We now describe the construction of the graph . Recall that the set of super-vertices consists of subsets of that share the same set of extant descendants. For every pair we have to decide whether . In order to do so, we pick a representative pair , where , and calculate three scores, corresponding to three putative relations of and : unrelated, siblings, and cousins. For each such relationship , we construct a pedigree by adding the relevant ancestry structure. For example, when considering the siblings relationship we construct by adding two common parents for and . For unrelated pairs we construct by adding a different pair of parents to each node (see Fig. 3).

We proceed by simulating inheritance on ; that is, the founders in are assigned unique haplotypes and we simulate the recombination process from top to bottom, with a recombination rate of . We then calculate IBD segments between each pair of extant descendants in and and calculate two IBD features: The number of IBD segments, and the total length of IBD sharing (we note that these features of IBD sharing were also considered by [15], a method for the inference of pair-wise family relationships). We repeat these simulations times for a specified parameter , thus obtaining an empirical estimate for the distribution of the IBD features. Using the above empirical distributions, we estimate the probability of observing the IBD features for each pair in under the relationship . Since the observed IBD features are typically not observed in any of the simulations, we use a smoothed form of the distribution using Gaussian kernel smoothing. Formally, let be the simulated IBD features in the simulations for a hypothesized relationship r. The density at point is calculated as:

Empirical tests led us to the conclusion that scaling the features to have equal variance and using a diagonal bandwidth matrix with a parameter in the range 1 to 8 gives the best results. The parameter compensates running time and accuracy. The accuracy stops improving near  = 50, which ends up with a very efficient analysis (See section 2.4 for more details).

Let be the observed IBD features between extant individuals and . The above procedure results in a probability , for every and every relationship in .

For each relationship , we define

We note that can be intuitively interpreted as a composite likelihood of . If is larger than and we add to with the weight

Fig. 4 shows the distribution of under different true relationships. Notice that cases where are distantly related (cousins, 2nd-cousins etc.) will tend to have a maximal score under . This is desirable, since we only seek to distinguish siblings from non-siblings at this point.

In the assignment stage, we are given the contracted siblings graph , and we search for an assignment of a sibling relation between super-vertices, depicted by an edge to a single sibling-relation between two individuals . Our assignment needs to obey the transitivity constraint of the full sibling relation. Recall that the weight of an edge corresponds to the strength of evidence for the existence of a sibling pair , where . We therefore formulate the edge assignment problem as follows:

In the following stage we define the half-sibling detection problem, where we attempt to detect groups of individuals with a single common-parent. First, we define the full-sibling relation, on individuals: . Notice that is defined as being reflective, and thus it is an equivalence relation on . is the quotient set of on , which in this case is simply the set of disjoint groups of full-siblings. We obtain from the edges in computed in section 2.2. is a clique cover, and so naturally describes an equivalence relation.

We define , which is the half-sibling relation, as a relation between equivalence classes in V, in respect to . Assuming the pedigree is known, HS is defined properly since if and are full siblings, and and are half-siblings, than and are half siblings. This allows us to simplify the half-sib detection problem, by constructing the polygamy graph , where s.t each vertex , represents a group of full-siblings, and each edge represents a half-sibling relation between and (see Fig. 6). The edges are added to , with a similar stage to 2.1, only the hypotheses tested this time are made for siblings groups , and are relevant to the half-sibling case (half-siblings,cousins,unrelated).

The graph has the convenient property that if a group of individuals have a single-common-parent then form a clique in . We thus assume by parsimony, that each clique in connects all of the children of a single parent , such that each is a full-sibling-group which contains the children of and a single mate. We therefore formulate the half-sib detection problem, as follows:

Simulating inheritance for the descendants of every two individuals during the graph constructions is very time consuming, and is the reason is impractical for large populations, or pedigrees deeper than 4 generations. Notice that if a pair of extant descendants has exactly the same ancestor structure in the pedigree, than the simulated IBD features are sampled from the same distribution. purposes caching individual pairs with identical inheritance paths, and introduces an accompanying dynamic programming algorithm for minimizing the number of operations.

In , we use a simplified version of this idea. For every pair of extant descendants, we calculate a least-common-ancestors (LCAs) vector , which is a list of the meiosis distances between and their least common ancestors. For example, all full-siblings will have the  = [1,1], since full-siblings always have two common ancestors, with one separating meiosis. We hash the simulated distribution for this LCA vector, where the key represents the vector itself, and the value is the distribution. We simulate inheritance only when needed, i.e. when have at least one descendant pair, without a hashed distribution, thus saving most of the redundant computation. Practically, the running time of is equivalent to the running time of , and is even slightly faster (see Table. 2). Although does not capture completely the ancestry structure for , we observed empirically (data not shown) that running simulations for each ancestry structure does not improve the reconstruction accuracy. Apparently, pairs of individuals with the same LCAs vector have similar IBD distributions. The similarity is large enough to make the repetition of inheritance simulation for two such pairs redundant.

The method, inheritance simulators, and quality evaluation tools are available at http://www.cs.tau.ac.il/heran/cozygene/software.shtml

Similarly to previous methods, we use a Wright-Fisher (WF) simulator that includes recombination and genders. We add several new features, which makes this simulator more flexible. First, we add the ability to control polygamy through a polygamy probability parameter , which controls the probability for an individual to have a child with more than one mate. Second, we add an option to simulate dynamic population sizes by specifying an initial population size and a final population size. The simulator calculates the required population change per generation and modifies the population size with that ratio in every generation.

Additionally, we experiment with a more realistic forward simulator that does not assume synchronized generations, and allows polygamy. We simulate inheritance as a function of time, where individuals can have children after the age of 20, and die at an age drawn from a capped exponential distribution with mean 50. The birthrate is changed according to the current population size, and is tuned to reach a predefined target population size. This simulator produces actual recombined haplotypes, from the haplotypes of 160 and HapMap representatives. More specifically, the simulation runs in 5 year iterations, and a pool of unmated mature individuals is maintained at all times. Every iteration, individuals from the pool are matched to uniformly drawn mates. A matching has probability to succeed. Every mated pair has a probability to have a child, where is initialized to be 1, and is modified in every iteration by +0.2 or -0.2 depending on whether the current population size is smaller or larger than the target population size. Polygamy is achieved through second-marriage, which can occur since once a mate dies, the individual is added back to the unmated pool. Finally, in order to include possible IBD detection errors, we detect IBD segments from simulated genotypes using , [17], and extract the IBD-features information from its output. This simulator also has the advantage of having a possible dynamic population size. The population grows or shrinks depending on the initial and target population sizes.

Many different measures can be accounted in evaluating the quality of reconstructed pedigrees. We first use a previously defined score, to compare to previous methods. For the large part of the presentation, we define and use other natural evaluation scores, which we deem as more relevant, and interpretable. In previous methods, a consensus-accuracy score, which counts the number of extant individual-pairs with the same minimal meiosis-distance as in the true pedigree was used [14]. This score treats correct detection of unrelated pairs and related pairs identically. This is problematic since the number of unrelated pairs dominates the score. For example, a trivial algorithm that outputs a pedigree where all individuals are unrelated receives a high consensus-accuracy score (see Fig. 9). As a new standard for pedigree-reconstruction evaluation, we suggest three types of scores: sensitivity, positive-predictive-value (PPV), and IBD-length prediction error.

We define sensitivity as the fraction of correctly constructed (distance wise) related pairs from the total number of related pairs in the original pedigree. PPV is defined as the fraction of correctly constructed related pairs from the total number of related pairs in the reconstructed pedigree. More formally, define as the reconstructed pedigree, as the original pedigree, as the minimal number of meiosis separating and in pedigree , and as the set of extant-individuals, which are related according to pedigree . Let . Then,

We run for generation, and compare the scores of reconstructed pedigrees for every generation against the first generations of the original pedigree. This way we can assess the accuracy of different relatedness degrees ( = 2 corresponds to siblings,  = 3 to siblings and first-cousins, etc.)

Scores such as sensitivity and PPV have the disadvantage of not weighing mistakes according to their magnitude. A second disadvantage is that the minimal meiotic distance does not capture the full complexity of a real pedigree (for example, double cousins detected as cousins will get a full scoring). For these reasons, we suggest to alternatively measure pedigree quality by calculating the root mean square IBD-length error ():where is the set of extant individuals in the population, is the observed total length of IBD segments between individuals and , and is the total length of IBD segments between individuals and , as given from simulating inheritance on the reconstructed pedigree . Since this score is dependent on the randomized scoring-simulation, we average the score of 5 runs. The can be interpreted as the expected prediction error (in Mbp) of the typical pair-wise total-IBD-length, given the reconstructed pedigree.

We tested the competing methods on monogamous Wright-Fisher simulated population, of constant sizes: 100, 200, 500, and 1000. When it was possible, we ran (up to 4 generations due to its high runtime complexity), and for larger populations we ran . was run in monogamous mode. Results on 100 and 200 individuals were similar, as well as results for 500 and 1000 individuals. In Fig. 10, we compare the three methods for small populations (200) and larger populations (1000). In all the scenarios we tested, was the most sensitive; for pedigrees of up to 5 generations (corresponding to 3rd cousins) and populations as small as 100 individuals. For the larger populations, the improvement in sensitivity is highest, where is able to build a pedigree which correctly predicts the minimal meiosis distance of more than 95% of 1st and 2nd degree relatives and more than 60% of relatives up to 3rd degree. At the same time, has a higher PPV up to pedigrees of 4 generations. In the 5th generation it gets a lower PPV than the other methods, but this disadvantage is not meaningful, since the sensitivity of these methods in the 5th generation is very low. gives better quality of results for larger populations, which is natural, since they tend to form simpler pedigrees with less multi-relationships between families, and less inbred families.

Considering scores, gets much better scores than the second best method, and is close to the optimal score, especially for larger populations. gets worse scores than / as a result of its practical tendency to over-predict inbreeding, which we observed during our experiments. An important feature of 's score is that it is non-increasing in the number of generations, similarly to the optimal score. In contrast, we do not see this behavior in other methods. Interestingly, the optimal scores decrease as the population size increases. We attribute this mainly to the increasing proportion of unrelated pairs in larger populations, which are easier to predict.

The simplified Wright-Fisher model that was used in pedigree reconstruction methods up to this day assumes a constant population size. Real populations sizes are obviously not constant, and it is known that population bottlenecks and expansion affect the IBD distribution in the population. We have conducted an experiment to test the effect of population size shifts on the distribution of chosen IBD features, and as a consequence on the quality of the resulting pedigree. We have run the Wright-Fisher simulation with changing initial population sizes of 100,200,300,400,500 and fixed the final population size at 500. By looking at the distribution of IBD features between all pairs of individuals, it is clear to see that the number of IBD segments and the mean IBD segment length have an inverse relationship with the initial population size. This corresponds to a higher proportion of relatives in the populations with smaller initial size. We have found that populations that grow from 100 to 500 individuals in five generations have similar IBD feature distributions to populations with constant population size of size 200. Interestingly the quality of the resulting pedigree of these populations remains unchanged when the initial population size is gradually decreased from 500 to 200. Only at initial size of 100 does the quality decrease. Sensitivity levels for initial population size of 100 are 0.96,0.75, and 0.54 for 2,3 and 4 generations. The largest decrease is for 3-generation pedigrees where the sensitivity is decreased by 10% on average. The PPV remains above 0.95 for generation 2,3 but is decreased from 0.85 to 0.71 in generation 4.

To asses the quality of on polygamous populations, we simulated polygamous populations of sizes 200 and 1000 with the Wright-Fisher model. In the simulated populations 33% of the siblings are half-siblings on average. Details regarding the execution of previous methods are the same as in section 3.3. was run with the polygamous mode. The results are summarized in Fig. 11. Once again is generally superior in terms of sensitivity, PPV and . A notable exception is 's relatively high sensitivity in generations 4 and 5 in smaller population sizes (200). Note however that this sensitivity comes at the cost of very low PPV and very high in these generations. The of is not shown in the graph since it is out of the charts, getting as high as 1500 Mbp. This result suggests that has a strong tendency to over-predict relationships in small polygamous populations.

Similarly to the monogamous case, achieves higher performance on larger, and as a result, more simply related populations. For a population size of 1000, is able to build a polygamous pedigree which correctly predicts the minimal meiosis distance of more than 97% of 1st degree relatives and more than 80% of 2nd degree relatives while maintaining a PPV greater than 80%. Polygamous populations pose a much greater challenge for pedigree reconstruction, and the performance is decreased in comparison to monogamous populations. According to our analysis, the difficulty in reconstructing polygamous pedigrees stems from the fact that the IBD feature distributions for the range of possible polygamous relationships have greater overlap than in monogamous relationships (See Fig. 12).

We test the performance of on populations produced by the polygamous, asynchronous forward simulator. We run the simulator for hundreds of simulation years, resulting in the mixing of the different generations, and reconstruct the last five generations. We use un-phased IBD segments, to account for the fact that our input is genotypes, and not haplotypes. As a necessary step, we aim to filter out cross-generation relationships, which are not currently modeled, by taking the genotypes from the youngest age stratum (Ages 0-20). We used the and HapMap genotypes as the founder population for our simulation. The results show a comparable success to the Wright-Fisher simulation, increasing our confidence that can be run on real populations. All accuracy measures show a decrease in accuracy compared to the Wright-Fisher simulation results. This is expected due to the addition of several factors (as discussed above), which adds to the complexity of the analysis (see Fig. 13).

We next use to reconstruct the historical pedigree for the HapMap MEX population. This population is of interest to us since it is known to contain several relatives, including a single 4-generation pedigree [5]. Age information is not publicly available for this dataset. Instead, we use known parent-offspring relationships to separate the population into three generations. The correct pedigree is not known, so we use previous relationship inference results by Stevens et al. to validate our results[18].

Running on the parent generation of HapMap phaseII+III genotypes, we are able to detect a single sibling relationship (NA19662,NA19685), three first-cousin relationships (NA19662,NA19664), (NA19664,NA19685), (NA19657,NA19786) and two second-cousin relationships (NA19657,NA19785), (NA19785,NA19786). We are able to reconstruct correctly the pedigree found by Kyriazopoulou et al. We do this fully automatically and without using the genotypes of the two known grandparents: (NA19662,NA19685) which makes the reconstruction a significantly harder task(see Fig. 14). Further more, all of the relationships inferred by except (NA19785,NA19786) are confirmed by Stevens et al.[18]. (NA19657,NA19786) are inferred as Third degree instead of first cousins, and (NA19657,NA19785) as Unknown degree instead of second cousins.