Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 59

Chapter 3
Bayesian Networks in the Presence of Temporal
Information

Abstract Real-world entities comprising a complex system evolve as a function
of time and respond to external perturbations. Dynamic Bayesian networks ex-
tend the fundamental ideas behind static Bayesian networks to model associations
arising from the temporal dynamics between the entities of interest. This has to
be contrasted with static Bayesian networks, which model the network structure
from multiple independent realizations of the entities of a snapshot of the pro-
cess. More importantly, incorporating the temporal signatures is useful in capturing
possible feedback loops that are implicitly disregarded in the case of static Bayesian
networks. Since feedback loops are ubiquitous in biological pathways, dynamic
Bayesian network modeling is expected to result in better representations of such
pathways.
In this chapter, we will introduce basic definitions and models for modeling asso-
ciations from multivariate linear time series using dynamic Bayesian networks. Ap-
plications include modeling gene networks from expression data. Two broad classes
of multivariate time series are considered: those whose statistical properties are in-
variant as a function of time and those whose properties do show change of time.

3.1 Time Series and Vector Auto-Regressive Processes

3.1.1 Univariate Time Series

A univariate time series is a sequence of random variables

{ X ( t ) } = {..., X ( t − 1 ) , X ( t ) , X ( t + 1 ) ,...} (3.1)

measured at successive time points, usually spaced at uniform time intervals. A
univariate time series { X ( t ) } is said to be second order or covariance stationary