The MPLN distribution (Aitchison and Ho, 1989) is a multivariate log normal mixture of independent Poisson distributions. The hidden layer of the MPLN distribution is a multivariate Gaussian distribution, which allows for the specification of a covariance structure. Further, the MPLN distribution can account for overdispersion in count data. Additionally, the MPLN distribution supports negative and positive correlations.

A mixture of MPLN distributions is introduced for clustering count data by Silva et al., 2019. Here, applicability is illustrated using RNA sequencing data. To this date, two frameworks have been proposed for parameter estimation: 1) an MCMC-EM framework by Silva et al., 2019 and 2) a variational Gaussian approximation with EM algorithm by Subedi and Browne, 2020.

Silva et al., 2019 used an MCMC-EM framework via Stan for parameter estimation. This method is employed in functions mplnMCMCParallel and mplnMCMCNonParallel.

Coarse grain parallelization is employed in mplnMCMCParallel, such that when a range of components/clusters (g = 1,…,G) are considered, each component/cluster size is run on a different processor. This can be performed because each component/cluster size is independent from another. All components/clusters in the range to be tested have been parallelized to run on a separate core using the parallel R package. The number of cores used for clustering is calculated using parallel::detectCores() - 1. No internal parallelization is performed for mplnMCMCNonParallel.

To check the convergence of MCMC chains, the potential scale reduction factor and the effective number of samples are used. The Heidelberger and Welch’s convergence diagnostic (Heidelberger and Welch, 1983) is used to check the convergence of the MCMC-EM algorithm. Starting values (argument: initMethod) and the number of iterations for each chain (argument: nIterations) play an important role for the successful operation of this algorithm.

Subedi and Browne, 2020 proposed a variational Gaussian approximation that alleviates challenges of MCMC-EM algorithm. Here the posterior distribution is approximated by minimizing the Kullback-Leibler (KL) divergence between the true and the approximating densities. A variational-EM based framework is used for parameter estimation. This algorithm is implemented in the function mplnVariational. The parsimonious family of models implemented by considering eigen-decomposition of covariance matrix in Subedi and Browne, 2020 is not yet available with this package.

Four model selection criteria are offered, which include the Akaike information criterion (AIC; Akaike, 1973), the Bayesian information criterion (BIC; Schwarz, 1978), a variation of the AIC used by Bozdogan (1994) called AIC3, and the integrated completed likelihood (ICL; Biernacki et al., 2000). Slope heuristics (Djump and DDSE; Arlot et al., 2016) could be used for model selection if more than 10 models are considered.

Starting values (argument: initMethod) and the number of iterations for each chain (argument: nInitIterations) play an important role to the successful operation of this algorithm. There maybe issues with singularity, in which case altering starting values or initialization method may help.

The Shiny app employing mplnVariational could be run and results could be visualized:

MPLNClust::runMPLNClust()

In simple, the runMPLNClust is a web applications available with MPLNClust.

Tutorials

For tutorials and plot interpretation, refer to the vignette:

browseVignettes("MPLNClust")

Citation for Package

citation("MPLNClust")

Silva, A., S. J. Rothstein, P. D. McNicholas, and S. Subedi (2019). A multivariate Poisson-log normal mixture model for clustering transcriptome sequencing data. BMC Bioinformatics. 20(1):394.

A BibTeX entry for LaTeX users is

  @Article{,
    title = {A multivariate Poisson-log normal mixture model for clustering transcriptome sequencing data},
    author = {A. Silva and S. J. Rothstein and P. D. McNicholas and S. Subedi},
    journal = {BMC Bioinformatics},
    year = {2019},
    volume = {20},
    number = {1},
    pages = {394},
    url = {https://pubmed.ncbi.nlm.nih.gov/31311497/},
  }

Package References

• Silva, A., S. J. Rothstein, P. D. McNicholas, and S. Subedi (2019). A multivariate Poisson-log normal mixture model for clustering transcriptome sequencing data. BMC Bioinformatics.

• Subedi, S., R.P. Browne (2020). A family of parsimonious mixtures of multivariate Poisson-lognormal distributions for clustering multivariate count data. Stat. 9:e310.

Other References

• Aitchison, J. and C. H. Ho (1989). The multivariate Poisson-log normal distribution. Biometrika.

• Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory, New York, NY, USA, pp. 267–281. Springer Verlag.

• Arlot, S., Brault, V., Baudry, J., Maugis, C., Michel, B. (2016). capushe: CAlibrating Penalities Using Slope HEuristics. R package version 1.1.1

• Biernacki, C., G. Celeux, and G. Govaert (2000). Assessing a mixture model for clustering with the integrated classification likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence 22.

• Bozdogan, H. (1994). Mixture-model cluster analysis using model selection criteria and a new informational measure of complexity. In Proceedings of the First US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach: Volume 2 Multivariate Statistical Modeling, pp.69–113. Dordrecht: Springer Netherlands.

• Ghahramani, Z. and Beal, M. (1999). Variational inference for bayesian mixtures of factor analysers. Advances in neural information processing systems 12.

• Robinson, M.D., and Oshlack, A. (2010). A scaling normalization method for differential expression analysis of RNA-seq data. Genome Biology 11, R25.

• Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics 6.

• Wainwright, M. J. and Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends® in Machine Learning 1.

Maintainer

• Anjali Silva ([email protected]).

Contributions

MPLNClust welcomes issues, enhancement requests, and other contributions. To submit an issue, use the GitHub issues.

Acknowledgments

• Dr. Marcelo Ponce, SciNet HPC Consortium, University of Toronto, ON, Canada for all the computational support.

• This work was funded by Natural Sciences and Engineering Research Council of Canada, Queen Elizabeth II Graduate Scholarship, and Arthur Richmond Memorial Scholarship.