A spectral theory for Wright’s inbreeding coefficients and related quantities

Abstract

Wright’s 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 Wright’s 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. The average value of Wright’s FST over all loci included in the genotype matrix can be obtained from the PCA of the between-population matrix. Assuming that a separation condition is fulfilled and for reasonably large data sets, this value of FST approximates the proportion of genetic variation explained by the first (K − 1) principal components accurately. The new definition of FST is useful for computing inbreeding coefficients from surrogate genotypes, for example, obtained after correction of experimental artifacts or after removing adaptive 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.

A geometric relationship of F 2 , F 3 and F 4 -statistics with principal component analysis

“…Previous interpretation of PCA in the context of population genetic models have focused on explicit models and aimed at directly interpreting the PCs in terms of population genetic parameters [ 16 , 23 , 46 , 48 ]. My interpretation here is different in that the utility of PCA is to simplify the geometry of the data, rather than attributing meaning to the produced PCs.…”

“…My interpretation here is different in that the utility of PCA is to simplify the geometry of the data, rather than attributing meaning to the produced PCs. One consequence is that the results here are less directly impacted by sample ascertainment, sample sizes or number of PCs, which are common concerns in the interpretation of PCA [ 23 , 46 – 48 ]; adding more PCs will provide a successively better approximation of the F -statistics.…”

“… 30 , 45 ], making use of the hundreds of thousands of loci available in most genome-scale datasets. The PCA decomposition has been studied for a number of explicit population genetic models including trees [ 16 ], spatially continuous structure [ 46 ], the coalescent [ 47 ] and discrete population models [ 48 ]. Here, in order to link PCA to F -statistics, I interpret both of them geometrically in allele frequency space , i.e.…”

Theoretical Analysis of Principal Components in an Umbrella Model of Intraspecific Evolution

“…The first PCs represent the axes of genetic variation which carry the most information, and the eigenvalues represent the variances of samples along the axes. The use of PCA dates back to the early days of human population genetics [5, 35], and it has been shown theoretically that it can reveal information about admixture and population differentiation among samples [37, 34, 38, 18].…”

smartsnp, an r package for fast multivariate analyses of big genomic data

“…summarizing population structure in the genomic era-essentially, because the underlying mathematical theory is conceptually simple (Fenderson et al, 2020) and PCA outputs have a clear genetic interpretation (François & Gain, 2021;McVean, 2009;Peter, 2021).…”