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299 nips-2012-Scalable imputation of genetic data with a discrete fragmentation-coagulation process

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Author: Lloyd Elliott, Yee W. Teh

Abstract: We present a Bayesian nonparametric model for genetic sequence data in which a set of genetic sequences is modelled using a Markov model of partitions. The partitions at consecutive locations in the genome are related by the splitting and merging of their clusters. Our model can be thought of as a discrete analogue of the continuous fragmentation-coagulation process [Teh et al 2011], preserving the important properties of projectivity, exchangeability and reversibility, while being more scalable. We apply this model to the problem of genotype imputation, showing improved computational efﬁciency while maintaining accuracies comparable to other state-of-the-art genotype imputation methods. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Scalable imputation of genetic data with a discrete fragmentation-coagulation process Lloyd T. [sent-1, score-0.338]

2 uk Abstract We present a Bayesian nonparametric model for genetic sequence data in which a set of genetic sequences is modelled using a Markov model of partitions. [sent-14, score-0.415]

3 The partitions at consecutive locations in the genome are related by the splitting and merging of their clusters. [sent-15, score-0.165]

4 We apply this model to the problem of genotype imputation, showing improved computational efﬁciency while maintaining accuracies comparable to other state-of-the-art genotype imputation methods. [sent-17, score-0.241]

5 Due to gene conversion and chromosomal crossover, genetic sequences exhibit a local ‘mosaic’-like structure wherein sequences are composed of prototypical segments called haplotypes [5]. [sent-23, score-0.332]

6 Locally, these prototypical segments are shared by a cluster of sequences: each sequence in the cluster is described well by a haplotype that is speciﬁc to the location on the chromosome of the cluster. [sent-24, score-0.601]

7 A continuous fragmentation-coagulation process (CFCP) has recently been proposed for modelling local mosaic structure in genetic sequences [9]. [sent-29, score-0.32]

8 The CFCP is a nonparametric models deﬁned directly on unlabelled partitions thereby avoiding both costly model selection and the label switching problem [8]. [sent-30, score-0.15]

9 Vertical axis represents clusters from last sample of an MCMC chain converging to DFCP posterior. [sent-36, score-0.147]

10 describes location-dependent unlabelled partitions such that at each location on the chromosome the clusters will split into multiple clusters which then merge to form the clusters at the next location. [sent-38, score-0.682]

11 The splitting and merging of clusters across the chromosome forms a mosaic structure of haplotypes. [sent-40, score-0.343]

12 Sections 4 and 5 report some experimental results showing good performance on an imputation problem, and in section 6 we conclude. [sent-43, score-0.139]

13 2 The discrete fragmentation-coagulation process In humans, most of the bases on a chromosome are the same for all individuals in a population. [sent-44, score-0.229]

14 At each SNP location, a particular chromosome has one of usually two possible bases (referred to as the major and minor allele). [sent-46, score-0.158]

15 Consequently, SNP data for a chromosome can be modelled as a binary sequence, with each entry indicating which of the two bases is present at that location. [sent-47, score-0.158]

16 In this paper we consider SNP data consisting of n binary sequences x = (xi )n , i=1 where each sequence xi = (xit )T is of length T and corresponds to the T SNPs on a segment of t=1 a chromosome in an individual. [sent-48, score-0.258]

17 The t-th entry xit of sequence i is equal to zero if individual i has the major allele at location t and equal to one otherwise. [sent-49, score-0.34]

18 We will model these sequences using a discrete fragmentation-coagulation process (DFCP) so that the sequence values at the SNP at location t are described by the latent partition πt of the sequences. [sent-50, score-0.292]

19 Each cluster in the partition corresponds to a haplotype. [sent-51, score-0.168]

20 The DFCP models the sequence of partitions using a discrete Markov chain as follows: starting with πt , we ﬁrst fragment each cluster in πt into smaller clusters, forming a ﬁner partition ρt . [sent-52, score-0.368]

21 Then we coagulate the clusters in ρt to form the clusters of πt+1 . [sent-53, score-0.27]

22 In the remainder of this section, we will ﬁrst give some background theory on partitions, and random fragmentation and coagulation operations and then we will describe the DFCP as a Markov chain over partitions. [sent-54, score-0.479]

23 Finally, we will describe the likelihood model used to relate the sequence of partitions to the observed sequences. [sent-55, score-0.15]

24 The fragmentation and coagulation operators are random operations on partitions. [sent-66, score-0.477]

25 The fragmentation F RAG(π, α, σ) of a partition π is formed by partitioning further each cluster a of π according to CRP(a, α, σ) and then taking the union of the resulting partitions, yielding a partition of S that is ﬁner than π. [sent-67, score-0.449]

26 Conversely, the coagulation C OAG(π, α, σ) of π is formed by partitioning the set of clusters of π (i. [sent-68, score-0.357]

27 , the set π itself) according to CRP(π, α, σ) and then replacing each cluster with the union of its elements, yielding a partition that is coarser than π. [sent-70, score-0.185]

28 The fragmentation and coagulation operators are linked through the following theorem by Pitman [12]. [sent-71, score-0.477]

29 Under t=1 the DFCP, the marginal distribution of the partition πt is CRP(S, µ, 0) and so µ controls the number of clusters that are found at each location. [sent-77, score-0.167]

30 Subsequently, we draw ρt |πt from F RAG(πt , 0, Rt ), which fragments each of the clusters in πt into smaller clusters in ρt , and then πt+1 |ρt from C OAG(ρt , µ/Rt , 0), which coagulates clusters in ρt into larger clusters in πt+1 . [sent-85, score-0.514]

31 In population genetics the CRP appears as (and was predated by) Ewen’s sampling formula [13], a counting formula for the number of alleles appearing in a population, observed at a given location. [sent-88, score-0.16]

32 Over a short segment of the chromosome where recombination rates are low, haplotypes behave like alleles and so a CRP prior on the number of haplotypes at a location is reasonable. [sent-89, score-0.412]

33 Further, since fragmentation and coagulation operators are deﬁned in terms of CRPs which are projective and exchangeable, the Markov chain is projective and exchangeable in S as well. [sent-90, score-0.542]

34 Chromosome replication is directional and so statistics for genetic processes along the chromosome are not reversible. [sent-94, score-0.287]

35 But the strength of this effect on SNP data is not currently known and many genetic models such as the coalescent with recombination [14] assume reversibility for simplicity. [sent-95, score-0.279]

36 The non-reversibility displayed by models such as fastPHASE is an artifact of their construction rather than an attempt to capture non-reversible aspects of genetic sequences. [sent-96, score-0.148]

37 3 Likelihood model for sequence observations Given the sequence of partitions (πt )T , we model the observations in each cluster at each location t=1 t independently. [sent-98, score-0.386]

38 (3) ρT −1 ρ2 π1 π2 ··· πT xi1 xi2 ··· xiT ∀1≤i≤n θ1a θ2a ∀ a ∈ π1 ∀ a ∈ π2 β1 β2 ··· θT a Figure 2: Left: Graphical model for the discrete fragmentation coagulation process. [sent-100, score-0.479]

39 For each sequence i, let ait ∈ πt be the cluster in πt containing i. [sent-104, score-0.217]

40 Let θta be the emission of cluster a at location t. [sent-105, score-0.177]

41 Let the mean of θta be βt (this is the latent allele frequency at location t). [sent-107, score-0.194]

42 We assume that conditioned on the partitions and the parameters, the observations xit are independent, and determined by the cluster parameter θta . [sent-108, score-0.347]

43 Since this distribution is heavy tailed, the βt variables will have more mass near 0 and 1 than they would have if γt were ﬁxed, adding sparsity to the latent allele frequencies. [sent-114, score-0.14]

44 Note that the prior gives more mass to values of Rt close to Rmin which we set close to zero, since we expect the partitions of consecutive locations to be relatively similar so that the mosaic haplotype structure can be formed. [sent-117, score-0.29]

45 4 Relationship with the continuous fragmentation-coagulation process The continuous version of the fragmentation-coagulation process [9], which we refer to as the CFCP, is a partition valued Markov jump process (MJP). [sent-121, score-0.209]

46 (The ‘time’ variable for this MJP is the chromosome location, viewed as a continuous variable. [sent-122, score-0.16]

47 There are two types of events in the CFCP: binary fragmentation events, in which a single cluster a is split into two clusters b and c at a rate of RΓ(#b)Γ(#c)/Γ(#a), and binary coagulation events in which two clusters b and c merge to form one cluster a at a rate of R/µ. [sent-124, score-1.082]

48 Then as ε → 0 the probability that the coagulation and fragmentation operations at a speciﬁc time step t induce no change in the partition structure πt approaches 1. [sent-127, score-0.499]

49 If we rescale the time steps by t → εt, then the expected number of binary events over a ﬁnite interval approaches ε times the rates given above and the expected number of all other events goes to zero, yielding the CFCP. [sent-129, score-0.155]

50 In the CFCP fragmentation and coagulation events are binary: they involve either one cluster fragmenting into two new clusters, or two clusters coagulating into one new cluster. [sent-130, score-0.794]

51 However, for the DFCP the fragmentation and coagulation operators can describe more complicated haplotype structures without introducing more latent events. [sent-131, score-0.585]

52 For example one cluster splitting into three clusters (as happens to the second haplotype from the top of Figure 1 after the 18th SNP) can be described 4 by the DFCP using just one fragmentation operator. [sent-132, score-0.549]

53 3 Inference with the discrete fragmentation coagulation process We derive a Gibbs sampler for posterior simulation in the DFCP by making use of the exchangeability of the process. [sent-134, score-0.561]

54 Each iteration of the sampler updates the trajectory of cluster assignments of one sequence i through the partition structure. [sent-135, score-0.273]

55 In this section, we derive the conditional distribution of trajectory i using the deﬁnition of fragmentation and coagulation and also the posterior distributions of the parameters Rt , µ which we will update using slice sampling [15]. [sent-138, score-0.545]

56 1 Conditional probabilities for the trajectory of sequence i −i We will refer to the projection of the partitions πt and ρt onto S − {i} by πt and ρ−i respectively. [sent-140, score-0.23]

57 t Let at (respectively bt ) be the cluster assignment of sequence i at location t in πt (respectively ρt ). [sent-141, score-0.371]

58 If the sequence i is placed in a new cluster by itself in πt (i. [sent-142, score-0.209]

59 , it forms a singleton cluster) we will denote this by at = ∅ and for ρ−i we will denote the respective event by bt = ∅. [sent-144, score-0.187]

60 Otherwise, if t −i the the sequence i is placed in an existing cluster in πt (respectively ρ−i ) we will denote this by t −i −i −i at ∈ πt (respectively bt ∈ ρt ). [sent-145, score-0.344]

61 Starting at t = 1, since the initial distribution is π1 ∈ CRP(S, µ, 0), the conditional cluster assignment of the sequence i in π1 is given by the CRP probabilities from (1): −i Pr(at = a|π1 ) = −i #a/(n − 1 + µ) if a ∈ πt , µ/(n − 1 + µ) if a = ∅. [sent-147, score-0.216]

62 (4) To ﬁnd the conditional distribution of bt given at , we use the deﬁnition of the fragmentation operation as independent CRP partitions of each cluster in πt . [sent-148, score-0.568]

63 If at = ∅, then the sequence i is in a cluster by itself in πt and so it will remain in a cluster by itself after fragmenting. [sent-149, score-0.305]

64 If at = a ∈ πt then bt must be one of the clusters in ρt into which a fragments. [sent-151, score-0.257]

65 This can be a singleton cluster, in which case bt = ∅, or it can be one of the clusters in ρ−i . [sent-152, score-0.292]

66 Since a is fragmented according to CRP(a, 0, R), when t the i-th sequence is added to this CRP it is placed in a cluster b ∈ Ft (a) with probability proportional to (#b − R) and it is placed in a singleton cluster with probability proportional to R#Ft (a). [sent-154, score-0.438]

67 Similarly, to ﬁnd the conditional distribution of at+1 given bt = b we use the deﬁnition of the coagulation operation. [sent-156, score-0.37]

68 If b = ∅, then the sequence i was not in a singleton cluster in ρ−i and so it t −i must follow the rest of the sequences in b to the unique a ∈ πt+1 such that b ⊆ a (i. [sent-157, score-0.277]

69 We will refer to the set of clusters in ρ−i that coagulate to form a by t Ct (a). [sent-160, score-0.148]

70 If b = ∅ then the sequence i is in a singleton cluster in ρ−i and so we can imagine it being the t last customer added to the coagulating CRP(ρt , µ/Rt , 0) of the clusters of ρt . [sent-161, score-0.388]

71 Hence the probability −i that sequence i is placed in a cluster a ∈ πt+1 is proportional to #Ct (a) while the probability that it −i forms a cluster by itself in πt+1 is proportional to µ/Rt . [sent-162, score-0.393]

72 2 Message passing and sampling for the sequences of the DFCP Once the conditional probabilities are deﬁned, it is straightforward to derive messages that allow us to conduct backwards-ﬁltering/forwards-sampling to resample the trajectory of sequence i in the DFCP. [sent-165, score-0.247]

73 This provides an exact Gibbs update for the trajectory of that sequence conditioned on the trajectories of all the other sequences and the data. [sent-166, score-0.188]

74 The messages we will deﬁne are the conditional distribution of all the data seen after a given location in the sequence conditioned on the cluster assignment of sequence i at that location. [sent-167, score-0.366]

75 As the fragmentation and coagulation conditional probabilities are only supported for clusters a, b such that b ⊆ a, these sums can be expanded so that only non-zero terms are summed over. [sent-176, score-0.61]

76 To sample from the posterior distribution of the trajectory for sequence i conditioned on the other trajectories and the data, we use the Markov property for the chain a1 , b1 , . [sent-180, score-0.179]

77 For subsequent bt and at+1 for locations t = 1, . [sent-187, score-0.152]

78 9 500 1000 1500 2000 runtime (seconds) 2500 3000 3500 Figure 3: Allele imputation for X chromosomes from the Thousand Genomes project. [sent-216, score-0.191]

79 Left: Accuracy for prediction of held out alleles for continuous (CFCP) and discrete (DFCP) versions of fragmentation-coagulation process and for popular methods BEAGLE and fastPHASE. [sent-217, score-0.173]

80 the posterior probabilities of µ and Rt given the partitions π1:T and ρ1:(T −1) are as follows: Pr(µ|π, ρ) ∝ Pr(µ) Pr(π1 |µ, R1 ) Pr(ρ1 |π1 , µ, R1 ) · · · Pr(πT |ρT −1 , µ, RT −1 ), ∝ Pr(µ) Γ(µ) µ−T + Γ(µ + n) T −1 T t=1 #πt t=1 Γ(µ/Rt ) . [sent-221, score-0.151]

81 (15) b∈ρt Experiments To examine the accuracy and scalability of the DFCP we conducted an allele imputation experiment on SNP data from the Thousand Genomes project1 . [sent-223, score-0.273]

82 We also compared the runtime of the samplers for the DFCP and CFCP on data simulated from the coalescent with recombination model [14]. [sent-224, score-0.168]

83 For the allele imputation experiment, we considered SNPs from 524 male X chromosomes. [sent-226, score-0.256]

84 The accuracies for the DFCP and CFCP were computed by thresholding the empirical marginal probabilities of the held out alleles at 0. [sent-232, score-0.157]

85 In order to match the expected number of clusters in the posterior, we also conducted allele imputation in the 50% missing data condition with µ ﬁxed at 10. [sent-236, score-0.412]

86 We then computed the accuracy of the samples by predicting held out alleles based on the cluster assignments of the sample. [sent-239, score-0.241]

87 7 In a second experiment we simulated datasets from the coalescent with recombination model consisting of between 10,000 and 50,000 sequences using the software ms [14]. [sent-241, score-0.176]

88 5 Results The accuracy of the DFCP in the allele imputation experiment was comparable to that of the CFCP and fastPHASE in all missing data conditions (Figure 3, left). [sent-243, score-0.307]

89 The CFCP has an arbitrary number of latent events between consecutive observations and it is likely that the runtime improvement shown by the DFCP is due to its reduced number of required message calculations. [sent-258, score-0.176]

90 The DFCP is a partition-valued Markov chain, where partitions change along the chromosome by a fragmentation operation followed by a coagulation operation. [sent-260, score-0.684]

91 The DFCP is designed to model the mosaic haplotype structure observed in genetic sequences. [sent-261, score-0.298]

92 We applied the DFCP to an allele prediction task on data from the Thousand Genomes Project yielding accuracies comparable to state-of-the-art methods and runtimes that were lower than the runtimes of the continuous fragmentation-coagulation process [9]. [sent-262, score-0.257]

93 0 ×104 Figure 4: Runtimes per iteration per sequence of The DFCP and CFCP induce different joint dis- DFCP and CFCP on simulated datasets consisttributions on the partitions at adjacent locations. [sent-272, score-0.17]

94 bitrary number of latent binary events wherein a single cluster is split into two clusters, or two clusters are merged into one. [sent-276, score-0.337]

95 The DFCP however can model any partition structure with one pair of fragmentation and coagulation operations. [sent-277, score-0.499]

96 Another avenue of future research is to understand how other genetic processes can be incorporated into the fragmentation-coagulation framework, including population admixture and gene conversion. [sent-280, score-0.178]

97 Although haplotype structure is a local property, the Markov assumption does not hold in real genetic data. [sent-281, score-0.233]

98 Properties of a neutral allele model with intragenic recombination. [sent-291, score-0.144]

99 Generating samples under a Wright-Fisher neutral model of genetic variation. [sent-375, score-0.175]

100 A uniﬁed approach to genotype imputation and haplotype-phase inference for large data sets of trios and unrelated individuals. [sent-393, score-0.205]

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