Relationship among evolutionary distance, variance-covariance matrix and principal component analysis

Principal component analyses (PCAs) are often used to visualize patterns of genetic variation in human populations. Previous studies showed a close correspondence between genetic and geographic distances. In such PCAs, the principal components are eigenvectors of the datas variance-covariance matrix, which is obtained by a genetic relationship matrix (GRM). However, it is difficult to apply GRM to multiallelic sites. In this paper, I showed that a PCA from GRM is equivalent to multidimensional scaling (MDS) from nucleotide differences. Therefore, a PCA can be conducted using nucleotide differences. The new method provided in this study provides a straightforward method to predict the effects of different demographic processes on genetic diversity.