Biology Reference
In-Depth Information
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Time
Fig. 3.1
Arabidopsis thaliana
gene expression time series from
Smith et al.
(
2004
)
•
ε
(
t
)
is a white noise vector of size
k
, with zero mean (E
(
ε
(
t
)) =
0) and time-
).
Similar to an auto-regressive process, a VAR(
p
)oforder
p
assumes a linear corre-
lation structure between the
k
variables observed at time points
t
and the
k
variables
observed at the
p
previous time points.
invariant positive definite covariance matrix (COV
(
ε
(
t
)) =
Σ
3.1.2.1 Covariance Stationarity of a VAR Process
AVAR(
p
) process can be written as a VAR(1) process via its
companion form
,
Y
(
t
)=
AY
(
t
−
1
)+
υ
(
t
)
,
(3.6)
with
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
.
A
1
A
2
...
A
p
−
1
A
p
Y
(
t
)
ε
(
t
)
I
0
...
00
.
.
.
0
.
0
0
I
...
00
Y
(
t
)=
,
A
=
,
υ
(
t
)=
(3.7)
.
.
.
.
.
.
.
(
−
+
)
Y
t
p
1
00
...
I
0
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