Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 67

Assumption 2 is also satisfied by construction provided the error covariance ma-
trix is diagonal (see Lebre 2009 ). Assuming uncorrelated errors between different

variables may not necessarily hold in real-world scenarios. Nevertheless, it is not
unreasonable. Assumption 3 is difficult to verify, but it is not too restrictive if the
variables included in the data set are distinct. Then from Theorem 3.1 ,aVAR(1)
process whose error covariance matrix

Σ

is diagonal can be represented by a dy-
namic Bayesian network whose arcs are identified by the nonzero elements of A .
For an illustration, any VAR(1) process with diagonal

Σ

where matrix A has the
following form (where the elements a ij refer to nonzero coefficients),

Σ

⎛ ⎞

a 11 a 12 0
a 21

⎝ ⎠ ,

A = 00 (3.14)

0 a 32 0

can be represented by the dynamic network in Fig. 3.2 c. For instance, the non-zero
coefficient a 12 implies the arc from X 2 to X 1 in Fig. 3.2 a.

3.3 Dynamic Bayesian Network Learning Algorithms

Several approaches have been covered in Chap. 2 for static Bayesian networks.
Learning a dynamic Bayesian network defining a VAR model from the given data
is a very different process and amounts to identifying the nonzero coefficients of
the auto-regressive matrix A . Under the homogeneity assumption (Assumption 4 in
Sect. 3.2.1 ), repeated time measurements can be used to perform linear regression.
Let k be the number of variables under study. Then each variable X i , i

= ,...,

1 k in

a VAR(1) process satisfies

k
j = 1 a ij X j ( t − 1 )+ b i + ε i ( t )

X i ( t )= where ε i ( t ) ∼ N ( 0 , σ i ( t )) . (3.15)

However, the classic ordinary least square estimates of the regression coefficients a ij
and b i can be computed only when n k , thus ensuring that the sample covariance

matrix has full rank. For real-world data, regularized estimators are required in most
cases.

3.3.1 Least Absolute Shrinkage and Selection Operator

The Least Absolute Shrinkage and Selection Operator or LASSO ( Tibshirani 1996 )
is a standard procedure, first applied to network inference by Meinshausen and
Buhlman ( 2006 ). This constrained estimation procedure tends to produce some
coefficients that are exactly zero by applying an L 1 norm penalty to their sum.
Variable

selection is then straightforward: only nonzero coefficients define