Case Study II: Constrained Unidimensional Scaling for Linkage Disequilibrium Maps
With the advance of the genotyping single nucleotide polymorphisms (SNPs) in mass scale of high density in a candidate region of the human genome, the linkage disequilibrium analysis can offer a much higher resolution of the biological samples than the traditional linkage maps. The advantages of LD maps include the revealing of the fine scale recombination patterns, the facilitating of the optimal SNP/marker spacing, and the increasing of the power for localizing disease genes etc. The first LD maps were proposed by Maniatis and colleagues (Maniatis et al., 2002). The derivation of this LD map is parametric and requires the estimation of three coefficient parameters. Nevertheless, these estimated parameters are found to have large variances among different populations.
We have formulated this LD mapping problem as a constrained unidimensional scaling problem. Our method, which is directly based on the measurement of LD among SNPs, is non-parametric. Therefore it is different from LD maps derived from the given Malecot model. We have proposed the quadratic programming approach for solving this constrained unidimensional scaling problem. Different from the classical metric unidimensional scaling problem, the constrained problem is not an NP-hard combinatorial problem. The optimal solution is determined by using the quadratic programming solver.
KeywordsIterative Algorithm Amplify Fragment Length Polymorphism Recombination Region Quadratic Programming Algorithm Linkage Disequilibrium
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