We then test the transcriptional homogeneity for each dimension of the identified multi-dimensional modules.
In this article, we further expanded the MBPLS method by imposing sparse constraints to identify multi-dimensional modules.
This result highlights the power of multi-dimensional modules in grouping functionally relevant factors from different regulatory layers.
To evaluate the biological relevance of those identified multi-dimensional modules, we first test the functional homogeneity for each dimension of them.
The simulation study showed that the sMBPLS method can accurately identify embedded multi-dimensional modules and remarkably outperforms the non-sparse approach.
All of the multi-dimensional modules lead to statistically significant interaction networks (P-value 1.0E-20 1.0E-20s analysis, which indicates the significant associations among them.
On the contrary, the sparsity penalty forces sMBPLS to focus on 'local' (i.e. across a small subsets of variables and samples) peaks in the covariance, which correspond to (multi-dimensional) modules of relatively small size.
In order to discover the multi-dimensional modules by using MBPLS, an intuitive two-step procedure can be performed: first applying MBPLS to the data, then selecting the top-ranking input and response variables to form a module by ordering the absolute values of the loadings and weight vectors.
In our sparse version of the MBPLS problem, we searched the sparse representations of loading vectors whose non-zero elements can form a multi-dimensional module.
The maximization of covariance between t and u can reveal the associations between from and Y, which lead to the discovery of a multi-dimensional module.
Specifically, we want to identify a multi-dimensional module whose input variables have the maximum covariance with response variables across a subset of samples.
