To infer the histories of population admixture, one important challenge with methods based on the admixture linkage disequilibrium (ALD) is to remove the effect of source LD (SLD), which is directly inherited from source populations. In previous methods, only the decay curve of weighted LD between pairs of sites whose genetic distance were larger than a certain starting distance was fitted by single or multiple exponential functions, for the inference of recent single- or multiple-wave admixture. However, the effect of SLD has not been well defined and no tool has been developed to estimate the effect of SLD on weighted LD decay. In this study, we defined the SLD in the formularized weighted LD statistic under the two-way admixture model and proposed a polynomial spectrum (p-spectrum) to study the weighted SLD and weighted LD. We also found that reference populations could be used to reduce the SLD in weighted LD statistics. We further developed a method, iMAAPs, to infer multiple-wave admixture by fitting ALD using a p-spectrum. We evaluated the performance of iMAAPs under various admixture models in simulated data and applied iMAAPs to the analysis of genome-wide single nucleotide polymorphism data from the Human Genome Diversity Project and the HapMap Project. We showed that iMAAPs is a considerable improvement over other current methods and further facilitates the inference of histories of complex population admixtures.
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