Fractal Dimension Calculation for Big Data Using Box Locality Index

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.