This section presents the proposed method for haplotype reconstruction. We discuss the statistical model employed and present an incremental algorithm for efficiently learning the model structure from genotype data. Finally, datasets and systems used in the experimental evaluation are described.
(Hidden) Markov models for haplotyping
We model the probability distribution on haplotypes by a left-right Markov model λ with 2·m states, with a state space as shown in Figure 1. A haplotype (of length 4 in the example) is sampled by traversing a path through the model from left to right. The Markov assumption P(h)=∏t=1mPt(h[t]|h[t−1],λ)
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<p>Figure 1</p>
A Markov model over haplotypes
A Markov model over haplotypes. The highlighted path encodes the haplotype "0110".
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Parameters are of the form Pt(h[t] | h[t - 1], λ), the probability of sampling the new allele h[t] at position t after observing the allele h[t - 1] at position t - 1. Note that separate (conditional) allele distributions Pt are defined for every sequence position t ∈ {1,...,m}, as linkage disequilibrium patterns will vary for different markers. This also means that the allele encoding at a given marker position, i.e., which allele is represented as '0' and which as '1', does not affect the distributions that can be represented.
This model is not directly applicable in haplotype reconstruction, because in reality only genotypes are observed whereas the phase information is hidden. The hidden phase information can be modeled by a hidden Markov model λ' as shown in Figure 2. A path through this model corresponds to sampling a pair of haplotypes (ordered allele pairs, in angle brackets), while the corresponding genotype (unordered pairs, in curly brackets) is emitted.
<p>Figure 2</p>
A hidden Markov model over genotypes
A hidden Markov model over genotypes. Possible paths for genotype observation '{0, 1}', '{1, 1}', '{0, 1}', '{0, 0}' are highlighted. The corresponding haplotype pairs are {(0100, 1110), (0110, 1100), (1100, 0110), (1110, 0100)}.
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To reflect the Hardy-Weinberg equilibrium assumption, constraints have to be placed on transition probabilities. A transition in this model corresponds to independently sampling two new alleles h1[t] and h2[t] at marker t based on their respective histories h1[t - 1] and h2[t - 1]. Therefore, the corresponding probability is actually the product of probabilities for sampling hi[t] after hi[t - 1]:
Pt(h1[t], h2[t] | h1[t - 1], h2[t - 1], λ') = Pt(h1[t] | h1[t - 1], λ)Pt(h2[t] | h2[t - 1], λ).
In this way, all parameters of λ' can be re-expressed as products of parameters of the model λ on haplotypes outlined above. Furthermore, λ' can be transformed into an equivalent HMM in which these constraints involving products of parameters are replaced with standard parameter tying constraints, which tie parameters in λ' to those in λ.
An advantage of this approach is that the model λ' can be trained directly from genotype data using Baum-Welsh algorithm 4, while implicitly estimating the distribution over haplotypes encoded in λ. Furthermore, the most likely reconstruction of a genotype can be directly obtained by the Viterbi algorithm 4. The presented idea of embedding a model on haplotypes into a model on genotypes in which the genotype phase is the hidden state information, and learning this model using EM, is related to the approaches used in the HIT 5 and fastPHASE 6 systems. In HIT, haplotypes are modeled as recombinations of a set of founder haplotypes, and an instance of the EM algorithm is derived to directly estimate the founders from genotype observations. In fastPHASE, haplotypes are modeled using local clusters, and cluster membership of a haplotype is determined by a hidden Markov model. Again, an instance of the EM algorithm for estimating the clusters directly from genotype data can be derived.
Higher-order models and sparse distributions
The main limitation of the model presented so far is that it only takes into account dependencies between adjacent markers. Expressivity can be increased by using a Markov model of order k > 1 for the underlying haplotype distribution 7:
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where h[j, i] is a shorthand for h[max{1, j}]...h[i]. Unfortunately, the number of parameters in such a model increases exponentially with the history length k. Fortunately, observations on real-world data (e.g., 8) show that only few conserved haplotype fragments from the set of 2k possible binary strings of length k actually occur. This can be exploited by modeling sparse distributions, where fragment probabilities which are estimated to be very low are set to zero. More precisely, let p = Pt(h[t] | h[t - k, t - 1]) and define for some small ε > 0 a regularized distribution
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If the underlying distribution is sufficiently sparse, P^
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SpaMM: a level-wise learning algorithm
Algorithm 1 The level-wise SpaMM learning algorithm.
Initialize k := 1
λ1 := INITIAL-MODEL()
λ1 := EM-TRAINING(λ1)
repeat
k := k + 1
λk := EXTEND-AND-REGULARIZE(λk-1)
λk := EM-TRAINING(λk)
until k = kmax
To construct the sparse order-k hidden Markov model, we propose a learning algorithm – called SpaMM for Sparse Markov Modeling – that iteratively refines hidden Markov models of increasing order (Algorithm 1). More specifically, the idea of SpaMM is to identify conserved fragments using a level-wise search, i.e., by extending short fragments (in low-order models) to longer ones (in high-order models), and is inspired by the well-known Apriori data mining algorithm 9. The algorithm starts with a first-order Markov model λ1 on haplotypes where initial transition probabilities are set to P˙
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The function EXTEND-AND-REGULARIZE(λk-1) takes as input a model of order k - 1 and returns a model λk of order k. In λk, initial transition probabilities are set to
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i.e., transitions are removed if the probability of the transition conditioned on a shorter history is smaller than ε. This procedure of iteratively training, extending and regularizing Markov models of increasing order is repeated up to a maximum order kmax.
Figure 3 shows the models learned in the first 4 iterations of the SpaMM algorithm on a real-world dataset. Note how some of the possible transitions are pruned, conserved fragments are isolated and the number of states in the final model is significantly smaller than for a full model of that order. Furthermore, the set of paths through the structure is a concise representation of all haplotypes that have non-zero probability according to the model.
<p>Figure 3</p>
Visualization of the SpaMM structure learning algorithm
Visualization of the SpaMM structure learning algorithm. Sparse models λ1,...,λ4 of increasing order learned on the Daly dataset are shown. Black/white nodes encode more frequent/less frequent allele in population. Conserved fragments identified in λ4 are highlighted.
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For a given genotype g, a reconstructed haplotype pair ⟨hg1
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The idea of using frequent fragments to build Markov models for haplotypes has also been used in the HaploRec method 7. In HaploRec, a set of fragments (of any length) that are frequent according to the current model is kept, and updated after each iteration of the EM algorithm.
Experimental methodology and evaluation
The proposed method was implemented in the SpaMM haplotyping system 10. We compared its accuracy and computational performance to several other state-of-the art haplotype reconstruction systems: PHASE version 2.1.1 11, fastPHASE version 1.1 6, GERBIL as included in GEVALT version 1.0 12, HIT 5 and HaploRec (variable order Markov model) version 2.0 13. All methods were run using their default parameters. The fastPHASE system, which also employs EM for learning a probabilistic model, uses a strategy of averaging results over several random restarts of EM from different initial parameter values. This reduces the variance component of the reconstruction error and alleviates the problem of local minima in EM search. As this is a general technique applicable also to our method, we list results for fastPHASE with averaging (fastPHASE) and without averaging (fastPHASE-NA).
The methods were compared using publicly available real-world datasets, and larger datasets simulated with the Hudson coalescence simulator 14. As real-world data, we used a collection of datasets from the Yoruba population in Ibadan, Nigeria 1, and the well-known dataset of Daly et al 8, which contains data from a European-derived population. For these datasets, family trios are available, and thus true haplotypes can be inferred analytically. Non-transmitted parental chromosomes of each trio were combined to form additional artificial haplotype pairs. Markers with minor allele frequency of less than 5% and genotypes with more than 15% missing values were removed. Note that if all trio members are heterozygous, the haplotype of the child can not be inferred. In this case, the genotype at this marker position is observed but the marker is ignored when computing the accuracy of the method.
For the Yoruba population, information on 3.8 million SNPs spread over the whole genome is available. We sampled 100 sets of 500 markers each from distinct regions on chromosome 1 (Yoruba-500), and from these smaller datasets by taking only the first 20 (Yoruba-20) or 100 (Yoruba-100) markers for every individual. There are 60 individuals in the dataset after preprocessing, with an average fraction of missing values of 3.6%. For the Daly dataset, there is information on 103 markers and 174 individuals available after data preprocessing, and the average fraction of missing values is 8%. Although results on a single dataset are not very meaningful, the Daly dataset was included because it has been used frequently in the literature.
The number of genotyped individuals in these real-world datasets is rather small. For most disease association studies, sample sizes of at least several hundred individuals are needed 15, and we are ultimately interested in haplotyping such larger datasets. Unfortunately, we are not aware of any publicly available real-world datasets of this size, so we have to resort to simulated data. We used the well-known Hudson coalescence simulator 14 to generate 50 artificial datasets, each containing 800 individuals (Hudson datasets). The simulator uses the standard Wright-Fisher neutral model of genetic variation with recombination. A chromosomal region of 150 kb was simulated. The probability of mutation in each base pair was set to 10-8 per generation, and the probability of cross-over between adjacent base pairs was set to 10-8. These values result in a mutation probability for the entire chromosomal region of μ = 0.0015 and cross-over probability of ρ = 0.0015. The diploid population size, N0, was set to the standard 10000, yielding mutation parameter θ = 4N0μ = 60, and the recombination parameter r = 60. For each data set, a sample of 1600 chromosomes was generated, and these were paired to form 800 genotypes. On average, one simulation produced approximately 493 segregating sites. For each data set, 50 markers were chosen from the segregating sites with minor allele frequency of at least 5%, such that marker spacing was as uniform as possible. The resulting average marker spacing was 3.0 kb. To come as close to the characteristics of real-world data as possible, some alleles were masked (marked as missing) after simulation. More specifically, the missing allele pattern found in the Yoruba datasets was superimposed onto the simulated data, shortening patterns to the size of the target marker map and repeating them as needed for additional individuals.
The accuracy of the reconstructed haplotypes produced by the different methods was measured by normalized switch error. The switch error of a reconstruction is the minimum number of recombinations needed to transform the reconstructed haplotype pair into the true haplotype pair. To normalize, switch errors are summed over all individuals in the dataset and divided by the total number of switch errors that could have been made.