text: We then implement a second function in Twigstats to compute, for 
  each target population and putative source populations, the propor- 
  tion of the genome ‘painted’ by each of the reference populations. 
  Given*k*reference populations, we denote by**a***i*the vector of length*k*
  storing these proportions for population*i*. We fitted our target popu- 
  lation as a mixture of putative source populations using a non-negative 
  least squares approach that finds a solution to the optimization prob- 
  lem min 
  , where**A**is a matrix storing ℓ**a**for puta- 
  tive source populations as its column vectors with ℓ indexing source 
  populations and**β**are non-negative mixture weights. 
  
  || 
  **a** −  **Aβ** || 
  *Σ*0≤  ≤1  target  2 
  **β**ℓ 
  
  We expect that this simulation model violates qpAdm assumptions 
  of no (or limited) gene flow after admixture between sources and ref- 
  erence groups. Consistent with this idea, qpAdm models are rejected 
  (*P*= 4 × 10−38 for migration rates of 0.001 and*P*= 5 × 10−8 for migration 
  rates of 0.005) when using Twigstats with a cut-off of 1,000 genera- 
  tions. However, these are not rejected using regular qpAdm, including 