Lower-dimensional transformations for high-dimensional minimum bounding rectangles

초록

There have been many research efforts on the lower-dimensional transformation of high-dimensional points. However, a lot of real data such as time-series sequences and streaming data can be modeled as an MBR (Minimum Bounding Rectangle) rather than a point in a high-dimensional space. To store and search those high-dimensional MBRs in a multidimensional index, we need to transform a high-dimensional MBR itself to a low-dimensional MBR directly. To tackle the problem, we first present a new notion of lower-dimensional MBR transformation. The lower-dimensional MBR transformation should construct a low-dimensional MBR by containing all the low-dimensional points to which an infinite number of high-dimensional points in the MBR are transformed. As the naive solutions for DFT (discrete Fourier transform) and DCT (discrete Cosine transform), we then propose DFTnaive and DCTnaive, respectively, which use all the vertex points on a high-dimensional MBR. These naive solutions, however, require a huge number of transformations as the dimension increases. To solve this problem, we further propose DFTadv and DCTadv, which use only two points (a lower-left and an upper-right points) rather than all the vertex points on an MBR. By presenting related theorems, we also formally prove that all the proposed solutions perform the lower-dimensional MBR transformation correctly. Experimental results show that our advanced solution outperforms the naive solution by up to 13,100 times, when the dimension is 16, and the improvement becomes larger as the dimension increases. These results indicate that the proposed notion of lower-dimensional MBR transformation provides a very practical framework for a variety of time-series data applications. © 2013 ICIC International.

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