# Quantification of intrinsic quality of a principal dimension in correspondence analysis and taxicab correspondence analysis

arXiv: https://arxiv.org/abs/2108.10685 · HTML: https://mlsanity.com/paper/2108.10685/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2021.

## Built on by (most cited first, 1 of 1)

- [Visualization of Extremely Sparse Contingency Table by Taxicab Correspondence Analysis: A Case Study of Textual Data](https://mlsanity.com/paper/2308.03079/) (2023)

## Abstract (arXiv)

Collins(2002, 2011) raised a number of issues with regards to correspondence analysis (CA), such as: qualitative information in a CA map versus quantitative information in the relevant contingency table; the interpretation of a CA map is difficult and its relation with the % of inertia (variance) explained. We tackle these issues by considering CA and taxicab CA (TCA) as a stepwise Hotelling/Tucker decomposition of the cross-covariance matrix of the row and column categories into four quadrants. The contents of this essay are: First, we review the notion of quality/quantity in multidimensional data analysis as discussed by Benz'ecri, who based his reflections on Aristotle. Second, we show the importance of unravelling the interrelated concepts of dependence/heterogeneity structure in a contingency table; and to picture them two maps are needed. Third, we distinguish between intrinsic and extrinsic quality of a principal dimension; the intrinsic quality is based on the signs of the residuals in the four quadrants, hence to the interpretability. Furthermore, we provide quantifications of the intrinsic quality and use them to uncover structure in particular in sparse contingency tables. Finally, we emphasize the importance of looking at the residual cross-covariance values at each iteration.