HaploRec: efficient and accurate largescale reconstruction of haplotypes
 Lauri Eronen^{1}Email author,
 Floris Geerts^{2} and
 Hannu Toivonen^{1, 3}
DOI: 10.1186/147121057542
© Eronen et al; licensee BioMed Central Ltd. 2006
Received: 29 June 2006
Accepted: 22 December 2006
Published: 22 December 2006
Abstract
Background
Haplotypes extracted from human DNA can be used for gene mapping and other analysis of genetic patterns within and across populations. A fundamental problem is, however, that current practical laboratory methods do not give haplotype information. Estimation of phased haplotypes of unrelated individuals given their unphased genotypes is known as the haplotype reconstruction or phasing problem.
Results
We define three novel statistical models and give an efficient algorithm for haplotype reconstruction, jointly called HaploRec. HaploRec is based on exploiting local regularities conserved in haplotypes: it reconstructs haplotypes so that they have maximal local coherence. This approach – not assuming statistical dependence for remotely located markers – has two useful properties: it is wellsuited for sparse marker maps, such as those used in gene mapping, and it can actually take advantage of long maps.
Conclusion
Our experimental results with simulated and real data show that HaploRec is a powerful method for the large scale haplotyping needed in association studies. With sample sizes large enough for gene mapping it appeared to be the best compared to all other tested methods (Phase, fastPhase, PLEM, Snphap, Gerbil; simulated data), with small samples it was competitive with the best available methods (real data). HaploRec is several orders of magnitude faster than Phase and comparable to the other methods; the running times are roughly linear in the number of subjects and the number of markers. HaploRec is publicly available at http://www.cs.helsinki.fi/group/genetics/haplotyping.html.
Background
The problem we consider is haplotype reconstruction: given the genotypes of a sample of individuals, the task is to predict the most likely haplotype pair for each individual. Computational haplotype reconstruction methods are based on statistical dependency between closely located markers, known as linkage disequilibrium. Many computational methods have been developed for the reconstruction of haplotypes. Some of these methods do not rely on the statistical modeling of the haplotypes [1–3], but most of them, like our proposed algorithm HaploRec, do [4–10]. For a review of these and other haplotyping methods we refer to [11–13]. Laboratory techniques are being developed for direct molecular haplotyping (see, e.g., [14, 15]), but these techniques are not mature yet, and are currently time consuming and expensive.
The need for a new haplotyping method is motivated by high throughput association analysis, where the goal is to locate a disease susceptibility gene by finding a haplotype fragment that is associated with the disease being studied. More and more often gene mapping studies use a large marker map spread over a long genomic region. A typical strategy for computationally haplotyping a long map is to first divide the map to small, overlapping windows, to reconstruct the haplotypes in each window separately, and then to combine haplotypes from the windows [16]. HaploRec is aimed to have the following important properties. First, increasing the window size should give relatively more accurate results since large windows contain more information, i.e., adding markers should improve accuracy (in the phases between the markers that already were there), whether the new markers are added between the old ones or not [17]. Second, the time complexity of the algorithm should be close to linear in the number of markers, in order to avoid unnecessary compromises when choosing the window size, and also close to linear in the number of genotypes to allow sample sizes of hundreds to thousands of individuals, as required by association analysis [18]. HaploRec produces accurate haplotype reconstructions, and scales to long marker maps (or windows) that span long genetic regions.
While the statistical models of HaploRec are novel, some of the algorithmic principles are similar to earlier work. HaploRec follows a likelihoodbased expectationmaximization (EM) haplotype inference strategy which was introduced in [4]. PLEM [5, 6] overcomes the computational complexity of the basic EM approach by using a a pruning strategy on the possible haplotype resolutions, called PartitionLigation (PL). The Snphap algorithm of Clayton [19] is also based on the EM algorithm, but uses a sequential pruning strategy; HaploRec also uses a similar pruning approach. The PL strategy is also used in the current version of Phase [7, 8]. PLEM and Snphap are based on a multinomial haplotype probability model with a uniform Dirichlet prior. Phase, however, uses a prior distribution based on coalescent theory (see [20] for a review) and uses Bayesian inference implemented with Gibbs sampling instead of EM. The underlying idea for our statistical models for haplotypes is that we derive an overall probability for a haplotype from the probabilities of its local fragments. We propose three probability models based on this idea: highorder Markov chains, variableorder Markov chains, and a segmentationbased model. Moreover, a novelty is that the lengths of the haplotype fragments utilized in the models vary, governed by a frequency threshold. In contrast to most other approaches, longrange dependencies between markers are not required for the methods to work, but they can be utilized where they do exist. Thus all these models scale naturally to the long and sparse marker maps often used in gene mapping.
Our segmentationbased model bears some resemblance to methods which combine haplotype block finding and haplotyping [9, 10]. However, whereas these models place universal block boundaries across the whole population, our model averages over all possible segmentations for each haplotype, without any fixed block boundaries.
Quite recently, Scheet and Stephens [21] introduced fastPhase, which models the population with a set of founder haplotypes (or clusters); the cluster memberships are allowed to change continuously along the chromosome, according to a hidden Markov model. Similar to our models, fastPhase allows for both "blocklike" patterns and gradual decline of linkage disequilibrium with distance.
In our extensive experimental evaluation in the Results section we will compare HaploRec with Phase, fastPhase, PLEM, Snphap, and Gerbil [10]. In the simulated settings, where sample sizes (number of subjects) are large enough for gene mapping, we observe the concrete benefits of the ability of HaploRec to improve its performance by haplotyping more markers at a time. The models we describe are relatively simple and are slightly outperformed by Phase when the number of markers is small, but when our method is given a longer map to be haplotyped, it can actually utilize the information contained in the additional markers to outperform Phase. HaploRec is in practice several orders of magnitude faster than Phase and has running times comparable to the other methods.
We next define the necessary notation.
Notation and problem statement
We assume a set (map, or window) M of ℓ markers 1,...,ℓ and denote the set of all observed alleles of marker i by A_{ i }. A haplotype H over M is then a vector of alleles: H ∈ Π_{i = 1,...,ℓ}A_{ i }. A genotype G over M is a vector of (unordered) allele pairs: G ∈ Π_{i = 1,...,ℓ}{{a_{1}, a_{2}}  a_{1}, a_{2} ∈ A_{ i }}. For SNP markers, A_{ i } = 2. Assuming that alleles are labeled "1" and "2", SNP haplotypes are vectors in {1, 2}^{ℓ} and SNP genotypes are vectors in {{1, 1}, {1, 2}, {2, 2}}^{ℓ}. We thus use terms haplotype and genotype to refer to data over the whole marker map, and not e.g. to just one marker.
The allele of haplotype H at marker i is denoted by H(i). Similarly, the unordered allele pair of a genotype G at marker i is denoted by G(i). Given a pair of haplotypes {H_{1}, H_{2}} and a genotype G such that G(i) = {H_{1}(i), H_{2}(i)} for all i, we say that {H_{1}, H_{2}} is compatible with G, or that {H_{1}, H_{2}} is a (possible) haplotype configuration for genotype G. Two haplotypes determine a unique compatible genotype in the obvious way. A genotype, on the other hand, can have several compatible haplotype configurations. For a genotype G with k heterozygous markers (i.e., k = {G(i) = {a_{1}, a_{2}}  a_{1} ≠ a_{2}}; a marker is homozygous if it is not heterozygous), there are 2^{k  1}different haplotype configurations. The set of all haplotype configurations for a genotype G is denoted by C_{ G }, with C_{ G } = 2^{k  1}. A single haplotype H is said to be compatible with a genotype G, if there exists a haplotype H' such that {H, H'} is compatible with G. The set of input genotypes is denoted by $\mathcal{G}$. The haplotype reconstruction problem is now defined as finding for each genotype G ∈ $\mathcal{G}$ the most plausible haplotype configuration {H_{1}, H_{2}} ∈ C_{ G }. Here, the interpretation of "most plausible" depends on each different haplotyping method; in most methods the most plausible haplotype configuration is the one with the highest estimated probability under the prior assumptions made by the method. In the case that multiple haplotype configurations are equally probable, we simply select one randomly.
Results
Haplotype probability models
Our focus is on data sets with a large number of relatively sparsely spaced markers. Under these conditions, recombinations between markers are common, and linkage disequilibrium between distant markers is weak. In this case it cannot be expected that complete haplotypes are shared between subjects; instead we aim to discover and utilize local regularities, or patterns in the haplotypes. We restrict our attention to a simple class of patterns: a haplotype fragment is a haplotype restricted to a (continuous) subrange of the original marker map. The idea is to model local linkage disequilibrium by estimating the frequencies of haplotype fragments, and to combine those frequencies into a probability model for complete haplotypes. We here very briefly describe three different haplotype probability models based on this idea; full specifications are given in the Methods section. We introduced the ideas for two of them (the Markovian models) in a preliminary conference paper [22], one (the segmentation model) is completely novel. The actual haplotyping algorithm has also been greatly improved, resulting in significantly more accurate results and reduced running times, while scaling to much larger data sets. An overview of the algorithm is given in the Methods section; a detailed description with pseudocode and complexity analysis is given in Additional file 1.
Let H(i, j) denote the sequence, or haplotype fragment, from the i th to the j th marker in a given haplotype H. In the variableorder Markov model the conditional probabilities at each marker i are estimated from fragments H(s_{ i }, i  1) of varying length:
The length of the context H(s_{ i }, i  1), and thus the order of the Markov chain, is individually adjusted for each position and each haplotype by choosing the longest matching context that has a predetermined minimum frequency.
The segmentation model considers each haplotype as a sequence of independent, nonoverlapping fragments, and defines the probability of a haplotype to be the product of fragment probabilities. A robust estimate is obtained by averaging over the set $\mathcal{S}$ of all possible segmentations of H into frequent fragments:
where S is a segmentation of H into (nonoverlapping) segments (s_{ i }, e_{ i }), q is a parameter for penalizing large numbers of segments, S is the number of segments in segmentation S, and C is a normalization factor,
As a simpler alternative, we also consider dorder Markov models with
More details on each of these models are given in the Methods section.
Experimental setting
Data simulation
Our main target application is data sets involving a large number of subjects (hundreds or thousands) and markers (hundreds or thousands per chromosome), because these are needed for associationbased gene mapping [18]. Such data is not yet publicly available for benchmarking, and thus our experiments are based mainly on simulated data. Real data from the HapMap project [23] is used to validate the general observations from simulations, and to test the method in slightly different settings, especially with small sample sizes.
Correspondence between marker spacing and linkage disequilibrium.
Marker spacing (kb)  D'  number of markers 

6.6 kb  0.88  500 k 
20 kb  0.73  166 k 
33 kb  0.64  100 k 
100 kb  0.45  33 k 
166 kb  0.36  20 k 
All experiments are run separately for 10 independently simulated data sets, and we report average results over them. Results from the different replicates are quite similar: while there is some variance in the accuracy between the different replicas, the relative performances of different methods are extremely similar. (See Additional file 2 for example results from all 10 replicas.) Unless otherwise stated, 100 markers were used in the experiments with HaploRec alone, and 30 markers in comparisons to other methods, in order to keep running times of some of the other methods reasonable.
Performance measure
As an accuracy measure, we primarily use relative switch accuracy, which is defined as the fraction of neighboring phases (between each pair of consecutive heterozygous markers) reconstructed correctly. An alternative measure would be absolute accuracy, which is the fraction of haplotype pairs reconstructed completely correctly (ignoring missing alleles). Absolute accuracy is problematic with long or sparse haplotypes, where some switch errors can be inevitable. It gives little information about the quality of a haplotype, except whether it is exactly correct or not; the switch measure is much more informative in this respect. Chromosomewide studies search for relatively short diseaseassociated haplotype fragments, and switch accuracy is almost directly related to the number of fragments correctly reconstructed. However, we also give some examples of absolute errors since they are widely used in the field (for short haplotypes).
Experimental results
Performance across different amounts of linkage disequilibrium
The three proposed haplotype probability models, Markov chain of variable order (VMM), the segmentation model (S), and as simpler baseline the Markov chain of fixed order (FMM), lead to different variants of HaploRec, subsequently abbreviated as HaploRecVMM, HaploRecS, and HaploRecFMM. The most important parameter of HaploRec is the one that indirectly specifies the size of the data structure used by the model (subsequently called model complexity parameter). For the variableorder Markov model and the segmentation model this parameter is the fragment frequency threshold, with the fixedorder Markov model, it is the order of the Markov model (for details, see the Methods section).
The segmentation model performs here best, variableorder Markov model is slightly less accurate, and the fixedorder model is clearly inferior. This and other experiments (not shown) show that the more complex models are good alternatives over the simple Markov model. For clarity of exposition, the fixedorder Markov model is excluded from the rest of the results as markedly inferior. We also considered a variant of the segmentation model where the maximum over different segmentations was used, instead of averaging over different segmentations. This alternative was slightly but consistently outperformed by the averaging model (results not shown).
Default values for model complexity parameters
Comparison of HaploRec with fastPhase, Gerbil, Phase, PLEM, and Snphap
Comparison of methods on simulated data sets with 33 kb marker spacing.
Method  Switch accuracy  No. of markers  Time per marker (sec.) 

HaploRecS  0.990  100  19.2 
HaploRecVMM  0.980  100  5.2 
(HaploRecS)  (0.969)  (30)  (6.6) 
(HaploRecVMM)  (0.970)  (30)  (2.5) 
Phase  0.981  30  1585 
fastPhase  0.858  20  61.3 
PLEM  0.938  30  0.9 
Snphap  0.921  30  0.4 
Gerbil  0.745  5  2.6 
An observation from this experiment is that the number of markers can have a significant effect on haplotyping accuracy. With the smallest tested numbers of markers, 5 and 10, all the methods achieve only a mediocre accuracy. When the number of markers is increased to 20 or 30, the accuracies of all methods except Gerbil improve clearly. The accuracy of both HaploRec variants increases monotonically with the number of markers. None of the other methods show this property. Phase's accuracy decreases slightly after 30 markers (and it was not able to handle more than 40 markers). Snphap and PLEM gain less from the increase in the number of markers, and also start to actually lose accuracy when the number of markers increases over 30. FastPhase only gains little and levels after 20 markers.
Snphap, the HaploRec variants, and Gerbil are relatively close to having linear running times in the number of markers (i.e., close to a horizontal line on the scale where yaxis gives the time per marker; Figure 3, right panel). Snphap is 1–2 orders of magnitude faster than PLEM, HaploRec and Gerbil; they are an order of magnitude faster than fastPhase, which in turn is 0–2 orders of magnitude faster than Phase. For clarity of exposition and fairness of comparison, we use 30 marker windows in the rest of the experiments for all methods. Based on the results above, it is about an optimal choice for Phase, fastPhase, PLEM, and Snphap, slightly suboptimal for HaploRec, and unfortunately bad for Gerbil. In particular, note that if a larger window size was chosen for HaploRec, HaploRecS could in many cases have given more accurate results than Phase (cf. Table 2).
Effect of marker density
Effect of sample size
For most methods, the running times are relatively close to linear in the number of genotypes, but Phase's running times increase somewhat more rapidly. We performed some additional experiments with fastPhase, trying out different parameter values (not shown). In particular, increasing the maximum number of clusters from the default value of 15 to 20 or 30 improved the accuracy at the cost of longer running times, but the accuracies did not reach those of HaploRec.
Effect of missing alleles and genotyping errors
Tests with larger data sets
HaploRecS without the maximum pattern length constraint has the best accuracy, but its high memory usage makes it unusable for the larger window sizes. On the other hand, the segmentation model does not work quite as well with the pattern length constraint. HaploRecVMM has a fairly good accuracy, and is almost unaffected by the pattern length constraint (the lines for the two variants are indistinguishable in the figure), making it practical for large data sets. Its accuracy stays approximately constant with 100 markers or more. FastPhase and PLEM maintain a lower but constant accuracy. Gerbil works up to 300 markers and maintains its (low) level of accuracy. While Snphap quickly loses accuracy with an increasing number of markers, it is very fast (its running times per marker are below a second). The running times of the HaploRec variants and PLEM are slightly superlinear (i.e., approaching a horizontal line; note that the yaxis shows the time per marker) and Gerbil clearly superlinear.
Experimental results with real data
To complement the systematic experimental analysis with several replicates of simulated data and reasonably sized samples, the methods were also tested on publicly available real data from the HapMap project [23].
The HapMap data we used consists of two separate populations: 30 trios from the Yoruba population in Ibadan, Nigeria, and another 30 trios from the CEPH population (Utah residents with ancestry from northern and western Europe). Both data sets (downloaded from the HapMap web site [25]) have the same set of 3.8 million SNPs spread over the whole genome. (For information on how HapMap data was processed for the experiments, see the Methods section.)
In the following experiments with this data, HaploRec was run with the same parameter values as before, chosen based on the first experiments with large samples of simulated data (Figures 1 and 2).
The younger CEPH population is clearly easier than the Yoruba population for all haplotyping methods. However, for both data sets, even the best methods only achieve a relatively low accuracy, which can probably be attributed to the small sample size. For the Yoruba data, Phase performs best across the range of tests, followed by the HaploRec variants and fastPhase, which are followed by Gerbil, PLEM and Snphap. For the CEPH data, the differences between the best methods (Phase, HaploRecS and fastPhase) are very small.
The above results show that HaploRec is competitive also on real data sets having a small sample size. On the other hand, the number of marker has a smaller effect on accuracy than in the simulated data sets. Also, increasing the number of markers to more than 40 does not improve accuracy here, unlike in the experiments with simulated data sets, where accuracy always improves with increasing number of markers. We believe the reason for this is the small sample size (60 genotypes vs. 1000 genotypes in the simulated data). In a larger sample, longer shared ancestral haplotype fragments (which are rarer than shorter ones) can be detected and utilized, leading to increased accuracy. To test this hypothesis, we ran HaploRec on (sparse) simulated data sets consisting of various numbers of genotypes and evaluated the effect of the number of markers (data not shown). The same effect was visible there as well: the improvement in accuracy gained by using more markers decreases with decreasing sample size.
Discussion
The HaploRec models and algorithm introduced in this paper are designed for haplotyping data sets in largescale disease association studies. The statistical models are relatively simple. Their central idea is to represent the probability of a long haplotype as a function of the probabilities of its local fragments. Our experimental results confirm that locality and simplicity were successful design choices: locality allows the models to actually benefit from large numbers of markers even when they are sparsely located, while simplicity allows an efficient implementation. In combination these properties result in a method that accurately and quickly haplotypes large windows of markers at a time.
We presented three different haplotype probability models: Markov chain of a fixed order, Markov chain of a variable order (HaploRecVMM), and a model based on segmenting each haplotype in frequent fragments (HaploRecS). Our experimental results show that the simple fixedorder Markov chain is clearly inferior, even if its order is chosen optimally for the data set. In our results for the more flexible and "selfadjusting" variableorder and segmentation models, the segmentation model gave generally more accurate results than the variableorder Markov model both in the simulated and real data sets, with the downside of being somewhat slower. Our experiments also indicated another interesting difference between the models: while the segmentation model is more sensitive to missing data than the variableorder Markov model, it is more robust against genotyping errors.
In experimental comparisons with existing haplotyping methods, the HaploRec models scale in a unique way to large data sets: Their accuracies improve both in the number of markers (Figure 3) and in the sample size (Figure 5) in an unparalleled way (Phase being an exception in some aspects), while being robust to sparse marker maps (Figure 4). This combination of properties makes HaploRec especially suitable for chromosome or genomewide association analysis, where large numbers of sparsely located markers are analyzed for hundreds or thousands of individuals. In such settings, HaploRec can outperform Phase in accuracy while being 2–3 orders of magnitude faster. Although Phase is very accurate and can also benefit from large samples, it is computationally very intensive, restricting its usability for large data sets. The recently proposed fastPhase method scales well but for large data sets, it seems to be clearly less accurate and also clearly slower than HaploRec.
The performance differences can be largely understood from the properties of the different models. Snphap and PLEM are based on a multinomial model, which is suitable when all markers are in strong linkage disequilibrium, but does not work well when the number of markers is large or the markers are spaced far apart. Gerbil solves this problem by dividing the marker map into blocks and modeling each block by a set of common "founder" haplotypes. This approach assumes strong linkage disequilibrium within each block, and does not account for more flexible patterns of linkage disequilibrium across block boundaries. Gerbil's relatively bad performance on simulated data can probably be attributed to two factors. Firstly, it only considers a small number of common haplotypes for each block, and thus cannot utilize the rarer shared haplotypes found in large samples. Secondly, there are no recombination hotspots in the simulated data. In fastPhase, each haplotype is a mosaic of founder haplotypes. This allows for both blocklike patterns of linkage disequilibrium and gradual decay of linkage disequilibrium with distance. However, it does not account for the fact that shared haplotypes are formed also in the later stages of the population history, forming more recent (and less frequent) patterns of linkage disequilibrium. The segmentation model of HaploRec bears some resemblance to the model used in fastPhase: each haplotype is a mosaic of frequent haplotype fragments, which may now be ancestral haplotype fragments from different stages of the population history. As old haplotypes are split by recombinations and new haplotypes are formed during the population history, longer shared fragments probably are of more recent origin and have a smaller frequency than shorter ones. HaploRec is not constrained to a fixed set of founders, but can more flexibly utilize (local) patterns of linkage disequilibrium present in the data using fragments of varying lengths. Being able to utilize the information contained in rare shared fragments is one possible explanation for the good performance of HaploRec on large samples. Phase works consistently well across most settings. Also its model accounts for recombinations, which explains why it performs well also on sparser marker maps. Its good accuracy especially on small samples is probably due to its more realistic prior compared to other methods, which becomes less important with increasing sample size. The accurate model employed by Phase has the downside of being computationally very intensive. As a result, the window size (number markers) it can handle is very limited, especially when the sample size is large. In the experiments with variable window size, we also observed that Phase's accuracy decreases when increasing the number of markers above 30; this may be because its default number of iterations is not sufficient to obtain mixing for larger window sizes.
Results with data from the HapMap project demonstrate that HaploRec works well also with real data (Figures 8 and 9). Unlike the assumed primary applications of HaploRec, these samples are very small (only 60 genotypes). Still, HaploRec is very competitive, even with its default parameter values chosen based on the simulated, larger data sets with sparser marker spacing. Experiments with the dense Daly data [26] (results not shown) indicate the same: among the tested methods, HaploRec was second only to Phase and fastPhase while being fastest of all methods. Future experiments with large real data sets, as they become publicly available, will be used to test the hypothesis that HaploRec will benefit more than the competing methods from larger sample sizes and from more markers.
For evaluating performance on reasonably large samples (1000 individuals), we used simulated data due to the lack of suitable public real data sets. Another option would have been to use pseudosimulated data, by randomly combining pairs of real haplotypes to form new pseudoindividuals (as done, e.g., in [10]). For the genetically long marker maps of association studies, this method would generate unrealistic data: multiple copies of complete haplotypes will appear in the final sample, when in reality the haplotypes would mostly have been segmented by recombinations. Such data would be unrealistically easy to haplotype, and already a simple multinomial model can give almost completely correct solutions.
Most haplotypebased association analysis methods assume haplotypes as input. There are several major fundamental issues related to this. The first questions the need to haplotype longer windows of markers maps at all, since gene mapping studies search for disease associations of relatively short genetic regions and for this it is sufficient to haplotype such a shorter segment at a time. The computational complexity can on one hand decrease – especially for methods that are not close to linear in the number of markers, such as Phase – but on the other hand there is an additional cost of redundantly haplotyping overlapping segments. Additionally, haplotyping accuracy improves with window size (cf. Figures 3 and 8), indicating that too short windows should be avoided.
The second issue relates to the use of estimated haplotypes in association analysis. It seems obvious that haplotype reconstruction tends to exaggerate linkage disequilibrium since haplotyping methods more or less directly aim to maximize it. It has been shown that estimated haplotypes can, indeed, lead to false positives [27, 28]. On the other hand, this does not always have to be the case. A simulation study shows that in association analysis, the haplotypes produced by HaploRec can be equally powerful to the true haplotypes, despite some inevitable phasing errors [29]. The locality of the statistical models has a subtle role here: in the casecontrol settings normally used in association studies, linkage disequilibrium is increased in the cases in the vicinity of the disease gene, making this most critical part of the marker map easier to haplotype. More work is needed to identify when statistically predicted haplotypes are useful and when not.
Another view to this issue is that some information is lost. As an extreme example, there may be several roughly equally likely haplotype configurations of which just one is chosen to the output. Since association analysis methods are typically based on frequencies of haplotypes (or fragments), the frequencies of different possible haplotypes – rather than the single most likely ones – should be more informative and fairly easily usable for many association methods. Most statistical haplotyping methods internally estimate haplotype frequencies; haplotype resolution can be seen as an extra step based on these frequencies. This also holds for HaploRec which actually already estimates frequencies of all haplotype fragments that have (an estimated) frequency above a small threshold. Ultimately, haplotype frequency estimation and association analysis could be combined into one model and process [27].
There are several possible directions for future work on improving the haplotype probability models. Currently, the variableorder Markov model only uses a simple frequency threshold to determine the context lengths. The set of contexts could be pruned further using the accuracy of predicting the next allele as a selection criterion [30]. Another possibility for refinement is to smooth the probability over several context lengths simultaneously [31]. Ideas from the models of Phase and Gerbil could be used to better account for mutations and genotyping errors. A possible approach would be to allow for a small number of mismatches between the haplotype fragments and complete haplotypes in the parameter estimation step, dividing some of the probability mass to fragments that are similar, but not identical with the observed ones.
Conclusion
Genotyping hundreds or even thousands of subjects for hundreds of thousands of markers is becoming technologically and economically feasible. It is estimated that data sets of this size start to be sufficiently powerful for genomewide disease association studies, depending on the disease and the population [18, 32]. However, many methods for associationbased gene mapping assume haplotype data. It has been shown, too, that haplotypes can be more powerful than single markers [33].
We presented models and methods for statistical haplotype reconstruction from genotypes of unrelated individuals, and specifically targeted large and sparse data sets, such as those needed in chromosome or genomewide disease association studies. We introduced three different haplotype probability models: Markov chain of a fixed order, Markov chain of a variable order, and a model based on segmenting each haplotype into frequent fragments. In Methods we give full specifications of the models and an concise description of the HaploRec algorithm; Additional file 1 contains a more detailed description of the algorithm and complexity analysis.
Experiments with simulated and real data demonstrate that these models and methods, collectively called HaploRec, are competitive with existing methods in terms of accuracy while being several orders of magnitude faster than the most accurate competitors. Of the two HaploRec models, the segmentation model is recommended as the default choice, as it generally gives more accurate results. However, for very large data sets, or when there is much missing data, the variableorder Markov model may be a better alternative, due to its smaller computational demands and smaller sensitivity to missing data.
Methods
Haplotype probability models
Recall that H(i, j) denotes the sequence (haplotype fragment) from the i th to the j th marker in a given haplotype H. We use the alternative notation frag(h, i, j) to denote a haplotype fragment from i to j, consisting of marker string h, when the fragment is not a projection from any particular haplotype H. We will denote H(i, i), a fragment consisting of a single marker, simply by H(i). Similarly, G(i, j) denotes the sequence of allele pairs from the i th to the j th marker in genotype G, called genotype fragment. Again, G(i, i) is denoted by G(i). We say that a fragment H(i, j) and a haplotype H' match if H(k) = H'(k) for all k : i ≤ k ≤ j. We say that a fragment H(i, j) and a genotype G match if there exists a string $\overline{H}$ ∈ Π_{k = i,...,j}A_{ k }such that {H(i, j), $\overline{H}$} is compatible with G(i, j). Given a set of haplotypes, the frequency of a fragment H(i, j) is defined as the number of haplotypes matching the fragment, and is denoted by $\mathcal{F}$(H(i, j)).
A simple model for haplotype probability is to consider the haplotype as a (firstorder) Markov chain in which the probability for a marker having a certain allele depends only on the preceding marker:
The obvious shortcoming of this model is that although linkage disequilibrium is normally strongest between neighbors, it is not limited to the immediate neighboring markers; a neighborhood of several markers is thus potentially more informative. More power in predicting the next allele can thus be obtained by increasing the order d of the Markov model:
With d = 1 we obviously have the standard Markov chain as a special case. Selecting a suitable value for d can be a problem. Increasing the order increases accuracy of predicting the next allele, but only to a certain extent; at some point, the conditional probability can no longer be reliably estimated from a limited sample of haplotypes. Another problem with fixing d is the fact that linkage disequilibrium may vary within the marker map; it is thus possible that no single value of d is suitable for all parts of the map. A more flexible alternative to the fixedorder Markov model is to use a variableorder Markov model, where the context is adjusted for each marker and haplotype individually. Informally, the goal is to find a flexible balance between generality and informativeness. We propose the following solution. When we estimate the probability of a haplotype H and consider the variableorder Markovian distribution at marker i, we find the longest observed fragment that (1) matches haplotype H and ends at marker i  1, and (2) has a frequency exceeding some given threshold, minfr. The fragments whose frequency does not exceed this threshold are considered uninformative. Using a frequency threshold is motivated by the fact that it is not likely that a long fragment of haplotypes is shared by different individuals unless it is inherited from the same ancestor; thus using only frequent fragments gives increased confidence in the fragments being identical by descent.
Given a frequency threshold minfr, we first compute the set of most frequent haplotype fragments, denoted by $\mathcal{F}$_{ minfr }, which will determine the sizes of contexts:
where h ranges over all possible fragments. Given a haplotype H, the longest matching fragments in $\mathcal{F}$_{ minfr }are then used to estimate the conditional probabilities at each marker i:
where s_{ i }= min{s  H(s, i  1) ∈ $\mathcal{F}$_{ minfr }}. The order of the Markov chain is thus individually adjusted for each position and each haplotype.
Although both fixed and variableorder Markov models have been extensively studied and used for many applications (see [34] for a review of variableorder Markov models), we are not aware of any previous applications to haplotype reconstruction. There is also a subtle difference between these models and typical applications of Markov chains. The models employed here are inhomogeneous, i.e., each marker has its own states and transition probabilities, whereas usually they are not dependent on the location in the sequence. Also unlike typical applications, we are simultaneously modeling two sequences whose entries are observed together as unordered pairs.
Another alternative of building a haplotype probability model from local fragments is to think of a complete haplotype as a mosaic of frequent fragments, originating from different founders or via different coalescence histories. In the segmentation model we consider nonoverlapping fragments to be independent, and consequently define the probability of a haplotype to be the product of fragment probabilities:
where S = (s_{1}, e_{1}), (s_{2}, e_{2}),..., (s_{ n }, e_{ n }) is a segmentation of H into consecutive nonoverlapping fragments (such that s_{1} = 1, s_{ i }= e_{i  1}+ 1 for all 1 <i ≤ n, and e_{ n }= ℓ).
The above formula leaves open the actual segmentation used for each haplotype. As the recombination history of the haplotypes is unknown, we of course have no way of deducing the "correct" segmentation. As a solution, we propose a model which averages over all possible segmentations:
where $\mathcal{S}$ is the set of all possible segmentations of H: $\mathcal{S}$ = {S : H(s, e) ∈ $\mathcal{F}$_{ minfr }for all(s, e) ∈ S}, and ${\mathcal{S}}^{\prime}$ is the set of all possible segmentations of the marker map:
A potential weakness of the model defined by Equation 8 is that there are more segmentations with a large number of segments than ones with a smaller number. To favor segmentations with less and longer segments, we introduce a penalty factor (controlled by an additional parameter q) for each additional fragment included in the segmentation:
where 0 <q ≤ 1, and S is the number of segments in segmentation S. Setting a smaller value to q will cause segmentations with a large number of fragments to have a smaller probability. Setting q = 1 corresponds to no penalty, in which case the formulation is equivalent to Equation 8. By experimenting we found that q = 0.1 works reasonably well in different settings. This value was used in all our experiments.
The HaploRec algorithm
Given one of the models we defined – fixedorder Markov chain, variableorder Markov chain, or the segmentation model – and its parameters, we can compute a probability for any given haplotype.
Assuming independence between the two haplotypes of an individual (HardyWeinberg equilibrium), the probability of any haplotype configuration for a genotype is just the product of the probabilities of its constituent haplotypes. The algorithmic problem is twofold: the haplotype reconstruction method has to simultaneously learn the model parameters and reconstruct the haplotypes of each individual.
Our algorithm, HaploRec, is a modified version of the the EM algorithm introduced in [4]. In the EM framework, the haplotype configuration underlying each genotype is considered as a latent variable, and the goal is to find a maximumlikelihood estimate for the model parameters $\mathcal{F}$. The likelihood of a single genotype is a sum of the probabilities of all its possible haplotype configurations (the values of latent variables), and the likelihood of the whole data is then just a product over all genotypes:
where P({H_{1}, H_{2}}  $\mathcal{F}$) = P(H_{1}$\mathcal{F}$)·P(H_{2}$\mathcal{F}$) is the probability of the haplotype pair {H_{1}, H_{2}}, given the model parameters $\mathcal{F}$. The EM algorithm works by iteratively improving estimates of the model parameters (the Mstep) and values of the latent variables (the Estep), until a (local) maximum of the likelihood is reached. The individual reconstructed haplotypes can then be obtained by just selecting, for each genotype, the compatible haplotype pair that has maximal probability according to the obtained maximumlikelihood estimate of the parameters.
The original algorithm [4] uses a multinomial model, where parameters are the frequencies of complete haplotypes. Our modifications to the EM algorithm consist of (1) replacing the multinomial model with one of our fragmentbased probability models and (2) using a sequential pruning strategy to overcome the exponential computational complexity of the Estep. Below, we give an outline of the HaploRec algorithm; a more detailed description, including pseudocode, handling of missing data and complexity analysis, is given in Additional file 1.
Representation of model parameters
Parameters of the three models are slightly different. In the Markov models, the parameters are the conditional allele probabilities; in the segmentation model, the parameters are the fragment probabilities. In practice, the conditional probabilities are derived from fragment probabilities as follows:
where d is the number of previous markers conditioned on and $\mathcal{F}$(H) denotes the estimated probability (frequency) of fragment H. As conditional probabilities can be straightforwardly derived from fragment frequencies, we always use the set of fragments as a representation for the model parameters, also in the case of the Markov models.
Estep
In the Estep of the EM algorithm, the aim is to calculate, for each G ∈ $\mathcal{G}$, and each compatible haplotype pair {H_{1}, H_{2}} ∈ C_{ G }, the probability that the genotype actually consists of that haplotype pair:
The model parameters, $\mathcal{F}$_{t  1}, are obtained from the previous parameter estimation step. The normalization just transforms the (prior) probabilities given by the model into a (posterior) probability distribution for configurations of the single genotype G. Exhaustively going through all possible configurations is feasible only when the number of heterozygous markers is small (≅ 20 or less). With more markers, we use a pruning strategy in which the set of possible haplotype configurations (separately for each genotype) is built up marker by marker, starting from a partial configuration containing only the allele pair at the leftmost marker. At each step, all partial configurations are extended with the allele pair at the next marker (for homozygous markers, each configuration is extended with the same allele pair; for heterozygous markers, there are two possible extensions for each configuration, and only the the B most probable configurations are propagated to the next step). In the final step, the C(≤ B) most probable configurations are returned. The approach is greedy; it is not guaranteed that the set of returned configurations consists exactly of the most probable ones. In preliminary experiments it was found that B = 25 and C = 10 give a reasonable computational efficiency, while increasing the parameters beyond these values does not significantly improve accuracy. These values were used in the experiments. Using a sequential pruning strategy to implement the Estep is conceptually simple, and has already been used in [19]. However, implementing it efficiently for our models is not trivial. In Additional file 1, we provide a detailed description of the data structures and algorithms for implementing the Estep for each of proposed models.
Mstep
In the Mstep of the EM algorithm, the model parameters are reestimated (based on the current haplotype estimates from the previous Estep), such that the likelihood of the data is improved from the previous iteration. The haplotype estimates from the previous Estep give the expected frequency of any haplotype in a given genotype. The estimated frequency of a haplotype fragment in a single genotype is obtained by a sum over all the haplotypes in its set of possible configurations that match the fragment, and the overall estimated frequency of a fragment h is obtained as an average over all genotypes:
where ${\delta}_{h,\{{H}_{1},{H}_{2}\}}$ = {i ∈ {1, 2}  h matches H_{ i }} ∈ {0, 1, 2} is the number of haplotypes in configuration {H_{1}, H_{2}} that match h. To compute the fragment frequencies, we first combine the sets of most probable haplotype configurations (from the previous Estep) from all genotypes into a weighted (multi)set of haplotypes, where the weight of each haplotype is the sum of the probabilities of its occurrences in the reconstructed configurations (note that the weights are thus mostly noninteger). Computing the set of frequent fragments is then done by by depthfirst search in the fragment containment lattice. First, all possible fragments of length one are generated, after which they are recursively extended to the right as long as the frequencies of resulting fragments stay above the given minimum frequency threshold and the fragments do not exceed a given maximum length. To improve the efficiency of the algorithm, the list of matching haplotypes is stored with each fragment during the execution of the algorithm (as depthfirst search is used, this does not significantly improve memory usage, because the list can be freed when the fragment is no longer in the stack). This way, the frequency of a each fragment can be computed by only matching its last allele to the list of haplotypes matching its prefix. The algorithm is guaranteed to find all frequent fragments, as frequency decreases monotonically when a fragment is extended.
Initialization
The initial frequencies are computed somewhat similarly as in the parameter estimation step. The difference is that there is not yet any information about the probability of different haplotype configurations, and thus all configurations of a genotype must be considered equally likely. The number of haplotype configurations compatible with a genotype G is exponential in the number of heterozygous markers in G, and enumerating all the possible configurations in C_{ G }is thus infeasible. Fortunately, the initial frequencies can be counted directly from the genotype data as follows, without explicitly generating the elements of C_{ G }:
where k_{G(i, j)}is the number of heterozygous markers in genotype fragment G(i, j) (note that a homozygous genotype has two identical haplotypes both matching the fragment, and thus weight 2). Computing the initial model can be implemented by a slightly modified version of the abovedescribed depthfirst search algorithm.
Using overlapping windows to haplotype very long marker maps
The HaploRec algorithm described above is directly suitable for windows (marker maps) containing up to 500–1500 markers. This practical limit depends on the number of genotypes, amount of missing data, complexity of the haplotype distribution, program parameters and the amount of available main memory. We have extended the implementation of the method by a simple but efficient feature that allows it to handle an arbitrarily large number of markers. Briefly, the idea is to sequentially haplotype a window of w markers at a time. In each window except the first one, phased haplotypes for the first o markers are obtained from the haplotypes in the previous, overlapping window. After a window is haplotyped, r of its last markers are discarded; they are only used to get better estimates for the w  o  r markers in the middle of the window. I.e., each window has length w = o + u + o + r, where u is the number of markers unique to this window, the first o represents the haplotypes obtained from the previous window and the second o those that are used by the next window. The window is moved o + u markers at a time. In experiments with Yoruba data (results not shown), HAPLOREC had practically identical accuracies with this extension and without it, at the expense of slightly larger running times per marker resulting from the overlapping windows.
Simulation of data
We used Hudson's coalescence simulator [24] to generate chromosomes under the standard WrightFisher neutral model of genetic variation with recombination. A long chromosomal region with 16666666 base pairs was simulated. The probability of a mutation in each base pair was set to 10^{8} per generation, and the probability of crossover between adjacent base pairs was set to 10^{8}. These values give the mutation probability for the entire chromosomal region μ = 0.16666666, and crossover probability ρ = 0.16666665. The diploid population size, N_{0}, was set to the standard 10000, giving the mutation parameter θ = 4N_{0}μ = 6666.6666, and the recombination parameter r = 6666.6665. This standard model simulates recombinations with a uniform distribution. As a result of the stochastic coalescence, however, the recombinations in the final population tend not to be uniformly distributed.
A sample of 2000 chromosomes was generated, and these were paired to form 1000 genotypes. On the average, one simulation produced approximately 55000 segregating sites. Markers were chosen from the segregating sites with minor allele frequency at least 5%, such that marker spacing was as uniform as possible. The actual number of markers and the position of the markers varied with different test settings. The amount of linkage disequilibrium between markers is perhaps the most important factor affecting the accuracy of populationbased haplotyping methods. Under the assumption of uniform recombination the amount of linkage disequilibrium is governed by the average distance between adjacent markers. To assess haplotyping performance with different amounts of linkage disequilibrium, marker spacing was varied between 6.6 and 166 kb. The average linkage disequilibrium between neighboring markers, measured with Lewontins D' measure, ranges respectively from 0.88 to 0.36 (Table 1). With 30 markers (the number of markers most commonly used in the following experiments), total map lengths are between 200 and 5000 kb. The largest data sets have 500 markers picked from the whole simulated region, giving an average marker spacing of 33 kb.
In real data, a fraction of alleles is practically always missing, and there may be genotyping errors. Therefore, in some of the experiments, part of the alleles were masked as missing. Either both or none of the homologous alleles of each marker were masked. This was done by fixing for each allele pair (single marker of a single genotype) a probability for having missing data. Likewise, genotyping errors were simulated by randomly changing each allele according to a given probability.
Processing of HapMap data
We used 30 trios from the Yoruba population in Ibadan, Nigeria, and another 30 trios from the CEPH population. Both data sets were downloaded from the HapMap web site [25].
The haplotypes of the children were inferred from the trios, and the nontransmitted parental chromosomes of each trio were combined to form additional artificial haplotype pairs (as is common in association studies if trios are available), resulting in a set of 60 genotypes for each population. Markers for which the phase could not be inferred, when all members of the trio were heterozygous, are included in the resulting data sets, but are not used in the switch accuracy calculations. In markers where only one allele was missing, the other was marked missing as well, since some of the tested programs could not handle markers where only one allele is missing. Markers with minor allele frequency less than 5% were discarded.
For testing, we first sampled 50 sets of markers from distinct regions of chromosome 1. The sampling was done by systematically taking (from the set of markers fulfilling the minor allele frequency threshold) markers 1–500 to the first data set, markers 501–1000 to the second, etc. In the resulting data sets, the average marker spacing is approximately 1.5 kb and the fraction of missing alleles is approximately 3.6%. Data sets with sparser maps were obtained by sampling markers from these; e.g., an average distance of 6 kb between markers was obtained by systematically choosing every fourth marker from the original samples.
For experiments with different numbers of markers, the markers were picked from the middle of each set of 500 markers. To reduce variance in results, caused by differences in the difficulty of haplotyping different markers, accuracy was always evaluated only on the 10 middlemost markers common to all the sets. This way, in each run, the test is the same and results are better comparable.
The same procedures were performed separately for the Yoruba and CEPH data sets, and the same set of experiments were performed for both populations. For each population, the reported experimental results are averaged over the 50 data sets to reduce variance in the results.
Availability and requirements

Project name: HaploRec

Project home page:http://www.cs.helsinki.fi/group/genetics/haplotyping.html.

Operating system: Platform independent

Programming language: Java

Other requirements: Java 1.5 or higher

License: Free for educational, research and nonprofit purposes

Any restrictions to use by nonacademics: License required
Declarations
Acknowledgements
This research has been financially supported by the Finnish Funding Agency for Technology and Innovation (Tekes). HT has been supported also by Alexander von Humboldt Foundation.
Authors’ Affiliations
References
 Clark AG: Inference of haplotypes from PCRamplified samples of diploid populations. Molecular Biological Evolution 1990, 7: 111–122.Google Scholar
 Gusfleld D: Haplotype Inference by Pure Parsimony. Proceedings of the 14:th Annual Symposium on Combinatorial Pattern Matching 2003.Google Scholar
 Ding Z, Filkov V, Gusfleld D: A LinearTime Algorithm for the Perfect Phylogeny Haplotyping (PPH) Problem. International Conference on Research in Computational Molecular Biology 2005 (RECOMB 2005), Volume 3500 of Lecture Notes in Computer Science, Springer 2005, 585–600.Google Scholar
 Excoffier L, Slatkin M: MaximumLikelihood estimation of molecular haplotype frequencies in a diploid population. Molecular Biological Evolution 1995, 12(5):921–927.Google Scholar
 Niu T, Qin ZS, Liu JS: Bayesian Haplotype Inference for Multiple Linked SingleNucleotide Polymorphisms. Am J Hum Genet 2002, 70: 17–169.View ArticleGoogle Scholar
 Qin ZS, Niu T, Liu JS: PartitionLigationExpectationMaximization Algorithm for Haplotype Inference with SingleNucleotide Polymorphisms. Am J Hum Genet 2002, 71: 1242–1247.PubMed CentralView ArticlePubMedGoogle Scholar
 Stephens M, Smith NJ, Donnelly P: A New Statistical Method for Haplotype Reconstruction from Population Data. Am J Hum Genet 2001, 68: 978–989.PubMed CentralView ArticlePubMedGoogle Scholar
 Stephens M, Scheet P: Accounting for Decay of Linkage Disequilibrium in Haplotype Inference and MissingData Imputation. Am J Hum Genet 2005, 76(3):449–462.PubMed CentralView ArticlePubMedGoogle Scholar
 Greenspan G, Geiger D: Modelbased inference of haplotype block variation. In Proceedings of the seventh annual international conference on Computational molecular biology. ACM Press; 2003:131–137.Google Scholar
 Kimmel G, Shamir R: GERBIL: Genotype resolution and block identification using likelihood. Proceedings of the National Academy of Sciences 2005, 102: 158–162.View ArticleGoogle Scholar
 Halldórsson BV, Bafna V, Edwards N, Lippert R, Yoosepth S, Istrail S: A Survey of Computational Methods for Determining Haplotypes. In Computational Methods for SNPs and Haplotype Inference, Volume 2983 of Lecture Notes in Computer Science. Berlin: Springer; 2004:26–47.Google Scholar
 Gusfleld D: An Overview of Combinatorial Methods for Haplotype Inference. In Computational Methods for SNPs and Haplotype Inference, Volume 2983 of Lecture Notes in Computer Science. Berlin: Springer; 2004:9–25.Google Scholar
 Salem R, Wessel J, Schork N: A comprehensive literature review of haplotyping software and methods for use with unrelated individuals. Human Genomics 2005, 2: 39–66.PubMed CentralView ArticlePubMedGoogle Scholar
 Yan H, Papadopoulos N, Marra G, Perrera C, Josef JiricnyCR, Boland, Lynch HT, Chadwick RB, de la Chapelle A, Berg K, Eshleman JR, Yuan W, Markowitz S, Laken SJ, Lengauer C, Kinzler KW, Vogelstein B: Conversion of diploidy to haploidy. Nature 2000, 403: 723–724.View ArticlePubMedGoogle Scholar
 Zhang K, Zhu J, Shendure J, Porreca GJ, Aach JD, Mitra RD, Church GM: Longrange polony haplotyping of individual human chromosome molecules. Nature Genetics 2006, 38: 382–387.View ArticlePubMedGoogle Scholar
 Eskin E, Sharan R, Halperin E: A Note on Phasing Long Genomic Regions using Local Haplotype Predictions. Journal of Bioinformatics and Computational Biology 2006, 4(3):639–647.View ArticlePubMedGoogle Scholar
 Coulonges C, Delaneau O, Girard M, Do H, Adkins R, Spadoni JL, Zagury JF: Computation of haplotypes on SNPs subsets: advantage of the "global method". BMC Genetics 2006, 7: 50.PubMed CentralView ArticlePubMedGoogle Scholar
 Wang WY, Barratt BJ, Clayton DG, Todd JA: Genomewide association studies: theoretical and practical concerns. Nature Reviews Genetics 2005, 6: 109–118.View ArticlePubMedGoogle Scholar
 Clayton D: SNPHAP: A program for estimating frequencies of large haplotypes of SNPs.[http://wwwgene.cimr.cam.ac.uk/clayton/software/snphap.txt]
 Hudson RR: Gene genealogies and the coalescent process. In Oxford surveys in evolutionary biology. Volume 7. Edited by: Futuyma D, Antonovics J. Oxford University Press, Oxford; 1990:1–44.Google Scholar
 Scheet P, Stephens M: A Fast and Flexible Statistical Model for LargeScale Population Genotype Data: Applications to Inferring Missing Genotypes and Haplotypic Phase. Am J Hum Genet 2006, 78(4):629–644.PubMed CentralView ArticlePubMedGoogle Scholar
 Eronen L, Geerts F, Toivonen H: A Markov chain approach to reconstruction of long haplotypes. In Pacific Symposium on Biocomputing 2004 (PSB 2004). Hawaii, USA: World Scientific; 2004:104–115.Google Scholar
 The International HapMap Consortium: The International HapMap project. Nature 2003, 426: 789–796.View ArticleGoogle Scholar
 Hudson RR: Generating samples under a WrightFisher neutral model of genetic variation. Bioinformatics 2002, 18: 337–338.View ArticlePubMedGoogle Scholar
 The HapMap database[http://www.hapmap.org/genotypes/2005–10/nonredundant]
 Daly MJ, Rioux JD, Schaffner SF, Hudson TJ, Lander ES: Highresolution haplotype structure in the human genome. Nature Genetics 2001, 29: 229–232.View ArticlePubMedGoogle Scholar
 Morris A, Whittaker J, Balding D: Little loss of information due to unknown phase for finescale linkagedisequilibrium mapping with singlenucleotidepolymorphism genotype data. The American Journal of Human Genetics 2004, 74: 945–953.View ArticlePubMedGoogle Scholar
 Curtis D, Sham PC: Estimated Haplotype Counts from CaseControl Samples Cannot Be Treated as Observed Counts. American Journal of Human Genetics 2006, 78: 729–730.PubMed CentralView ArticlePubMedGoogle Scholar
 Hintsanen P, Sevon P, Onkamo P, Eronen L, Toivonen H: An empirical comparison of case – control and triobased study designs in highthroughput association mapping. Journal of Medical Genetics 2006, 43: 617–624.PubMed CentralView ArticlePubMedGoogle Scholar
 Ron D, Singer Y, Tishby N: The Power of Amnesia. In Advances in Neural Information Processing Systems. Volume 6. Edited by: Cowan JD, Tesauro G, Alspector J. Morgan Kaufmann Publishers, Inc; 1994:176–183.Google Scholar
 Kermorvant C, Dupont P: Improved Smoothing for Probabilistic Suffix Trees Seen as Variable Order Markov Chains. Proceedings of the 12th European Conference on Machine Learning, Springer 2002, 185–194.Google Scholar
 Kruglyak L: Prospects for wholegenome linkage disequilibrium mapping of common disease genes. Nature Genetics 1999, 22: 139–144.View ArticlePubMedGoogle Scholar
 Akey J, Jin L, Xiong M: Haplotypes vs single marker linkage disequilibrium tests: what do we gain? European Journal of Human Genetics 2001, 9: 291–300.View ArticlePubMedGoogle Scholar
 Begleiter R, ElYaniv R, Yona G: On Prediction Using Variable Order Markov Models. Journal of Artificial Intelligence Research 2004, 22: 385–421.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.