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Extended Data Fig. 3 | Simulations. a, Theoretically computed r 2 stratified  
by minor allele frequency between true genotypes and genotypes with 
randomly introduced errors (Methods), emulating imputation accuracy plots. 
b, Admixture simulation where sources P1 and P2 split 250 or 500 generations 
ago and a pulse admixture event gives rise to a target population PX 50 
generations ago. We vary demographic history, error rates, and sample sizes 
and simulate 20 replicates for each scenario (see Methods for simulation 
details). Admixture proportions are computed using an f 4-ratio statistic and the 
Twigstats cutoff is set to twice the source split time and the rare variant cutoff 
is 5%. We plot two standard errors around the mean. c, qpAdm simulation taken 
from21,23, as well as an adapted version where all population split times and  
the admixture date are divided by 5. The Twigstats cutoff time is chosen to be 
1200 generations (top) and 600 generations (bottom). We simulate 10 replicates 
and plot two standard errors around the mean. d , Simulation adapted from 23 of 
a stepping stone model with 9 populations organised on a 1-dimensional grid as 
shown, where individuals are able to migrate between adjacent fields. We run a 
rotational qpAdm to fit population 4 using other pairs of populations to the left 
and right as sources. We run 50 replicates and set the p-value of models with 
inferred proportions outside of [0,1] to 0. We then compute the proportion 
where a given pair achieves the best p-value (top) and show the median p-value 
across these replicates (bottom). In all simulations in b, c, d, we sample N = 20 
haploid sequences per population, except for one simulation in b , where we 
sample N = 100 sequences.