectiveness and efficiency in fractal calculation of a big synthetic dataset. In summary, this work provides an efficient solution for estimating the intrinsic dimension of big data, which is essential for many machine learning methods and data analytics including feature selection and dimensionality reduction.


Fractal dimension Intrinsic dimension Box-counting Box locality index MapReduce Spark 



This study was supported, in part, by the National Science Foundation and the Environmental Protection Agency under Cooperative Agreement No. DBI-0830117. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the Environmental Protection Agency. This work has not been subjected to EPA review and no official endorsement should be inferred. Support by the UCLA Water Technology Research Center is also acknowledged. R. Rallo is supported by the Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of the Environment and SustainabilityUniversity of CaliforniaLos AngelesUSA
  2. 2.Advanced Computing, Mathematics, and Data DivisionPacific Northwest National LaboratoryRichlandUSA
  3. 3.Chemical and Biomolecular Engineering DepartmentUniversity of CaliforniaLos AngelesUSA
  4. 4.Center for Environmental Implications of NanotechnologyUniversity of CaliforniaLos AngelesUSA

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