In many real-life applications, data are often hierarchically structured at different levels of granulations. A multi-scale information table is a special hierarchical data set in which each object can take on as many values as there are scales under the same attribute. An important issue in such a data set is to select optimal scale combination in order to keep certain condition for final decision. In this paper, by employing Shannon’s entropy, we study the selection of optimal scale combination to maintain uncertain measure of a knowledge from a generalized multi-scale information table. We first review the concept of entropy and its basic properties in information tables. We then introduce the notion of scale combinations in a generalized multi-scale information table. We further define entropy optimal scale combination in generalized multi-scale information tables and generalized multi-scale decision tables. Finally, we examine relationship between the entropy optimal scale combination and the classical optimal scale combination. We show that, in either a generalized multi-scale information table or a consistent generalized multi-scale decision table, the entropy optimal scale combination and the classical optimal scale combination are equivalent. And in an inconsistent generalized multi-scale decision table, a scale combination is generalized decision optimal if and only if it is a generalized decision entropy optimal.

在许多实际应用中，数据通常在不同粒度级别上进行分层结构。多尺度信息表是一种特殊的分层数据集，其中每个对象可以采用与同一属性下的尺度一样多的值。这样的数据集中的一个重要问题是选择最佳的比例组合，以便为最终决策保留一定的条件。在本文中，通过运用香农熵，我们研究了最佳尺度组合的选择，以维持来自广义多尺度信息表的知识的不确定度量。我们首先在信息表中回顾熵的概念及其基本属性。然后，我们在广义的多尺度信息表中引入尺度组合的概念。我们进一步在广义多尺度信息表和广义多尺度决策表中定义熵最优尺度组合。最后，我们研究了熵最优尺度组合和经典最优尺度组合之间的关系。我们表明，在广义多尺度信息表或一致的广义多尺度决策表中，熵最优尺度组合和经典最优尺度组合是等效的。并且在不一致的广义多尺度决策表中，当且仅当尺度组合是广义决策熵最优时，尺度组合才是广义决策最优。我们表明，在广义多尺度信息表或一致的广义多尺度决策表中，熵最优尺度组合和经典最优尺度组合是等效的。并且在不一致的广义多尺度决策表中，当且仅当尺度组合是广义决策熵最优时，尺度组合才是广义决策最优。我们表明，在广义多尺度信息表或一致的广义多尺度决策表中，熵最优尺度组合和经典最优尺度组合是等效的。并且在不一致的广义多尺度决策表中，当且仅当尺度组合是广义决策熵最优时，尺度组合才是广义决策最优。