Scale Transformations in Functional Biplots: Effects on Representational Fidelity and Graphical Interpretability

The Biplot, as a graphical technique for visualizing multivariate data, can be extended to the functional data setting using the scores and principal functions obtained through Functional Principal Component Analysis (FPCA). However, differences in their magnitudes may hinder joint visualization and interpretation. This study evaluates the effects of graphical scale transformations on ordinal agreement and geometric fidelity, and introduces a grid-independent scale-balancing criterion based on the root-mean-square magnitude of principal functions over a fixed functional domain. Six representations are examined using rural population percentages from 20 Latin American countries (1960–2025): the original scale, standardization, three reference-range adjustments, and reciprocal scaling. Spearman’s correlation measures agreement between observed country rankings and Biplot projections, while mathematical analysis and Monte Carlo simulations assess reconstruction and marker-distance properties under varying sample sizes, temporal resolutions, noise levels, and generating structures. The original scale achieves the highest empirical ordinal agreement; among non-preserving alternatives, Case 2 performs best in both empirical and simulated comparisons despite substantial reconstruction changes. Cases 1 and 3 yield identical rankings, whereas reciprocal scaling preserves the truncated Karhunen–Loève reconstruction and projection rankings. These findings underscore the need to distinguish ordinal agreement, geometric fidelity, and graphical readability.