OLAP的形式代数

摘要：

联机分析处理（OLAP）包括允许查询多维数据库的工具和算法。它基于多维模型，其中数据可以被视为立方体，其中每个单元格包含一个或多个可以沿着维度聚合的度量。尽管在这个领域有大量的工作，但仍然需要一种标准的OLAP语言，因为对于许多通常的OLAP操作来说，没有定义良好的、公认的语义。在本文中，我们解决这个问题，并提出了一套操作的数据立方体。我们清楚地定义了这些操作的语义，并证明它们可以组成，产生一个强大的语言，足以表达复杂的OLAP查询。我们将这些操作表示为一个固定多维矩阵上的原子变换序列，该矩阵的单元格包含一个测量序列。每一个原子变换都产生一个新的测度。当一系列转换定义OLAP操作时，将生成一个标志，指示哪些单元格必须被视为下一个操作的输入。这样，一个优雅的代数被定义。我们的主要贡献，在该领域的其他类似的努力是，第一次，正式证明的操作的正确性，从而为他们提供了一个明确的语义。我们相信，目前的工作将作为一个基础，建立更坚实的实用工具，数据分析。

英文摘要：

Online Analytical Processing (OLAP) comprises tools and algorithms that allow querying multidimensional databases. It is based on the multidimensional model, where data can be seen as a cube, where each cell contains one or more measures can be aggregated along dimensions. Despite the extensive corpus of work in the field, a standard language for OLAP is still needed, since there is no well-defined, accepted semantics, for many of the usual OLAP operations. In this paper, we address this problem, and present a set of operations for manipulating a data cube. We clearly define the semantics of these operations, and prove that they can be composed, yielding a language powerful enough to express complex OLAP queries. We express these operations as a sequence of atomic transformations over a fixed multidimensional matrix, whose cells contain a sequence of measures. Each atomic transformation produces a new measure. When a sequence of transformations defines an OLAP operation, a flag is produced indicating which cells must be considered as input for the next operation. In this way, an elegant algebra is defined. Our main contribution, with respect to other similar efforts in the field is that, for the first time, a formal proof of the correctness of the operations is given, thus providing a clear semantics for them. We believe the present work will serve as a basis to build more solid practical tools for data analysis.

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