H15
HW7 (due Mar. 20).
Exercises 12.1, 12.2 (a, d) from the textbook
In many cases, at each time
t,
several related quantities are observed and, therefore,
we want to study these quantities simultaneously by grouping them to form a vector.
By doing so we have a vector or
multivariate
process
.
We consider first the case of
bivariate
processes.
Suppose we are given two processes,
.
,1
,2
{}
,
{
tt
XX
}
}
}
}
}
μ
We say that
is a
stationary bivariate process
(or that
are
jointly stationary
)
if
,1
{,
,1
,
{
(a)
and
are each (univariate) stationary processes, and
,1
{
t
X
{
t
X
(b)
is a function of
h
only.
cov{
,
}
th
t
+
The individual ACVFs are defined in the usual way,
The corresponding ACFs are then
11
1
1
22
, 2
2
, 2
2
()
{
}
{
}
{
}
{
}
t
t
hE
X
X
X
X
γμ
+
+
⎡⎤
=−
−
⎣⎦
−
11
11
11
22
22
22
/
(
0
)
/
(
0
)
hh
ρ
γγ
ργ
γ
=
=
The
crosscovariance function
describes the correlation structure between the processes,
and is defined by
12
1
2
() c
o
v
{
,
}
{
}
{
}
t
t
hX
X
E
X
X
++
⎡
⎤
==
−
−
⎣
⎦
.
Its normalized version is the
crosscorrelation function
[]
12
12
1/2
11
22
(0)
(0)
h
h
=
.
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 Spring '09
 gUR
 Covariance and correlation, stationary bivariate process, bivariate AR

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