Abstract

Principal Component Analysis (PCA) biplots provide a framework for low-dimensional visualisation of multivariate data matrices X:n×p of numeric data. Categorical PCA (CatPCA) allows for the inclusion of categorical variables by assigning scores to the category levels. The method is based on an iterative alternating least-squares algorithm that begins with initial category scores and proceeds by alternately estimating an optimal low-rank approximation and updating the category scores. When the category scores are held fixed, PCA yields the best-fitting low-rank approximation; conversely, when the approximation is fixed, new category scores are obtained through optimisation using Lagrange multipliers.
In a CatPCA biplot, observations are represented as points, typically in a two-dimensional display. Variables are simultaneously represented by calibrated axes for numerical variables and piecewise linear axes for categorical variables. This dual representation provides an informative graphical summary that facilitates the simultaneous interpretation of relationships among observations, variables, and category levels. As discussed by Gower, Lubbe and Le Roux (2011), the CatPCA biplot can be regarded as a natural multivariate extension of the classical scatterplot, retaining its intuitive interpretability while substantially increasing its analytical capabilities.
More recently, Lubbe and Brito (in press) extended PCA biplot methodology to the broader framework of Symbolic Data Analysis (SDA). In contrast to conventional data analysis, where each observational unit is described by a single value for each variable, SDA accommodates more complex data structures, including interval-valued and histogram-valued variables, thereby capturing intrinsic variability within observational units. For a comprehensive overview of SDA, see Brito (2014).

Building on these developments, the present paper extends PCA biplot visualisation for numerical, interval-valued, and histogram-valued variables to incorporate categorical and modal variables within a unified symbolic data framework. The resulting methodology broadens the scope of biplot analysis and provides a flexible graphical tool for the exploration and interpretation of heterogeneous symbolic datasets.

References
Brito, P. 2014. Symbolic data analysis: another look at the interaction of data mining and statistics. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 4(4), pp. 281-295.
Gower, J. C., Lubbe, S., and Le Roux, N. J. 2011. Understanding Biplots. John Wiley & Sons.
Lubbe, S. and Brito, P. In press. Principal component analysis biplots for symbolic data analysis.