In probability theory and statistics, given a stochastic process
X
=
(
X
t
)
, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. With the usual notation E for the expectation operator, if the process has the mean function
μ
t
=
E
[
X
t
]
, then the autocovariance is given by
C
X
X
(
t
,
s
)
=
cov
(
X
t
,
X
s
)
=
E
[
(
X
t
−
μ
t
)
(
X
s
−
μ
s
)
]
=
E
[
X
t
X
s
]
−
μ
t
μ
s
,
where t and s are two time periods or moments in time.
Autocovariance is closely related to the more commonly used autocorrelation of the process in question.
In the case of a multivariate random vector
X
=
(
X
1
,
X
2
,
.
.
.
,
X
n
)
, the autocovariance becomes a square n by n matrix,
C
X
X
, with entry
i
,
j
given by
C
X
i
X
j
(
t
,
s
)
=
cov
(
X
i
,
t
,
X
j
,
s
)
and commonly referred to as the autocovariance matrix associated with vectors
X
t
and
X
s
.
If X(t) is a weakly stationary process, then the following are true:
μ
t
=
μ
s
=
μ
for all
t,
s
and
C
X
X
(
t
,
s
)
=
C
X
X
(
s
−
t
)
=
C
X
X
(
τ
)
where
τ
=
|
s
−
t
|
is the lag time, or the amount of time by which the signal has been shifted.
When normalizing the autocovariance, C, of a weakly stationary process with its variance,
C
X
X
(
0
)
=
σ
2
, one obtains the autocorrelation coefficient
ρ
:
ρ
X
X
(
τ
)
=
C
X
X
(
τ
)
σ
2
with
−
1
≤
ρ
X
X
(
τ
)
≤
1
.
The autocovariance of a linearly filtered process
Y
t
Y
t
=
∑
k
=
−
∞
∞
a
k
X
t
+
k
is
C
Y
Y
(
τ
)
=
∑
k
,
l
=
−
∞
∞
a
k
a
l
C
X
X
(
τ
+
k
−
l
)
.
