Civil Engineering Reference
In-Depth Information
Fig. 2.5
The auto covariance function
As illustrated in Fig. 2.5 the auto covariance function is the mean value of the time
series multiplied by itself at a time shift equal to
τ
. Theoretically
τ
may vary between
()
Cov
0 and
T
, but the practical significance of
τ
is in the
vicinity of
T
. The reason is that while it in theoretical developments is convenient to
consider
τ
seizes to exist long before
x
()
x t
as a continuous function, it will in practical calculations only occur as a
discrete and finite vector of random values
x
, usually taken at regular intervals
t
Δ
. If
t
N
Tt
/
T
is large and
, in
which case the continuous integral in Eq. 2.15 may be replaced by its discrete
counterpart
Δ
is small, then the number of elements in this vector is
≈Δ
