HAPLOFREQ – Estimating Haplotype Frequencies Efficiently

Halperin, Eran; Hazan, Elad

doi:10.1007/11415770_42

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…In the maximum likelihood approach to PHP, one assumes an a priori probability p h for every possible haplotype h (inferred, e.g., from genotype frequencies [6]), and seeks the most likely set H of haplotypes explaining the observed genotypes, where the likelihood of a set H is given by L(H) = h∈H p h . In the special case when all a priori haplotype probabilities are equal, likelihood maximization recovers the maximum parsimony approach to PHP [7,8], in which one seeks the smallest set H of haplotypes explaining G.…”

Section: Maximum Likelihood Population Haplotypingmentioning

confidence: 99%

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Hajiaghayi¹,

Jain

Lau

et al. 2006

Computational Science – ICCS 2006

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Abstract. In this paper we consider the minimum weight multicolored subgraph problem (MWMCSP), which is a common generalization of minimum cost multiplex PCR primer set selection and maximum likelihood population haplotyping. In this problem one is given an undirected graph G with non-negative vertex weights and a color function that assigns to each edge one or more of n given colors, and the goal is to find a minimum weight set of vertices inducing edges of all n colors. We obtain improved approximation algorithms and hardness results for MWMCSP and its variant in which the goal is to find a minimum number of vertices inducing edges of at least k colors for a given integer k ≤ n.

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Section: Maximum Likelihood Population Haplotypingmentioning

confidence: 99%

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Hajiaghayi¹,

Jain

Lau

et al. 2006

Computational Science – ICCS 2006

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“…Similarly, we can optimize Equation (8) by the EM algorithm. As proved by Halperin and Hazan (2006), the likelihood of Equation (8) is concave. The EM algorithm will converge to global optimal solution, and it will be a good initial guess for the objective function in Equation (4).…”

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confidence: 90%

CNVeM: Copy Number Variation Detection Using Uncertainty of Read Mapping

Wang

Hormozdiari

Yang

et al. 2012

Lecture Notes in Computer Science

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Copy number variations (CNVs) are widely known to be an important mediator for diseases and traits. The development of high-throughput sequencing (HTS) technologies has provided great opportunities to identify CNV regions in mammalian genomes. In a typical experiment, millions of short reads obtained from a genome of interest are mapped to a reference genome. The mapping information can be used to identify CNV regions. One important challenge in analyzing the mapping information is the large fraction of reads that can be mapped to multiple positions. Most existing methods either only consider reads that can be uniquely mapped to the reference genome or randomly place a read to one of its mapping positions. Therefore, these methods have low power to detect CNVs located within repeated sequences. In this study, we propose a probabilistic model, CNVeM, that utilizes the inherent uncertainty of read mapping. We use maximum likelihood to estimate locations and copy numbers of copied regions and implement an expectation-maximization (EM) algorithm. One important contribution of our model is that we can distinguish between regions in the reference genome that differ from each other by as little as 0.1%. As our model aims to predict the copy number of each nucleotide, we can predict the CNV boundaries with high resolution. We apply our method to simulated datasets and achieve higher accuracy compared to CNVnator. Moreover, we apply our method to real data from which we detected known CNVs. To our knowledge, this is the first attempt to predict CNVs at nucleotide resolution and to utilize uncertainty of read mapping.

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“…For instance, the special case when the convex set is a polytope, lies at the core of non-linear optimization and has been studied for its applications in operations research and economics [8]. The special case of the problem where the convex set is a simplex arises in computational biology for analyzing genomic frequencies from incomplete data [19].…”

Section: P Quadratic Grothendieck Maximization Problemmentioning

confidence: 99%

Bypassing UGC from some Optimal Geometric Inapproximability Results

Guruswami¹,

Raghavendra²,

Saket³

et al. 2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this work we bypass the UGC assumption in inapproximability results for two geometric problems, obtaining a tight NP-hardness result in each case.The first problem known as the L p Subspace Approximation is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a 1 , . . . , a m } ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the sum of the p th powers of the distances to the points. For p = 2, k = n − 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p (2 < p < ∞) it is NP-hard to approximate this problem to within a factor of γ p − for constant > 0, where γ p is the pth moment of a standard Gaussian variable. This matches the factor γ p approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi [12], who also showed the same hardness result under the UGC.The second problem we study is the related L p Quadratic Grothendieck Maximization Problem, considered by Kindler, Naor and Schechtman [29]. Here, the input is a multilinear quadratic form ∑ n i, j=1 a i j x i x j and the goal is to maximize the quadratic form over the p unit ball, namely all x with ∑ n i=1 |x i | p = 1. The problem is polytime solvable for p = 2. We show that for any constant p (2 < p < ∞), it is NP-hard to approximate Val p (A) to within a factor of γ 2 p − for any > 0. The same hardness factor was shown under the UGC in [29]. We also obtain a γ 2 p -approximation algorithm for the problem using the convex relaxation of the problem defined by [29]. A γ 2 p approximation algorithm has also been independently obtained by Naor and Schechtman [33].These are the first approximation thresholds, proven under P = NP, that involve the Gaussian random variable in a fundamental way. Note that the problem statements themselves have no mention of Gaussians.

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HAPLOFREQ – Estimating Haplotype Frequencies Efficiently

Cited by 5 publications

References 16 publications

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

CNVeM: Copy Number Variation Detection Using Uncertainty of Read Mapping

Bypassing UGC from some Optimal Geometric Inapproximability Results

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