Goodness-of-fit filtering in classical metric multidimensional scaling with large datasets

Metric multidimensional scaling (MDS) is a widely used multivariate method with applications in almost all scientific disciplines. Eigenvalues obtained in the analysis are usually reported in order to calculate the overall goodness-of-fit of the distance matrix. In this paper, we refine MDS goodness-of-fit calculations, proposing additional point and pairwise goodness-of-fit statistics that can be used to filter poorly represented observations in MDS maps. The proposed statistics are especially relevant for large data sets that contain outliers, with typically many poorly fitted observations, and are helpful for improving MDS output and emphasizing the most important features of the dataset. Several goodness-of-fit statistics are considered, and both Euclidean and non-Euclidean distance matrices are considered. Some examples with data from demographic, genetic and geographic studies are shown.

度量型多维标度（Metric Multidimensional Scaling, MDS）是一种应用广泛的多元统计方法，几乎覆盖所有科学研究领域。分析所得的特征值通常会被报告，用于计算距离矩阵的整体拟合优度。本文对MDS拟合优度的计算方法进行了改进，提出了新增的单样本与成对样本拟合优度统计量，可用于筛选MDS可视化映射中拟合效果欠佳的观测样本。所提出的统计量尤其适用于包含异常值、且往往存在大量拟合效果欠佳样本的大型数据集，有助于优化MDS输出结果并凸显数据集的核心特征。本文共考量了多种拟合优度统计量，同时覆盖欧氏距离与非欧氏距离矩阵两类场景。文末辅以人口学、遗传学与地理学领域的研究数据集案例进行演示。