Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 70

Fig. 3.3 Time-varying structures of some gene regulation networks learned with ARTIVA. 50
targets ( dark grey ) and 5 transcription factors (parents, light grey ) are modeled in each network.
From an exploratory analysis of the gene expression profiles of the parents and the targets ( bottom
left panel ), three temporal phases are identified and delimited by the changepoints CP1 and CP2
( dashed lines ). The regulatory models for the different temporal phases are represented clockwise
in the top left , top right ,and bottom right panels. The plots of the networks around each parent
were obtained using the IGRAPH library ( Csardi and Nepusz 2006 ). Note that a variable can be
simultaneously parent and target, thus allowing the identification of auto-regulation mechanisms
as shown here for TF3 (regulatory model 2)

where

• ε ( represents the experimental noise and is assumed to follow a p -dimensional

Gaussian distribution with zero mean and covariance matrix

t )

diag ( σ

2 , i

Σ = )

h 2

ε i ( t ) ∼ N ( 0 , ( σ

i )

) ;

B h

• represents the baseline value of the variables X 1 ( t ) ,..., X p ( t ) in phase h and

does not depend on the parents activities;

A h p matrix such that the coefficient a ij represents the regulatory

interaction between the target X i and the parent X j in phase h .

a ij ) • =( is a p ×