Fig. 1: Correcting arch and shrinkage effects.

a–c Coalescent simulation of allele frequencies drifting through time in a random mating population. Samples with ages ranging from 0 (present-day) to 4000 years (past) were simulated. a Covariance matrix for observed samples. In covariance matrices (a–c), blue colors indicate lower covariance values whereas the yellow and gray colors indicate higher values. b FA and PC plots of individual samples (present-day: gray color, past: dark blue color). c Theoretical covariance matrix defined as the minimum of forward times for pairs of samples (Brownian model). d FA and PC plots for a coalescent simulation of a two-population model (left: population 1, right: population 2, present-day: gray color, past: dark blue color). e–f Factor analysis corrects for a shrinkage effect visible in projections of ancient samples (salmon and blue colors) onto PCs of present-day admixed individuals (yellow color). Gray crosses represent population centers from which admixture estimates are computed.