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

Area of Science:

• Population Genetics
• Genomics
• Bioinformatics

Background:

• Wright's inbreeding coefficient (FST) traditionally assesses population structure at specific loci.
• Principal Component Analysis (PCA) is increasingly used for unsupervised population structure analysis with large genomic datasets.

Purpose of the Study:

• To elucidate the theoretical relationships between FST and PCA in a model of K discrete populations.
• To provide an equivalent definition of FST derived from genotype matrix decomposition.

Main Methods:

• Decomposition of the genotype matrix into between- and within-population components.
• Application of PCA to the between-population matrix to derive average FST.
• Theoretical analysis linking FST to the spectrum of the genotype matrix.

Main Results:

• An equivalent definition of FST is derived from genotype matrix decomposition.
• Average FST is shown to be obtainable from PCA of the between-population matrix.
• FST approximates the proportion of genetic variation explained by the first (K-1) principal components under specific conditions.

Conclusions:

• PCA results can be interpreted through population genetic concepts like FST.
• The new FST definition facilitates analysis with surrogate or corrected genotypes.
• Extends population genetic analyses to incorporate temporal, geographical, and environmental factors.