Mathematical Sciences: Methods for Ranked Data

在排序数据中，每个观测值都是n个项目的排列。 由于排列没有自然的线性排序，许多常见的统计方法不适合排名数据。 在n-1维欧氏空间中，n个元素的排列可以自然地放在内接于球面的多面体的顶点上。 这种几何结构提出了新的图形和分析方法，用于分析完整和部分排名的数据。 特别感兴趣的是概率密度直方图，频谱分析，核平滑方法和置信区域。 提出了分析排序数据的统计方法。 当一组“法官”被要求评估一组项目，然后按偏好顺序排列时，就会出现排名数据。 因此，每个“法官”都陈述了他的第一，第二，等等，以及他所评估的项目中的最后选择。 以图形方式显示和分析排名数据的适当方法非常重要。 这种类型的数据在社会科学、市场研究和选举中经常遇到。 一个不太为人所知，但同样重要的领域，其中遇到的排名数据是专家系统的军事应用。

In ranked data each observation is a permutation of n items. Because the permutations do not have a natural linear ordering, many common statistical methods are inappropriate for ranked data. However, the permutations of n items can be naturally placed on the vertices of a polytope inscribed in a sphere in n-1 dimensional Euclidean space. This geometry suggests new graphical and analytical methods for analyzing full and partially ranked data. Of particular interest are probability density histograms, spectral analysis, kernal smoothing methods, and confidence regions. Statistical methods are proposed to analyze ranked data. Ranked data occur when a group of "judges" are asked to evaluate a set of items and then to rank them in order of preference. Hence, each "judge" states his first, second, etc., and last choices among the items that he has evaluated. Proper methods to graphically display and analyze ranked data are very important. This type of data is encountered frequently in the social sciences, in market research, and in elections. A less well known, but equally important area in which ranked data are encountered is expert systems for military applications.