How to summarise a collection of trees that came from a Bayesian analysis

After running a Bayesian phylogenetic analysis we are usually left with a large collection of trees, that came from the posterior distribution of the model given our data. Then if we want to work with a single tree - that is, to have a point estimate of this posterior distribution of trees - the most usual ways are to calculate the consensus tree or to select the most frequent tree. There are other ways, but let's fix on those by now.

We might not be aware of it, but when we choose for one or another summary we are in fact deciding for the tree estimate that minimizes its distance to all other trees in the set, and in expectation this will be the closest to the true tree under this distance metric (the so called Bayes estimator). This depends on what exactly do we mean by "distance" between trees, and that's what the article "Bayes Estimators for Phylogenetic Reconstruction" (doi 10.1093/sysbio/syr021) is about.

For example, the majority-rule consensus tree is the best we can get if we assume that the Robinson-Foulds distance (RF distance) is a good way of penalizing trees far away from the true one (I won't dwell into the meaning of "truth"; for us, the True tree^® is the one that originated the data). To be more explicit, the consensus tree is the one whose RF distance to all trees in the sample is the shortest possible. This will be the closest we can get to the true tree for this sample, if by "close" we mean "with a small RF distance".

Now suppose I don't like the RF metric because I can only count to two: if the trees are the same the distance is zero, but if they are different then the distance is not zero, and I don't care how different they are (think of apples and oranges). In this case the best representative of my sample is the one that appears more often, known as modal value or Maximum A Posteriori (MAP) value, since our sample comes from a posterior distribution. Is it the closest I can get to the true tree for this distribution? Yes, for this particular definition of distance: the MAP tree is the tree that maximizes the expected coincidence with the true tree.

In the article they also mention that if you want to find the tree that minimizes the expected quartet distance to the true value, then the quartet puzzling method will find this tree for you. But the quartet puzzling tree is not as easy to calculate as the consensus or MAP tree, and there is no straightforward way to find the tree that minimizes other distances in general (e.g. the dSPR, the geodesic distance or the Gene Tree Parsimony). Therefore the authors offer the well-known hill-climbing heuristics for finding the best tree, and use the squared path difference as an example of distance.

Below you can find the presentation I gave to my group last week about this paper, it contains basically some background information and a summary of their method. One thing that is absent from the slides are the results, which I briefly summarize below:

• their method (called "Bayes" in the figures or "BE") always used the path difference as distance measure; this is the overall distance they were trying to minimize.
• they simulated many data sets with several levels of sequence divergence, and reconstructed the phylogeny using Maximum Likelihood, Neighbor-Joining, and Bayesian analysis. From the Bayesian posterior distribution they elected as point estimates the consensus tree, MAP tree, and used their method to find the BE under the path difference.
• Figures 3 and 5 show the distance between the inferred and the true trees, where on figure 3 this distance is the path difference and in figure 5 it is the RF distance. As expected,  the Bayes estimator is better than any other measure at minimizing the path difference distance to the true tree, while the consensus tree wins if we want the closest in terms of RF distance.
• this result is rephrased in figures 8 and 9, which now look specifically at the distances between BE or MAP trees and the true tree. What they plot is distance(BE, true) - distance(MAP, true) for a different definition of distance(,) in each case. The MAP tree is correlated to the consensus tree (if the MAP frequency is larger than 50% they are equal, for instance). Therefore it should come as no surprise that if we define closeness to the true tree in terms of RF distance, the MAP tree will be closer than the BE as shown in figure 9. Because BE assumes that closeness to true is calculated in terms of the path difference, which is reinforced in figure 8.
• The authors wisely avoid offering the "best" Bayes estimator, since it depends on your judgment of how to penalize trees different from the true one.

Journal Club @ UVigo 2011.07.22

View more presentations from Leonardo de Oliveira Martins

OBS: This was my first time using beamer for Latex (after all these years, I know), so the slides are not prime time material. This is also my first submission to slideshare, and I like the idea of an embedded presentation within the blog post. I use latex a lot, and I think it would be easier for me to prepare a post with figures, equations and text within a presentation, and then simply embed it here with a minimum of extra text.

Maybe I'll try this next time, a presentation but with much more text than the recommended - in real life presentations the slides should support and complement but not replace the lecturer. Then you tell me if you would like to read on such a format or if you prefer a more traditional article-ish post.

Reference:
Huggins, P., Li, W., Haws, D., Friedrich, T., Liu, J., & Yoshida, R. (2011). Bayes Estimators for Phylogenetic Reconstruction Systematic Biology, 60 (4), 528-540 DOI: 10.1093/sysbio/syr021

Distribution of recombination distances between trees - poster at SMBE2010

(Post exported from my old blog, now defunct -- published originally on July 12, 2010)

I just came back from SMBE2010, where I presented a poster about our recombination detection software and had the chance to see awesome research other people are doing. The poster can be downloaded here (1.MB in pdf format) and I'm distributing it under the Creative Commons License. Given the great feedback I got from other participants of the meeting, I thought it might be a good opportunity to comment about the work, guided by the poster figures. Please refer to the poster to follow the figures and the explanation, I'll try to reproduce here my presentation taking into account the commentaries I received.

The motivation for the development of the model was to be able to say, for a given a mosaic structure, if the breakpoints can be explained by one recombination event or several. The recombination mosaic structure is usually inferred assuming the parental sequences (those not recombining) are known beforehand - in the figure they are the subtypes B and F - and recombination is then inferred when there is a change in the parental closest to the query sequence. Another problem is that it is common to analyze each one of the query sequences independently against the parentals - if all one wants is the "coloring", then this might be enough. For the above figure I analyzed each query sequence against one reference sequence from the subtypes B, F and C (thus comprising a quartet for each analysis). And we know that these mosaics don't tell the whole story.

If we know the topologies for both segments separated by the recombination breakpoint, then we can say, at least in theory, the minimum number of recombination events necessary to explain the difference (the real number can be much larger since we only detect those that lead to a change in the topology...). This minimum number is the Subtree Prune-and-Regraft distance, and is related to the problem of detection of horizontal gene transfers. In our case we devised an approximation to this distance based on the disagreement between all pairs of bipartitions belonging to the topologies: at each iteration we remove the smallest number of "leaves" such that the topologies will become more similar, and our approximate "uSPR distance" will be how many times we iterate this removal.

It is just an approximation, but it is better (closer to the SPR distance) than the Robinson-Foulds or the (complementary) Maximum Agreement Subtree distances, which compete in speed with our algorithm. For larger topologies it apparently works better than for smaller, but this is an artifact of the simulation - one realized SPR "neutralizes" previous ones, and it happens more often for small trees.

Our Bayesian model works with a partitioning of the alignment, where recombination can only occur between segments and never within. This doesn't pose a problem in practice since it will "shift" the recombinations to the border - the idea is that several neighboring breakpoints are equivalent to one breakpoint with a larger distance. This segments could be composed of one site each, but for computational reasons we usually set it up at five or ten base pairs. The drawback is the loss of hability to detect rate heterogeneity within the segment.

Each segment will have it own toplology (represented by Tx, Ty and Tz in the figure), which will coincide for many neighboring segments since we have the distance between them as a latent variable penalizing against too many breakpoints:

This is a truncated Poisson distribution, modified so that it can handle underdispersion - the parameter w will make the Poisson sharper around the mean - and each potential breakpoint has its own lambda and w.

The posterior distribution will have K terms for the segments (topology likelihood and evolutionary model priors) and K-1 terms for the potential breakpoints (distances between segments), as well as the hyper-priors. I use the term "potential breakpoint" because if two consecutive segments happen to have the same topology (distance equals zero) then we don't have an actual breakpoint. Again, considering only the recombinations that change the topology. This posterior distribution is calculated through MCMC sampling in the program called biomc2.

To test the algorithm, we did simulations with eight and twelve taxa datasets, simulating one (for the eight taxa datasets) or two recombinations per breakpoint. We present the output of the program biomc2.summarise, which interprets the posterior samples for one replicate: based on the posterior distribution of distances for each potential breakpoint, we neglect the actual distances and focus on whether it is larger than or equal to zero (second figure of the panel). Based on this multimodal distribution of breakpoints we infer the regions where no recombination was detected (that we call "cold spots"), credible intervals around each mode (red bars on top) or based on all values (red dots at bottom, together with the cold spots).

We also looked at the average distance for each potential breakpoint per replicate dataset, and show that indeed the software can correctly infer the location and amount of recombination for most replicates. It is worth remembering that we were generous in our simulations, in that there is still phylogenetic signal preserved on alignments with many mutations and a few recombinations. If recombination is much more frequent, then any two sites might represent distinct evolutionary histories and our method will fail.

We then analyzed HIV genomic sequences with a very similar mosaic structure, as inferred by cBrother (an implementation of DualBrothers). Here it is important to say that we used cBrother only to estimate the mosaic structure of the recombinants, doing an independent analysis for each sequence against three reference parental sequences. Therefore the figure is not a direct comparison of the programs, contrary to what its unfortunate caption might induce us to think. The distinction is between analyzing all sequences at once or independently, in quartets of sequences. If we superpose the panels it might become clearer to compare them:

click on the figure for a larger version (this one is not on the poster)

The curve in blue shows the positions where there is a change in closest parental for the query sequence, if each query sequence is analyzed neglecting the others. In red we have our algorithm estimating recombinations between all eleven sequences (eight query and three parental sequences). We can see that:

1. all breakpoints detected by the independent analysis were also detected by our method;


2. many recombinations were detected only when all sequences were analyzed at once, indicating that they do not involve the parental sequences - de novo recombination;


3. if we look at the underlying topologies estimated by our method (figure S2 of the PLoS ONE paper), we see that those also detected by the independent analysis in fact involve the parentals while the others don't;


4. biomc2 not only infers the location of recombination, but also its "strength" - given by the distance between topologies;

Finally, we show two further developments of the software: a point estimate for the recombination mosaic, and the relevance of the chosen prior over distances. The point estimate came from the need of a more easily interpretable summary of the distribution of breakpoints: instead of looking at the whole multimodal distribution, we may want to pay attention only to the peaks, or some other similar measure. This is a common problem in bioinformatics: to represent a collection of trees by a single one or to find a protein structure that represents best an ensemble of structures. In our case we have a collection of recombination mosaics (one per sample of the posterior distribution), and we elect the one with the smallest distance from all other mosaics - we had to devise a distance for this as well...

To show the importance of the prior distribution of distances, we compared it with simplified versions, like setting the penalty parameter w fixed at a low or high value. The overall behavior for all scenarios is lower resolution around breakpoints, and for weaker penalties we reconstruct the topologies better than for stronger ones, at the cost of inferring spurious breakpoints more often. We also compared the original model with a simplification where the topological distance is neglected and the prior considers only if the topologies are equal or not. This is similar to what cBrother and other programs do, and by looking at the top panel we observe that the results were also equivalent (blue lines labeled "cBrother" and "m=0"). In the same panel we plot the performance using our original ("unrestricted") model as a gray area.

I also submitted the poster to the F1000 Poster Bank.

References:
de Oliveira Martins, L., Leal, É., & Kishino, H. (2008). Phylogenetic Detection of Recombination with a Bayesian Prior on the Distance between Trees PLoS ONE, 3 (7) DOI: 10.1371/journal.pone.0002651
de Oliveira Martins, L., & Kishino, H. (2009). Distribution of distances between topologies and its effect on detection of phylogenetic recombination Annals of the Institute of Statistical Mathematics, 62 (1), 145-159 DOI: 10.1007/s10463-009-0259-8

Fault-tolerant conversion between sequence alignments

(Post exported from my old blog, now defunct -- published originally on May 17, 2010)

Despite I'm very charitable when testing my own programs, I'm not so nice when asked to scrutinize other people's work. That's why I was happy to see the announcement about the ALTER web server being published at Nucleic Acids Research (open access!).

I am not involved in the project, but I was in the very comfortable position of being one of the beta testers: all I needed to do is to find the largest and most obscure datasets I had and try them; then complain to the authors about the minimal details. I tried some big datasets (I think it was influenza H3N2 HA and HIV-1 complete genomes from South America, around 2 and 4 Mbytes each), and my simulated alignments created "by hand" from PAML. And ALTER could handle them in the end: they even sent me a report explaining how each one of my commentaries was used to improve the software, and asking me to try again until I feel satisfied.

The ALTER web server is a converter between multiple sequence alignment (MSA) formats, for DNA or protein, focused not only on the format itself (like FASTA or NEXUS) but more on the softwares that generated the alignment and the software where the alignment is going to be used in (e.g. clustal or MrBayes). They mention that this program-oriented format conversion is necessary since all useful softwares eventually violate the (outdated) format specification. In their own words

[D]uring the last years MSA's formats have `evolved' very much like the sequences they contain, with mutational events consisting of long names, extra spaces, additional carriage returns, etc.

The web service can automatically recognize the input format, and generate an output for several programs, in several formats. I found it very easy to use, as you proceed it automatically shows you the possible next steps in the same page. Another very nice feature is the possibility of collapsing duplicate (identical) sequences, working then only with the haplotypes (unique sequences). If later you need the information about the collapsed duplicates check out the "info" panel on the bottom of the screen (inside the "log" window).

The obvious case when this elimination of duplicates is useful is when doing phylogenetic reconstruction (in many cases you can safely remove identical sequences), but another option offered by ALTER is to remove very similar sequences, where you can define the threshold of similarity. Sometimes when I'm doing a preliminary analysis on a dataset, I want to discard sequences too similar in order to get an overall picture of the data, and some other times I must remove closely-related sequences since my recombination-detection program has a limitation on the number of taxa...

Besides the user-friendly web service, they also offer a geek-friendly API - if you want your program to communicate directly with the service - and the source code, licensed under the LGPL.

Reference:
Glez-Pena, D., Gomez-Blanco, D., Reboiro-Jato, M., Fdez-Riverola, F., & Posada, D. (2010). ALTER: program-oriented conversion of DNA and protein alignments Nucleic Acids Research DOI: 10.1093/nar/gkq321

The specialization of novel genes

(Post exported from my old blog, now defunct -- published originally on April 27, 2010)

Recently a paper about the software MANTiS called my attention, and I've been trying to write about it for a while. This announcement at the EvolDir list seemed like the perfect opportunity. I must warn you though that I've never used the software and I don't have any intimacy with the underlying databases, but the article is easy to follow.

The main result of the paper, published in Genome Biology and Evolution, is that there is a correlation between the mean number of anatomical systems (human tissues or cell types) where the gene is expressed and the time when the gene appeared on the phylogeny of the species. In other words, recent gene families are expressed in fewer anatomical systems (are more specific) than ancient ones. An anatomical system is a hierarchical classification of human tissues (e.g. the first level of the hierarchy: nervous, dermal, embryo, etc) available from gene expression data. So the age of appearance of a gene is an indicator of its specificity. Since the genes are subject to duplication we may have more than one member of the gene family in the same species, and the authors show that this correlation is maintained if we consider the appearance of the gene itself (as a result of duplication) or the appearance of the whole gene family to which the gene belongs.

They worked with gene families identified by MANTiS, which is a pipeline that 1) downloads data from metazoan genomes at ENSEMBL, 2) infers the gene tree based on the protein alignment of the gene family and 3) detects duplications through a reconciliation with a given species tree. Each gene tree is produced by EnsemblCompara which, as I understand, employs an extension of "reciprocal best hits" (that allow for many-to-many relations) to find the members of the family, and then maximum likelihood to find the tree itself. I will talk more about the gene tree/species tree reconciliation in the future, but it is enough to say that it's the minimal list of nodes on the gene tree that represent duplications. We have an example of such a reconciled gene tree below, where the duplications are represented by the red boxes:

extracted from Bioinformatics 2008 24(2):151-157

MANTiS creates a new character (the brown polygons, that I think of as an orthologous group) for each duplication event, and the phylogenetic profile generated by these characters is then used to calculate the branch lengths of the species tree through a least squares approach. The phylogenetic profiles are represented by 0's and 1's in the inlet figure above, from which a distance matrix must be calculated in order to have the branch lengths.

In the study two datasets were created for the presence/absence of genes: one called "families only" composed of one character for each single gene and for each protein family, and another called "with duplications" where a new character is created for each duplication event. Both analyses were necessary since gene gain through duplications is important in explaining genome size increase.

MANTiS creates a database relating each gene to its biological function and anatomical system: the biological processes and molecular functions (ontology terms) of protein families are given by the PANTHER database for human, mouse, rat and D. melanogaster, while the gene expression data (related to the anatomical systems) comes from eGenetics, GNF and HMDEG. When comparing the time of appearance of the gene (as explained above) and the expression data for the genes we have a figure like the following:

modified from Genome Biology and Evolution Vol. 2010:13

We must notice that in this graph the X axis is inverted (that is, left is older with the present day at the right) giving the impression of a negative correlation. So older gene families - or duplications - are expressed in more cell types in humans.  Similar results were obtained using rat expression data - since the expression datasets had information for both - or using the other expression datasets.

The authors say that a possible explanation for this behaviour is the increase in the number of distinct cell types (blue line, notice the inverted axis again :D), where new genes are likely to be more specific to a cell type (which may have appeared recently itself). Associated with this explanation is the subfunctionalization of duplicated genes, and the tendency to subfunctionalize ("specialize") can explain the decreased extent of expression. The subfunctionalization process itself might be related to the generation of a new cell phenotype.

One shortcoming of the analysis is that the gene family inference might fail to detect distantly related genes, and therefore what appears to be a gene gain (the "birth" of a new gene family) might be in fact a duplication of a more ancient single gene family. For example if after the duplication number 3 on the first figure the sequences diverged too much, we might wrongly classify them as two gene families. But to be free from this problem is a tall order. The authors also call our attention to the problem of low coverage of some genomes and taxonomic bias.

References:
Milinkovitch, M., Helaers, R., & Tzika, A. (2009). Historical Constraints on Vertebrate Genome Evolution Genome Biology and Evolution, 2010, 13-18 DOI: 10.1093/gbe/evp052
Tzika, A., Helaers, R., Van de Peer, Y., & Milinkovitch, M. (2007). MANTIS: a phylogenetic framework for multi-species genome comparisons Bioinformatics, 24 (2), 151-157 DOI: 10.1093/bioinformatics/btm567

Using System-on-a-Chip hardware to speed up alignments

(Post exported from my other blog, now defunct -- originally published on March 21, 2010)

In recent years there has been an explosion of parallel algorithms for solving bioinformatics problems, namely phylogenetic reconstruction and sequence alignment. These algorithms follow the growth of new hardware solutions like  Field-Programmable Gate Arrays (integrated circuits capable of  performing simple instructions in parallel), Cell microprocessors (like the one inside Playstation 3), Graphics Processing Units (nvidia and ATI powerful graphic cards) and massively parallel cluster architectures (like the IBM BlueGene). There is now an article describing a parallelized Needleman–Wunsch  alignment algorithm for the the Tile64 RISC processor.

TILE64 Processor Block Diagram (click to enlarge)

The Tile64 card is composed of 64 core processors, with each core running its own Linux OS and standard programs, and communicating using the Tilera API.  The Tile64 is a System on Chip (SyC), that therefore can be plugged into a PCI slot and be used independently from the CPU. On the other hand it can handle only integer number instructions, which limits its usability for numerical computations.

The Needleman–Wunsch algorithm is used for global sequence alignment. That is, for given two sequences it tries to maximize the score by including as few insertions as possible in each one of the sequences. It is closely related to the Smith-Waterman algorithm for local alignment, which tries to find the longest subsequence with positive score – where the score function is almost the same as for Needleman–Wunsch.

Both algorithms are a dynamic programming method where a matrix is built with the scores for all possible pairwise combinations (the solution is found by backtrack after the matrix is complete). After initialization of the matrix (first row and first column) the score of a cell can be calculated by looking at its immediate top and left neighbor cells, represented by the arrows in the figure below. For example the score of cell q4d4 depends only on q4d3, q3d3 and q3d4.

Alignment matrix for a pair of sequences, adapted from ref. 2 (click to enlarge)

In the article they use an implementation of the FastLSA algorithm, a parallel version of Needleman–Wunsch where instead of storing the whole matrix it stores one row/column combination per block, since depending on the sequence length the memory requirements for the whole matrix can become prohibitive. In other words it stores the score values only for a grid of rows and columns (e.g. at every ten sites). In [1] they claim that this implementation is therefore well suited for very long sequences, which cannot be handled for instance by the “needle” application of the EMBOSS package or the CUDA implementation of the Smith­Waterman algorithm [2].

The parallelism is achieved if we notice that the cells belonging to the same anti-diagonal (one such anti-diagonal represented in gray) can be calculated independently. Thus distinct cores can calculate the score of these cells at the same time with the so-called wavefront parallelism. Their solution achieved gains of 20 times over similar programs – even though their SyC implementation is in C and the other CPU implementations are in Java.

references:

[1] Galvez, S., Diaz, D., Hernandez, P., Esteban, F., Caballero, J., & Dorado, G. (2010). Next-generation bioinformatics: using many-core processor architecture to develop a web service for sequence alignment Bioinformatics, 26 (5), 683-686 DOI: 10.1093/bioinformatics/btq017

[2] Manavski, S., & Valle, G. (2008). CUDA compatible GPU cards as efficient hardware accelerators for Smith-Waterman sequence alignment BMC Bioinformatics, 9 (Suppl 2) DOI: 10.1186/1471-2105-9-S2-S10

F1000 Poster Bank

The poster presented at SMBE 2010 is now also available at the F1000 Poster Bank.

Poster presented at SMBE2010 available for download

I uploaded the poster presented at SMBE2010 about biomc2 and wrote some comments on it at the methodology blog.

The poster is under the Creative Commons License.

New article describing priors on biomc2

Our new paper is out!

Leonardo de Oliveira Martins and Hirohisa Kishino (2010) Distribution of distances between topologies and its effect on detection of phylogenetic recombination. Annals of the Institute of Statistical Mathematics 62(1): 145--159. doi:10.1007/s10463-009-0259-8

Unfortunately I had to transfer the copyright to the ISM, but I am still allowed to maintain the accepted manuscript at my personal homepage [1]. In this article we describe in more details theoretical aspects that we couldn't explore before (because it would divert the audience's attention). The novelties are:

• describing the topology distance parameter as an augmented variable. It seems contradictory that our method assumes that the number of recombination break-points is variable, and at the same time we claim that the number of parameters is constant. This is because we have a variable (we could think of it as an indicator variable) that tells if there is recombination or not. Actually it is not just an indicator variable since it doesn't simply say if there is recombination or not, but actually approximates the amount of recombination. I like the analogy with linear models (the infamous Y=B0 + B1 X1 + B2 X2 + ... + Bn Xn), where despite the number of variables is constant (n+1) those parameters too close to zero mean that the "effective" number of variables is smaller.
• describing the mini-sampler procedure. There is a strategy in reversible-jump MCMC (rjMCMC) that allows the chain to walk a little bit before accepting/rejecting, that we refer to as the mini-sampler. As we just saw our model doesn't need a rjMCMC since the number of parameters is constant, but still we employ the same strategies, making sure that the moves respect the detailed balance of the chain. This mini-sampler is necessary because on the one hand each step changes the topology just a little, and on the other hand we want to be able to handle cases where neighboring topologies are very different.
• showing the importance of the modified Poisson distribution as a prior for the distances. We explicitly work with two scenarios:
  
  1. forcing the penalty hyperparameter to a fixed value. This shows the relevance of the hierarchical modelling.
     
  2. using a simplified distance that can only detect presence/absence of recombination witout being able to quantify it. This is a model analogous to other procedures that don't explicitly take the distance into account, and is equivalent to the "indicator variable" described above.
  
  
• description of the ensemble of mosaics and how to choose the most representative one. Each MCMC sample is one set of topologies (one per segment) that we call the mosaic structure. We then devise the calculation of a distance between these mosaics to quantify how similar two samples are, and find the centroid mosaic - the sample most similar to all other samples. It is worth noticing that
  
  1. this is done after the MCMC sampling is finished, and is not part of the Bayesian model per se
  2. this distance between samples is completely unrelated to the distance between topologies (that we call dSPR).
     
  
  

In the meanwhile I'm updating the software site and upgrading the program: nothing special, but I realized that many libraries were unnecessary and that biomc2.summarise was painfully slow for large topologies. Now it is only slow.

[1] the first time that you try to access any file hosted on https://corn.ab.a.u-tokyo.ac.jp/ your browser will complain about my self-signed certificate - some lobby is not happy about it. Please neglect the terrorist warnings.

version 1.9 available for download

I've just uploaded the source code together with compiled versions for Mac OSX (intel) and for Debian/Ubuntu (AMD64) of the most recent "stable" version of the software biomc2. They can be downloaded from here, and the installation instruction as well as an explanation about inputs and outputs can be seen here.

I had a problem compiling the static versions because of the cairo library, so the binaries are dynamically linked  - which means they might be useless if you are lacking some library... and notice that the Mac OSX executables are simple command line programs, without any fancy resources.

Just for the sake of completeness, here is where the files are physically located.

I believe the documentation is insufficient, but I will work on that soon. Your feedback is welcome!

New version coming

I hope I can release the new version of the software this week. The program biomc2.summarise was completely rewritten, and now we don't have loads of tree files hindering the maximum number of segments.

I will try also to include some information about the software on the address http://www.biomcmc.org.