Part mutual information for quantifying direct associations in networks

Abstract

Quantitatively identifying direct dependencies between variables is an important task in data analysis, in particular for reconstructing various types of networks and causal relations in science and engineering. One of the most widely used criteria is partial correlation, but it can only measure linearly direct association and miss nonlinear associations. However, based on conditional independence, conditional mutual information (CMI) is able to quantify nonlinearly direct relationships among variables from the observed data, superior to linear measures, but suffers from a serious problem of underestimation, in particular for those variables with tight associations in a network, which severely limits its applications. In this work, we propose a new concept, "partial independence," with a new measure, "part mutual information" (PMI), which not only can overcome the problem of CMI but also retains the quantification properties of both mutual information (MI) and CMI. Specifically, we first defined PMI to measure nonlinearly direct dependencies between variables and then derived its relations with MI and CMI. Finally, we used a number of simulated data as benchmark examples to numerically demonstrate PMI features and further real gene expression data from Escherichia coli and yeast to reconstruct gene regulatory networks, which all validated the advantages of PMI for accurately quantifying nonlinearly direct associations in networks.conditional mutual information | systems biology | network inference | conditional independence B ig data provide unprecedented information and opportunities to uncover ambiguous correlations among measured variables, but how to further infer direct associations, which means two variables are dependent given all of the remaining variables (1), quantitatively from those correlations or data remains a challenging task, in particular in science and engineering. For instance, distinguishing dependencies or direct associations between molecules is of great importance in reconstructing gene regulatory networks in biology (2-4), which can elucidate the molecular mechanisms of complex biological processes at a network level. Traditionally, correlation [e.g., the Pearson correlation coefficient (PCC)] is widely used to evaluate linear relations between the measured variables (2, 5), but it cannot distinguish indirect and direct associations due to only relying on the information of co-occurring events. Partial correlation (PC) avoids this problem by considering additional information of conditional events and can detect the direct associations. Thus, PC becomes one of the most widely used criteria to infer direct associations in various areas. As applications of PC to network reconstruction (6), recently Barzel and Barabási (7) proposed a dynamical correlation-based method to discriminate direct and indirect associations by silencing indirect effects in networks, and Feizi et al. (8) developed a network deconvolution method to distinguish direct dependencies by removing the combined effect of al...

Personalized characterization of diseases using sample-specific networks

“…Actually, instead of the correlation or PCC , we can similarly use partial correlation, conditional mutual information or part mutual information to construct a direct association network (13,59,60), which will be addressed in future studies.…”

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“…Consequently, ceRNAs exhibit indirect and positively correlated expression mediated by miRNAs, and several powerful tools could be used to quantify dependencies among them, such as correlation coefficient, partical correlation coefficient, mutual information or conditional mutual information (20). In recent years, several computational algorithms have been developed, especially for disease conditions (15,16,21,22), where Ala et al .…”

Network perturbation by recurrent regulatory variants in cancer

“…This pattern was not found when the TDs of liver cancer were projected onto the breast network (S3A Fig). We also constructed other types of regulatory networks that show regulatory associations but not regulatory directions [23,32,33] for each cancer. Again, a high connectivity of the TDs was observed (Fig 1D, S3B~S3D Fig).…”

“…Choice of the longest or shortest path did not make significant differences. For non-directional association networks, we employed ARACNe (Algorithm for the Reconstruction of Accurate Cellular Networks) [23] and PCA-PMI (Part Mutual Information-based PC-algorithm) [33]. We applied the available tools (http://califano.c2b2.columbia.edu/aracne [45] and http://www.sysbio.ac.cn/cb/chenlab/software/PCA-PMI/) for gene expression data in breast cancer and liver cancer separately.…”