论文标题
在混合模型中可识别的必要条件
Necessary and sufficient conditions for identifiability in the admixture model
论文作者
论文摘要
我们考虑来自N个个体的M SNP数据，这些人是K未知的古代人群的混合。令$π_{si} $为snp s个单个i的参考等位基因的频率。因此，二倍体个体的SNP s的参考等位基因的数量是在二元分布的，该参数为2和$π_{si} $。我们假设$π_{si} = \ sum_ {k = 1}^kf_ {kf_ {sk} q_ {ki} $，其中$ f_ {sk {sk {sk {ki {ki} $是snp s snp s snp s snp s snp s的等位基因k，$ q_ {ki} $是个人i的人口k的比例。 我对F和Q的可识别性感兴趣，直到对古代人群的重新贴出来。在什么条件下，当$π= f^1q^1 = f^2q^2 $是$ f^1 $和$ f^2 $和$ q^1 $和$ q^2 $等于？我表明，一个矩阵上的锚固条件（Cabreros和Storey，2019年），另一个矩阵上的独立条件足以识别。 我将争辩说，Cabreros和Storey中必要条件的证明是不正确的，我将提供正确的证据，此外，这不需要了解祖先人口的数量。 我还将为可识别性提供抽象的必要条件。我将证明，如果不会失去可识别性，就不能大大偏离锚固条件。最后，我为非混合案例显示了必要和足够的条件。
We consider M SNP data from N individuals who are an admixture of K unknown ancient populations. Let $Π_{si}$ be the frequency of the reference allele of individual i at SNP s. So the number of reference alleles at SNP s for a diploid individual is binomially distributed with parameters 2 and $Π_{si}$. We suppose $Π_{si}=\sum_{k=1}^KF_{sk}Q_{ki}$, where $F_{sk}$ is the allele frequency of SNP s in population k and $Q_{ki}$ is the proportion of population k in the ancestry of individual i. I am interested in the identifiability of F and Q, up to a relabelling of the ancient populations. Under what conditions, when $Π=F^1Q^1=F^2Q^2$ are $F^1$ and $F^2$ and $Q^1$ and $Q^2$ equal? I show that the anchor condition (Cabreros and Storey, 2019) on one matrix together with an independence condition on the other matrix is sufficient for identifiability. I will argue that the proof of the necessary condition in Cabreros and Storey, 2019 is incorrect, and I will provide a correct proof, which in addition does not require knowledge of the number of ancestral populations. I will also provide abstract necessary and sufficient conditions for identifiability. I will show that one cannot deviate substantially from the anchor condition without losing identifiability. Finally, I show necessary and sufficient conditions for identifiability for the non-admixed case.
