Table 1 Matrix notation

$G=\left[\begin{array}{l}{g}_{11}{g}_{12}\cdots g_{1N}\\ g_{21}{g}_{22}\cdots g_{2N}\\ ⋮⋮\ddots ⋮\\ g_{M1}{g}_{M2}\cdots g_{\mathit{\text{MN}}}\end{array}\right]$

$P=\left[\begin{array}{l}{p}_{11}{p}_{12}\cdots p_{1K}\\ p_{21}{p}_{22}\cdots p_{2K}\\ ⋮⋮\ddots ⋮\\ p_{M1}{p}_{M2}\cdots p_{\mathit{\text{MK}}}\end{array}\right]$

$Q=\left[\begin{array}{l}{q}_{11}{q}_{12}\cdots q_{1N}\\ q_{21}{q}_{22}\cdots q_{2N}\\ ⋮⋮\ddots ⋮\\ q_{M1}{q}_{M2}\cdots q_{\mathit{\text{KN}}}\end{array}\right]$

g _li  ∈ {0, 1, 2} : number of reference alleles at l th locus for i th individual.

$0\le P_{\mathit{lk}}\le 1$: percentage of reference alleles at l th locus in k th population.

$q_{\mathit{\text{ki}}}\ge 0,\underset{k=1}{\overset{K}{\Sigma }}{q}_{\mathit{\text{ki}}}=1$: fraction of i th individual’s genome originating from k th population.

M = number of loci (markers)

$1\le l\le M$

N = number of individuals

$1\le i\le N$

K = number of populations

$1\le k\le K$