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首页> 外文期刊>International Journal of Innovative Computing Information and Control >LOWER-DIMENSIONAL TRANSFORMATIONS FOR HIGH-DIMENSIONAL MINIMUM BOUNDING RECTANGLES
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LOWER-DIMENSIONAL TRANSFORMATIONS FOR HIGH-DIMENSIONAL MINIMUM BOUNDING RECTANGLES

机译：高维最小边界矩形的低维变换

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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.
机译：关于高维点的低维变换，已经进行了许多研究工作。但是，许多实际数据（例如时间序列和流数据）可以建模为MBR（最小边界矩形）而不是高维空间中的点。为了在多维索引中存储和搜索那些高维MBR，我们需要将高维MBR本身直接转换为低维MBR。为了解决该问题，我们首先提出了一种低维MBR变换的新概念。低维MBR转换应通过包含所有低维点来构造低维MBR，在MBR中将无数个高维点转换为该低维点。作为DFT（离散傅里叶变换）和DCT（离散余弦变换）的幼稚解决方案，我们分别提出了DFTnaive和DCTnaive，它们使用了高维MBR上的所有顶点。但是，随着尺寸的增加，这些幼稚的解决方案需要大量的转换。为解决此问题，我们进一步提出了DFTadv和DCTadv，它们仅使用两个点（左下角和右上角点），而不使用MBR上的所有顶点。通过提出相关定理，我们也正式证明了所有提出的解决方案都能正确执行低维MBR变换。实验结果表明，当尺寸为16时，我们的高级解决方案的性能比原始解决方案高出13,100倍，并且随着尺寸的增加，改进效果也会更大。这些结果表明，提出的低维MBR转换概念为各种时间序列数据应用程序提供了非常实用的框架。

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