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Graphical models in inference of biological networks

Farnoudkia, Hajar
In recent years, particularly, on the studies about the complex system’s diseases, better understanding the biological systems and observing how the system’s behaviors, which are affected by the treatment or similar conditions, accelerate with the help of the explanation of these systems via the mathematical modeling. Gaussian Graphical Models (GGM) is a model that describes the relationship between the system’s elements via the regression and represents the states of the system via the multivariate Gaussian (normal) distribution. This distribution also explains the structure of biological systems by means of its "conditional independence" feature. Therefore, in the inverse of the covariance matrix of the multivariate normal distribution, the "zero" value implies no functional interaction, and the "non-zero" value stands for the interaction between the proteins in the estimate of the system’s structure. In this study, as the novelty, we use the Copula Gaussian Graphical Models (CGGM) in modeling the steady-state activation of the biological networks and make the inference of the model parameters under the Bayesian setting. We suggest the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm to estimate the plausible interactions (conditional dependence) between the systems’ elements which are proteins or genes. Several data sets are used to illustrate the out-performance of the proposed RJMCMC in comparison with most of its alternatives. Also, we used some semi- Bayesian RJMCMC method to estimate the autoregressive coefficient matrix where GGM repeated through time. We improved the model by full-Bayesian approach and followingly, by a tuning parameter to increase the accuracy of the estimated matrices. Some simulated data sets are used to show the accuracy of the different proposed methods. Finally, we suggested a method to discover the relationships between variables through copula which is more flexible and it is more appropriate for the nonsymmetric or tail dependent cases. We applied the suggested ways in four real data set and we saw that copula can discover the joint density structure in addition to the available relationships in terms of the shape of the joint distribution to see whether it is symmetric or non-symmetric or even tail dependent or not.