Environmental Engineering Reference
In-Depth Information
values are very hard to obtain or they cannot be determined in practice. A possi-
ble solution for this problem is obtained if the expected values may be estimated
through time averages of realizations (measurements) of random functions or ran-
dom sequences (which are stochastic processes). These time averages may be quite
meaningless unless certain conditions are met.
4.3.2.1 Random Functions or Stochastic Processes
A real random function (stochastic process) is a function
z
of two arguments,
mapping a set of outcomes Ω of a probability space - the elements of which are
denoted here by ξ -toaspace
H
of functions or sequences if
t
is discrete [47].
Figure 4.14 illustrates this mapping. Then for a given value ξ
(
t
,
ξ
)
ξ
ξ
)
is a
function of time,
e.g.
, a sequence if
t
is discrete. This function is called a realization
of
z
=
,
z
(
t
,
. In practice a realization is the resulting measurement made on a variable
in a plant, out of all possible measurements that could have occurred for different
ξ, which represents the random nature of the plant variable. On the other hand, for
a particular
t
(
t
,
ξ
)
t
,
z
t
,
t
,
ξ
)
=
(
ξ
)
is random variable and
z
(
is a real number [47].
ξ
)
In industrial plants usually only one realization
z
(
t
,
of a random function
(
,
)
z
is available through measurement of a variable in a given time interval. Thus
certain statistics used for plant analysis, modeling and control must be obtained from
single measurements, in particular, correlations between a set of plant variables,
expected values and variances of plant variables,
etc.
It turns out that in the stochastic process
z
t
ξ
argument ξ is almost always
omitted in the technical literature, and is only implicitly considered by stating that
z
(
t
,
ξ
)
is a random sequence or random function (random process).
A real random variable (which might be better denoted as a function rather than a
variable) is a function η
(
t
)
which maps a set of outcomes Ω of a probability space
to the real line ℜ and satisfies certain conditions that insure that the distribution
function
F
(
ξ
)
may be obtained [47].
A random function (or random sequence)
z
(
x
)
(
t
,
ξ
)
is wide sense stationary if [47]:
1.
E
{
z
(
t
,
ξ
)} =
z
is constant,
i.e.
, does not depend on time
t
;
2.
E
{
z
(
t
,
ξ
)
z
(
t
−
τ
,
ξ
)} =
R
zz
(
τ
)
,
i.e.
, its autocorrelation depends only on the dis-
placement τand not on
t
.
Similar definitions apply to joint wide sense stationarity [47]. For example, let
the discrete time model of a plant be given by
ϕ
T
y
(
i
,
ξ
)=
(
i
,
ξ
)
θ
+
w
(
i
,
ξ
),
(4.116)
where
w
(
i
,
ξ
)
is a white noise random sequence, and
T
ϕ
(
i
,
ξ
)=[
ϕ
1
(
i
,
ξ
) ...
ϕ
m
(
i
,
ξ
)]
.
(4.117)
i
,
According to the above definition, each ϕ
1
(
i
,
ξ
)
is a random sequence, ϕ
1
(
ξ
)
is
ξ
)
a real random variable, ϕ
1
(
i
,
is one of the sequences that results (
e.g.
, through
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