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VMCMC: a graphical and statistical analysis tool for Markov chain Monte Carlo traces
BMC Bioinformatics volume 18, Article number: 97 (2017)
Abstract
Background
MCMCbased methods are important for Bayesian inference of phylogeny and related parameters. Although being computationally expensive, MCMC yields estimates of posterior distributions that are useful for estimating parameter values and are easy to use in subsequent analysis. There are, however, sometimes practical difficulties with MCMC, relating to convergence assessment and determining burnin, especially in largescale analyses. Currently, multiple software are required to perform, e.g., convergence, mixing and interactive exploration of both continuous and tree parameters.
Results
We have written a software called VMCMC to simplify postprocessing of MCMC traces with, for example, automatic burnin estimation. VMCMC can also be used both as a GUIbased application, supporting interactive exploration, and as a commandline tool suitable for automated pipelines.
Conclusions
VMCMC is a free software available under the New BSD License. Executable jar files, tutorial manual and source code can be downloaded from https://bitbucket.org/rhali/visualmcmc/.
Background
Bayesian inference using Markov chain Monte Carlo (MCMC) is today a common and trusted approach in molecular phylogenetics (see, e.g., in [1–4]), and is used by phylogeny inference software such as MrBayes [5], BEAST [6], BAliPhy [7], PrIMe [8] and JPrIME [9]. A typical goal in phylogenetics is to determine evolutionary relationships, for a set of species or for genes of interest, but researchers may also be interested in other parameters, for example parameters related to substitution patterns or duplication/loss processes. An advantage with Bayesian phylogenetic inference is that you can obtain posterior distributions of evolutionary parameters, conditional on your data, where the evolutionary parameters can be classified as discrete parameters (e.g., phylogenetic trees) or continuous (e.g., duplication and loss rates) [10]. For phylogenetics, Bayesian inference is often complicated by the continuous parameters’ dependency on discrete parameters such as trees, that due to their structure can have problems with mixing.
The output from MCMC, the trace, is a conditional sampling of unknown parameters, needing postprocessing to yield the desired posterior distributions. Users need to inspect the trace for possible nonconvergence, estimate burnin (how many of the first samples should be discarded?), assess mixing (does it look like a random sampling of the posterior?), and compute parameter statistics. Postprocessing can, however, be constrained by practical difficulties, in particular due to presence of both discrete and continuous parameters in the trace, and due to the need to manually extract useful information from MCMC runs, currently using multiple software (for example Tracer [6], AWTY [11] and CODA [12]). Userfriendly interfaces supporting graphical and interactive parameter exploration currently has potential to improve.
When converged, MCMC samples are drawn from the underlying stationary distribution and the trace represents a sample from the posterior distribution of analyzed parameters. After removing burnin samples, a converged chain displays little correlation between remaining samples, which also indicates good mixing. Convergence is, in theory, guaranteed in MCMC if the chain is allowed enough iterations. However in practice, it is not possible to estimate how many iterations are needed, nor is it possible to determine whether the chain has converged. Hence, heuristics are used to assess nonconvergence and the initial “burnin”. Software like CODA [12], AWTY [11], and Tracer [6] have support for this, and more, and for example MrBayes [5] has a convergence diagnostic based on standard deviation of split frequencies built in. These heuristics analyze a single parameter of the chain at a time and the decision of convergence is, usually, left to visual analysis of the trace and the heuristic value proposed for the parameter. For example, Tracer [6] suggests the heuristic that the effective sample size (ESS) [13] is greater than 200, for all parameters, as well as manual analysis of the trace to ascertain that the chain has indeed converged. Please note that the recommendation/heuristic of ESS greater than 200 by Tracer does not have a theoretical justification or a systematic study to support it and using a sufficiently large ESS for assessing convergence for highdimensional MCMC is recommended by Gong and Flegal [14]. In the convergence analysis of large data sets, e.g., for genomewide analysis, the manual inspection of convergence becomes a bottleneck and one would like to rely on automated nonconvergence assessment and burnin estimation based on multiple convergence diagnostics. Therefore, we see a need for simplified largescale convergence analysis.
There are several reasons to scrutinize individual traces. First, one may doubt automatic convergence/burnin assessment, e.g., when different convergence diagnostics give contradictory assessments. Second, there may be doubts regarding the MCMC proposals, in particular, when the sampled posterior distribution is multimodal and seemingly converged multiple runs provide very different samples. Third, there may be questions regarding mixing of an MCMC chain, e.g., when a trace is stuck in a local optimum for a long time and the proposal acceptance ratios of parameters are low. Fourth, surprising results, e.g., when a new result contradicts previous studies, a trace should be scrutinized to establish that the MCMC has not behaved strangely. Mixing of parameters is an important aspect of a successful MCMC run, in particular for phylogeny analysis where one wants to make sure that the tree posterior is sampled well. Although trace statistics can give important indications, various visualizations are often needed to better understand the different aspects of a chain and its trace. We have experimented with visualizations that focus on trees and their relation to the trace, that we have not found in Tracer, AWTY, and CODA, and believe the community can benefit from them.
We present VMCMC, a tool for phylogenetic MCMC analysis, with support for analysis and exploration of chain convergence, burnin estimation, trace visualization, parameter estimation, graphical display of parameter traces, which can run both as a commandline tool and as an application with a graphical userinterface (GUI).
Implementation
VMCMC presents MCMC traces statistically and visually, and enables both automatic and interactive analysis. VMCMC also supports output from many popular MCMC programs, e.g., PrIMe, JPrIME, BEAST, and MrBayes.
There are three use cases for VMCMC that may be of particular general interest:

Largescale automated analysis: Applying MCMC on genomewide data, with perhaps hundreds or thousands of phylogenies to estimate, users typically make arbitrary burnin decisions and, e.g., throw away the initial 25% of a trace. VMCMC can be used on the command line, easily integrated in scripts, and can be asked to estimate a burnin, assess nonconvergence, and estimate parameters automatically.

Detailed trace exploration: When trying to understand a single MCMC trace, and decide whether it is plausible that it has converged, it can be illuminating to see where in a trace certain trees are found, determine majority rule consensus trees and other parameters for different parts of a trace, and have tree space visualized. VMCMC supports a high degree of interactivity for such exploration.

Simultaneous handling of both real and tree parameters: Currently, there is no software that simultaneously analyzes both tree and continuous parameters of a chain. The available MCMC analysis software either handle the continuous parameter (CODA and Tracer) or the tree parameter (AWTY) but not both simultaneously. VMCMC supports analysis of both types of parameters simultaneously and can, e.g., show the occurrence of a specific tree topology in the trace of a real parameter or show the effect of changing burnin on both the real and tree parameter simultaneously.
Features
Like AWTY, Tracer, and CODA, VMCMC can also be used for graphical trace visualization, standard parameter statistics, and posterior distribution estimation. Some features unique to VMCMC are given below:

Graphical trace visualization: VMCMC employs a userfriendly GUI to display the trace of a selected real parameter, which helps in seeing trends for a particular parameter as well as visually inspect whether this parameter may have converged (Fig. 1).

Tree analysis: VMCMC presents a posterior distribution of trees, sorted by probability and hence easy access to the tree with maximum a posteriori probability and the closest alternatives. Selected trees are shown in Newick format and are visualized (Fig. 2) in a panel using code from forester [15]. A majority rule consensus tree can be computed, also restricted to selected trees, thus simplifying comparative tree analysis.

Tree space analysis: VMCMC calculates unweighted RobinsonFoulds distance [16] between each pair of trees and displays a twodimensional projection (Fig. 3) using a multidimensional scaling technique [17–19], where similar trees are expected to cluster. Point sizes proportional to probabilities inform the user of the estimated posterior.

Visualizing tree mixing and parameter dependence: VMCMC can indicate where a selected tree is found in the trace, thus help visualizing tree mixing and how parameters and trees are correlated (Fig. 4).

Convergence and burnin: Convergence assessment and burnin estimation can be performed using Geweke’s convergence diagnostic [20], GelmanRubin’s convergence diagnostic [21], and HöhnaSahlin’s ESSbased estimator [22, 23].

Analyzing parallel chains: Currently, traces from two parallel chains can be analyzed and visualized together (Fig. 5). Convergence can be tested on numerical posteriors by applying a MannWhitney U test, and tree split distributions [24] can be compared using a chisquare test for two samples. VMCMC provides the flexibility to perform parallel chain analysis using userspecified burnins for both chains. The parallel chains are appended to each other after removing specified burnins for both chains, and statistics and posterior distribution is calculated for joint traces. This is a useful feature in cases with poor convergence or high computational demands.

Command line tool: VMCMC can be used both as a GUI application and as a command line tool suitable as a component when writing scripts. Command line output is given in JSON format for easy parsing.
VMCMC tries to hide the computational costs inherent in the problem and utilizes multithreading to perform computations (e.g., in loading of graphics, and in calculation of statistics and of convergence diagnostics) independently from the user interface. Some calculations are delayed: the “Tree Analysis” tab is not prepared until the user asks for it to avoid unnecessary calculation of the needed, and expensive, twodimensional RobinsonFoulds distance matrix. Furthermore, in order to show trees with nonnegligible posterior frequency only and to speed up the loading of analysis on “Tree Analysis” tab, we have limited the computation of distance matrix to unique trees with posterior probability at least 0.2%. While VMCMC can work with any number of unique trees in the posterior, a warning for expecting a delay in loading the tab is generated if the posterior contains more than 45 unique trees with frequency greater than 0.2%.
We have applied VMCMC to real datasets and experienced response times that we feel are reasonable given the trace sizes. Table 1 displays the system time taken by VMCMC from specifying the input file to loading of graphical results on a standard MacBook Pro with 2.6 GHz Intel Core i5 processor and 8 GB RAM.
Results and discussion
For an example application of VMCMC, we used a tetrapod 18S ribosomal RNA dataset, named “DS1”, collected by Hedges et al. [25]. DS1 is known to be problematic for convergence and mixing of the tree parameter [26, 27]. We ran JPrIME twice with the same settings (using the JC69 model and default parameters; the species tree was dated using TimeTree.org [28] and is available as Additional file 1), each time for 10 million iterations and sampling every 100 iterations, yielding two traces containing 10^{5} samples which we name trace 1 and trace 2 (available in Additional files 2 and 3). Here we present how VMCMC can be used to evaluate mixing of the tree parameter for trace file (Additional file 3). The tutorial for VMCMC is available as Additional file 4.
Figure 1 shows the trace plot of DLRModelDensity parameter (this is the probability reported from the DLR submodel [8]) for trace 1, where the fuzzy caterpillarlike trace indicates that the trace does not show nonstationary behavior for DLRModelDensity parameter (i.e., the chain does not seem to be “stuck” in a state, nor still continuously improving). In the “Tree” tab, we see a summary of the tree posterior on the left. As an example, we have selected the top five (by frequency) trees and computed a majorityrule consensustree for them (Fig. 2). The consensus tree has five edges with support values lower than 1.0, indicating where the selected trees differ. Looking at the “Tree analysis” tab (Fig. 3), the size of each topology (represented by a colored circle) is proportional to the frequency of each topology in the trace. The “Tree analysis” tab also shows that the MCMC chain is repeatedly sampling T0 and similar trees (e.g., T2 and T5), or T1 and similar trees (e.g., T3 and T4), but the transition to other regions is rare. The clear partitioning of tree space into two disjoint parts is indicative of a bimodal distribution. This conclusion is in agreement with previous studies [26, 27] that suggest the data is bimodal. If we select the five top trees, push the “Mark selection in graph” button, and view the “Graph” tab again (Fig. 4), the regions where the selected trees are found in the trace are marked in different colors. We observe intermixing of all five colored lines and no long undisturbed stretches of a single color.
Comparing two parallel chains, run independently on the same data, is another way to investigate MCMC mixing. VMCMC can load two traces and superimpose plots. Figure 5 displays superimposed trace plots, from the DS1 dataset, where one can see that the red trace is different from blue trace at many places. Such aberrations are also checked through statistical hypothesis testing, and we note that the MannWhitney U test (left sidebar in Fig. 5) rejects the hypothesis that the traces are from the same distribution and a twosample chisquare test on trees’ split frequencies indicates the same (not shown here). We conclude that the two parallel chains are sampling from two different posterior distributions and have either not reached the stationary distribution, or need more iterations to be able to safely conclude convergence.
VMCMC has implemented various convergence diagnostics commonly used by phylogenetics community, e.g., GelmanRubin’s convergence diagnostic [21], effective sample size (ESS)based diagnostics and Geweke convergence diagnostic [20]. Please note that some of these diagnostics are outdated and these will either be removed or be replaced in the newer releases of VMCMC by more accurate and recent diagnostics like FixedWidth Stopping Rule (FWSR) [14]. To elucidate this point, Gong and Flegal [14] have shown that Geweke convergence diagnostic [20] is misleading and outdated for assessing convergence. Further calculations based on Gong and Flegal’s work [14] reveal that an ESS of 6000 measured by a 95% confidence interval corresponds to computational uncertainty of approximately 5% the size of the posterior standard deviation while the ESS recommendation of 200 made by Tracer and measured by a 95% confidence interval is equivalent to having a computational uncertainty of 28% the size of the posterior standard deviation. Therefore, the estimated variances used in calculations with Tracer’s heuristic can be unstable for the sample size of 200 and the simulation has, probably, not converged yet. Thus, we are looking forward to connect ESS based tests available in VMCMC to computational uncertainty using FWSR in the newer releases of VMCMC.
Conclusions
VMCMC can be applied to trace files from several molecular phylogenetics MCMC tools. Assessing whether a chain has converged and is sampling from the stationary distribution is a nontrivial task. As demonstrated by the example, VMCMC can help identify issues with mixing and convergence of the MCMC run for all parameters. The graphical user interface supports interactive data exploration and the command line interface enables largescale automated application.
VMCMC simplifies tasks in MCMC analysis that we encounter in our work and we believe that our software can be valuable to the community as well.
An executable jar file, tutorial, and source code can be downloaded from https://bitbucket.org/rhali/visualmcmc/. The tutorial is also available as Additional file 4.
Abbreviations
 ESS:

Effective sample size
 FWSR:

Fixedwidth stopping rule
 GUI:

Graphical user interface
 JSON:

JavaScript object notation
 MCMC:

Markov chain Monte Carlo
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Acknowledgments
The authors thank Kristoffer Sahlin for help on convergence statistics and burnin estimation methods.
Funding
RHA was funded by a stipend from the higher Education Commission (HEC) of Pakistan. SAM was funded by EuroSPIN (an Erasmus Mundus joint doctoral program) and Swedish eScience Research Center (SeRC). These funding bodies did not influence our design study, analysis, data interpretation, or manuscript.
Availability of data and materials
An executable jar file, tutorial, and source code of VMCMC can be downloaded from https://bitbucket.org/rhali/visualmcmc/. The datasets analysed during this study are included in this published article and its supplementary information files.
Authors’ contributions
The first prototype was developed as Bachelors thesis project by JM and MB guided by LA. Further development has been done by RHA as part of his PhD project and he has been responsible for design, implementation, drafting the manuscript and the tutorial, and coordination between other coauthors. SAM implemented the parallel chain analysis part. RMA implemented the handling of input generated from MrBayes and BEAST software. JS has been involved with the tree analysis tab and RobinsonFoulds distance metric. SMZ has incorporated tree visualization code from forester. LA has supervised the project, generated the idea and drafted the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Additional files
Additional file 2
Sample run file generated from JPrIME. Can be opened using standard text editors or through VMCMC. (MCMC 71065.6 KB)
Additional file 3
Sample run file generated from JPrIME using the same data as Additional file 2. Can be opened using standard text editors or through VMCMC. (MCMC 71065.6 KB)
Additional file 4
Tutorial file containing all information about input, output, commands to run VMCMC from command line, GUI options, FAQ and even a list of features we have identified as interesting and which we will target for implementation in future. It is in pdf format and can be open using Adobe Acrobat Reader. (PDF 1710.08 KB)
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Ali, R.H., Bark, M., Miró, J. et al. VMCMC: a graphical and statistical analysis tool for Markov chain Monte Carlo traces. BMC Bioinformatics 18, 97 (2017). https://doi.org/10.1186/s1285901715053
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DOI: https://doi.org/10.1186/s1285901715053