Biplots for the correlation matrix

The visualization of the correlation matrix by means of biplots is considered. The classical centering operations, either by the overall mean, the column means, or row and column means are shown to be problematic for the visualisation of the correlation matrix, and sub-optimal in terms of goodness-of-fit. More flexible adjustments are possible by using a single scalar adjustment, a set of column scalars or both row–and–column scalars using a weighted alternating least squares algorithm. Recently, correlation biplots with a single scalar adjustment have been advocated and outperform the usual correlation biplots made by principal component analysis. This article presents an iterative algorithm for a column adjustment of the correlation matrix with the goal of improving the goodness-of-fit over the the use of a single scalar adjustment and studies its usefulness in practical data analysis. The resulting biplots are harder to read but can be made more interpretable by using correlation tally sticks. Correlation tally sticks are advocated for improving the visualization of correlation structure. The weighted root mean squared error is used to compare low-dimensional approximations to the correlation matrix across methods.

本文探讨了利用双标图(biplots)对相关矩阵进行可视化的研究主题。研究表明，经典的中心化操作（包括总体均值中心化、列均值中心化，或行与列均值中心化）在相关矩阵可视化中存在缺陷，且在拟合优度(goodness-of-fit)方面表现为次优。通过加权交替最小二乘算法(weighted alternating least squares algorithm)，采用单一标量调整、一组列标量，或同时使用行与列标量，可实现更灵活的调整方案。近年来，带有单一标量调整的相关双标图被提出，其表现优于通过主成分分析(principal component analysis)构建的常规相关双标图。本文提出了一种用于相关矩阵列调整的迭代算法，旨在相较单一标量调整方案进一步提升拟合优度，并研究了该算法在实际数据分析中的应用价值。由此得到的双标图可读性有所下降，但通过使用相关计数棒图(correlation tally sticks)可提升其可解释性。相关计数棒图被提出用于优化相关结构的可视化效果。本研究采用加权均方根误差(weighted root mean squared error)，对不同方法下相关矩阵的低维近似效果进行对比。