Claim-Specific Admissibility of PCA Biplot Interpretations: Target Alignment, Spectral Identifiability, and Projection Adequacy

Abstract

Unit-variance standardisation is often applied routinely before principal component analysis (PCA), although it replaces covariance geometry by correlation geometry. A resulting biplot may be computed correctly yet fail to support a scientific interpretation about the original-scale phenomenon. We therefore make the scientific statement, rather than the decomposition alone, the unit of methodological assessment. A PCA-biplot claim is representationally well posed only when the declared scientific target justifies the operator analysed, the invoked axis or invariant subspace is identifiable at the stated level, and the displayed projection preserves the prespecified relationships within a substantively justified tolerance; failure of any condition makes the claim ill posed for the stated interpretation. A projector formulation yields basis-invariant diagnoses for repeated-eigenvalue blocks, and a residual-Gram bound quantifies pairwise error from omitted coordinates. Six controlled population scenarios provide exact representational truth, including zero target association with arbitrary projected angles, collapse of a full-space 60-degree relationship to 0 degrees, and non-identifiable named axes. The contribution is not another reminder that scaling matters: it establishes that computational correctness is necessary but not sufficient for scientific interpretability and provides a formal procedure for retaining, reformulating, qualifying, or rejecting a PCA-biplot statement. A real-data illustration and bootstrap extension are provided as supplementary material.