Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 60

if the first two moments, i.e., the mean E ( X ( t )) and covariance COV ( X ( t )) ,are

invariant as a function of time:

∀ t , E ( X ( t )) = μ , (3.2)

∀ , , ( ( ) , ( − )) = (( ( ) − μ )( ( − ) − μ )) = γ .

t i COV X t X t i E X t X t i (3.3) i

In other words, the first two moments of covariance-stationary time series are invari-
ant over time. In this section, stationary time series implicitly refer to covariance-
stationary time series.
Stationary univariate time series X are often modeled as auto-regressive pro-

cesses, where the value at a given time t is given as a linear combination of those at
earlier time points, X

( t )

( t − i ) , i = 1 ,..., p :

∀ t p , X ( t )= a 1 X ( t − 1 )+ ··· + a i X ( t − i )+ ··· + a p X ( t − p )+ b + ε ( t ) (3.4)

where

• X ( t ) is the random variable observed at time t ;

• p is the lag or order of the time series;

• a i ∈ R p , are the coefficients associated with the random variables

observed at the previous p time points, i.e., t

, i = 1 ,...,

− 1 , t − 2 ,..., t − p ;

• b ∈ R is the baseline measurement, i.e., the intercept;

2

• ε ( t ) is a Gaussian white noise, i.e., ε ( t ) ∼ N ( 0 , σ ) .

3.1.2 Multivariate Time Series

Multivariate time series (MTS) are sequences of multivariate random variables mea-
sured at successive time points. MTS data are commonly encountered in real-world
settings where the objective is to understand the associations between multiple en-
tities from their temporal signatures. An example of MTS from Smith et al. ( 2004 ),
representing the expression profiles of a set of genes, is shown in Fig. 3.1 .
Multivariate time series are commonly modeled as vector auto-regressive (VAR)
process. A VAR process is essentially a multivariate extension of an auto-regressive
process. A vector auto-regressive process VAR( p )oforder p , the variables observed
at any time t p areassumedtosatisfy

X ( t )= A 1 X ( t − 1 )+ ··· + A i X ( t − i )+ ··· + A p X ( t − p )+ B + ε ( t ) (3.5)

where

• X ( t )=( X i ( t )) , i = 1 ,..., k , is the vector of k variables observed at time t ;

• A i , i = 1 ,..., p are matrices of coefficients of size k × k ;

• B is a vector of size k representing the baseline measurement for each variable;