Mocapy++ - A toolkit for inference and learning in dynamic Bayesian networks
© Paluszewski and Hamelryck; licensee BioMed Central Ltd. 2010
Received: 2 October 2009
Accepted: 12 March 2010
Published: 12 March 2010
Mocapy++ is a toolkit for parameter learning and inference in dynamic Bayesian networks (DBNs). It supports a wide range of DBN architectures and probability distributions, including distributions from directional statistics (the statistics of angles, directions and orientations).
The program package is freely available under the GNU General Public Licence (GPL) from SourceForge http://sourceforge.net/projects/mocapy. The package contains the source for building the Mocapy++ library, several usage examples and the user manual.
Mocapy++ is especially suitable for constructing probabilistic models of biomolecular structure, due to its support for directional statistics. In particular, it supports the Kent distribution on the sphere and the bivariate von Mises distribution on the torus. These distributions have proven useful to formulate probabilistic models of protein and RNA structure in atomic detail.
A Bayesian network (BN) represents a set of variables and their joint probability distribution using a directed acyclic graph [1, 2]. A dynamic Bayesian network (DBN) is a BN that represents sequences, such as time-series from speech data or biological sequences . One of the simplest examples of a DBN is the well known hidden Markov model (HMM) [4, 5]. DBNs have been applied with great success to a large number of problems in various fields. In bioinformatics, DBNs are especially relevant because of the sequential nature of biological molecules, and have therefore proven suitable for tackling a large number of problems. Examples are protein homologue detection , protein secondary structure prediction [7, 8], gene finding , multiple sequence alignment  and sampling of protein conformations [9, 10].
Here, we present a general, open source toolkit, called Mocapy++, for inference and learning in BNs and especially DBNs. The main purpose of Mocapy++ is to allow the user to concentrate on the probabilistic model itself, without having to implement customized algorithms. The name Mocapy stands for Markov chain Mo nte Ca rlo and Py thon: the key ingredients in the original implementation of Mocapy (T. Hamelryck, University of Copenhagen, 2004, unpublished). Today, Mocapy has been re-implemented in C++ but the name is kept for historical reasons. Mocapy supports a large range of architectures and probability distributions, and has proven its value in several published applications [9–13]. This article serves as the main single reference for both Mocapy and Mocapy++.
Some popular Free BN packages with an API. Extracted from Murphy .
Harva et al. 
Lunn et al. 
Elvira Consortium 
U. Pittsburgh 
Blimes, Zweig 
Winn and Minka 
BP, EP, Gibbs, VB
Paluszewski and Hamelryck
In bioinformatics, models are typically trained using large datasets. Some packages in Table 1 only provide exact inference algorithms that are often not suitable for training models with large datasets. Other packages have no or little support for DBNs, which is important for modelling biomolecular structure. To our knowledge none of the publically available open source toolkits support directional statistics, which has recently become of crucial importance for applications in structural bioinformatics such as modelling protein and RNA structure in 3D detail [9, 10, 12, 15]. Furthermore, Mocapy++ is the only package that uses the stochastic EM [16–18] algorithm for parameter learning (see the Materials and Methods section). These features make Mocapy++ an excellent choice for many tasks in bioinformatics and especially structural bioinformatics.
Mocapy++ is implemented as a program library in C++. The library is highly modular and new node types can be added easily. For object serialization and special functions the Boost C++ library  is used. All relevant objects are serializable, meaning that Mocapy++ can be suspended and later resumed at any state during training or sampling. The LAPACK library  is used for linear algebra routines.
Mocapy++ uses CMake  to locate packages and configure the build system and can be used either as a static or shared library. The package includes a Doxygen configuration file for HTML formatted documentation of the source code. An example of a Python interface file for SWIG http://www.swig.org is also included in the package.
Most of the internal data is stored in simple Standard Template Library (STL)  data structures. However, STL or other public libraries offer little support for multidimensional arrays when the dimension needs to be set at run-time. In Mocapy++ such a multidimensional array is for example needed to store the conditional probability table (CPT) of the discrete nodes. The CPT is a matrix that holds the probabilities of each combination of node and parent values. For example, a discrete node of size 2 with two parents of sizes 3 and 4, respectively, will have a 3 × 4 × 2 matrix as its CPT. Mocapy++ therefore has its own implementation of a multidimensional array, called MDArray. The MDArray class features dynamic allocation of dimensions and provides various slicing operations. The MDArray is also used for storing the training data and other internal data.
Specifying a DBN in Mocapy++
Mocapy++ supports several node types, each corresponding to a specific probability distribution. The categorical distribution (discrete node), multinomial (for vectors of counts), Gaussian (uni- and multivariate), von Mises (uni- and bivariate; for data on the circle or the torus, respectively) , Kent (5-parameter Fisher-Bingham; for data on the sphere)  and Poisson distributions are supported. Some node types, such as the bivariate von Mises and Kent nodes, are to our knowledge only available in Mocapy++. The bivariate von Mises and Kent distributions are briefly described here. These distributions belong to the realm of directional statistics, which is concerned with probability distributions on manifolds such as the circle, the sphere or the torus [23, 25].
where x is a random 3D unit vector that specifies a point on the 2D sphere.
The various parameters can be interpreted as follows:
κ: a concentration parameter. The concentration of the density increases with κ.
β: determines the ellipticity of the equal probability contours of the distribution. The ellipticity increases with β. If β = 0, the Kent distribution becomes the von Mises-Fisher distribution on the 2D sphere.
γ1: the mean direction.
γ2: the main axis of the elliptical equal probability contours.
γ3: the secondary axis of the elliptical equal probability contours.
The Kent distribution can be fully characterized by 5 independent parameters. The concentration and the shape of the equal probability contours are characterized by the κ and β parameters, respectively. Two angles are sufficient to specify the mean direction on the sphere, and one additional angle fixes the orientation of the elliptical equal probability contours. The latter three angles are in practice specified by the three orthonormal γ vectors, which form a 3 × 3 orthogonal matrix.
Bivariate von Mises Distribution
where C(κ1, κ2, κ3) is the normalizing factor and ϕ, ψ are random angles in [0, 2π[. Such an angle pair defines a point on the torus.
The distribution has 5 parameters:
μ and ν are the means for ϕ and ψ respectively.
κ1 and κ2 are the concentration of ϕ and ψ respectively.
κ3 is related to their correlation.
Inference and Learning
Mocapy++ uses a Markov chain Monte Carlo (MCMC) technique called Gibbs sampling  to perform inference, i.e. to approximate the probability distribution over the values of the hidden nodes. Sampling methods such as Gibbs sampling are attractive because they allow complicated network architectures and a wide range of probability distributions.
Parameter learning of a DBN with hidden nodes is done using the expectation maximization (EM) method, which provides a maximum likelihood point estimate of the parameters. In the E-step, the values of the hidden nodes are inferred using the current DBN parameters. In the subsequent M-step, the inferred values of the hidden nodes are used to update the parameters of the DBN using maximum likelihood estimation. The E- and M-step cycle is repeated until convergence. Parameter learning using the EM algorithm requires a method to perform inference over the possible hidden node values. If one uses a stochastic procedure to perform the E-step (as in Mocapy++), a stochastic version of the EM algorithm is obtained. There are two reasons to use a stochastic E-step. First, deterministic inference might be intractable. Second, certain stochastic versions of the EM algorithm are more robust than the classic version of EM . EM algorithms with a stochastic E-step come in two flavors [1, 17]. In Monte Carlo EM (MC-EM), a large number of samples is generated in the EM step. In Stochastic EM (S-EM) [16–18] only one sample is generated for each hidden node, and a 'completed' dataset is obtained. In contrast to MC-EM, S-EM has some clear advantages over deterministic EM algorithms: S-EM is less dependent on starting conditions, and has a lower tendency to get stuck at saddle points, or insignificant local maxima. Because only one value needs to be sampled for each hidden node in the E-step, S-EM can also be considerably faster than MC-EM. S-EM is especially suited for large datasets, while for small datasets MC-EM is a better choice. Mocapy++ supports both forms of EM.
Results and Discussion
Hamelryck et al. sample realistic protein Cα-traces using an HMM with a Kent output node. Boomsma et al. extend this model to full atomic detail using the bivariate von Mises distribution . In both applications, Mocapy was used for parameter estimation and sampling. Zhao et al. used Mocapy for related work. Mocapy has also been used to formulate a probabilistic model of RNA structure  (Figure 1) and to infer functional interactions in a biomolecular network .
In practice, the most time consuming step in parameter learning is Gibbs sampling of the hidden nodes. The running time for one sweep of Gibbs sampling for a hidden discrete node is O(l × s) where l is the total number of slices in the data and s is the size of the node. The largest model that, to our knowledge, has been successfully trained with Mocapy++ is an extension of TorusDBN . The dataset consisted of 9059 sequences with a total of more than 1.5 million slices. The model has 11897 parameters and one EM-iteration takes 860 seconds. The number of S-EM iterations needed for likelihood convergence is around 100.
Toolkits for inference and learning in Bayesian networks use many different algorithms and are implemented in a variety of computer languages (Matlab, R, Java,...); comparisons are thus necessarily unfair or even irrelevant. Therefore, we feel it suffices to point out that Mocapy++ has some unique features (such as the support for directional statistics), and that the benchmarks clearly show that its performance is more than satisfactory for real life problems - both with respect to speed and data set size.
Future Directions of Mocapy++
The core of Mocapy++ described here is not expected to change much in future versions of Mocapy++. However, Mocapy++ is an evolving project with room for new features and additions. We therefore encourage people to propose their ideas for improvements and to participate in the development of Mocapy++. Potential directions include:
Additional probability distributions
Graphical user interface
Plugins for reading data in various formats
Mocapy++ has a number of attractive features that are not found together in other toolkits : it is open source, implemented in C++ for optimal speed efficiency and supports directional statistics. This branch of statistics deals with data on unusual manifolds such as the sphere or the torus , which is particularly useful to formulate probabilistic models of biomolecular structure in atomic detail [9–12]. Finally, the use of S-EM for parameter estimation avoids problems with convergence [16, 17] and allows for the use of large datasets, which are particularly common in bioinformatics. In conclusion, Mocapy++ provides a powerful machine learning tool to tackle a large range of problems in bioinformatics.
Availability and Requirements
Project name: Mocapy++
Project home page: http://sourceforge.net/projects/mocapy
Operating system(s): Linux, Unix, Mac OS X, Windows with Cygwin
Programming language: C++
Other requirements: Boost, CMake and LAPACK, GNU Fortran
License: GNU GPL
The authors thank the colleagues at the Bioinformatics centre who have helped in the development of Mocapy++: Christian Andreetta, Wouter Boomsma, Mikael Borg, Jes Frellsen, Tim Harder and Kasper Stovgaard. We also thank John T. Kent and Kanti Mardia, University of Leeds, UK and Jesper Ferkinghoff-Borg, Technical University of Denmark for helpful discussions. We acknowledge funding from the Danish Council for Strategic Research (Program Commission on Nanoscience, Biotechnology and IT, Project: simulating proteins on a millisecond time scale, 2106-06-0009).
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