A Spectral Theory for Wright's Inbreeding Coefficients and Related Quantities

Wrights inbreeding coefficient, FST, is a fundamental measure in population genetics. Assuming a predefined population subdivision, this statistic is classically used to evaluate population structure at a given genomic locus. With large numbers of loci, unsupervised approaches such as principal component analysis (PCA) have, however, become prominent in recent analyses of population structure. In this study, we describe the relationships between Wrights inbreeding coefficients and PCA for a model of K discrete populations. Our theory provides an equivalent definition of FST based on the decomposition of the genotype matrix into between and within-population matrices. Assuming that a separation condition is fulfilled, our main result states that the proportion of genetic variation explained by the first (K - 1) principal components can be accurately approximated by the average value of FST over all loci included in the genotype matrix. This equivalent definition of FST can be used to evaluate the fit of discrete population models to the data. It is also useful for computing inbreeding coefficients from surrogate genotypes, for example, obtained after correction of experimental artifacts or after removing genetic variation associated with environmental variables. The relationships between inbreeding coefficients and the spectrum of the genotype matrix not only allow interpretations of PCA results in terms of population genetic concepts but extend those concepts to population genetic analyses accounting for temporal, geographical and environmental contexts.