Cross-Validation as a Bandwidth Criterion for Geographically Weighted PCA: An Ethics of Ambiguity

Geographically weighted principal component analysis (GWPCA) summarizes spatial multivariate structure through local eigen decompositions whose output depends strongly on a bandwidth, conventionally chosen by leave-one-out cross-validation (CV). While locating the bandwidth that minimizes CV, several competing minima are frequently observed. We focus on the implications of multiple ambiguous minima as the object of study. We formalize it through the CV-equivalent set ($H_{\mathrm{CV}}$) of bandwidths that the data do not distinguish from the CV minimum ($h_{CV}$) and show that bandwidths within this set differ materially in their bias--variance composition and loading structure. Simulations across eight data-generating processes show that wherever the covariance subspace varied locally, the equivalent set spanned roughly a quarter of the candidate range grid, and the oracle bandwidth typically lay inside the set while differing from $h_{CV}$. In applications to the Dublin voter turnout and the Kola geochemical survey, the same ambiguity appeared, and within-set disagreement between local loadings concentrated where the local eigengap was small. We offer these findings as the first in-depth study of the characteristics of CV as used in GWPCA, and recommend reporting standards to include the equivalent set, and mapping the local eigengap.