Environmental Engineering Reference
In-Depth Information
z(t ,
ξ
)
H
z(t,
ξ
)
t
Ω
ξ
ξ
t
z :
Ω
H
z(t,
ξ
)
(
,
)
Figure 4.14
A random function (process) is a function
z
of two arguments mapping a set
of outcomes Ω of a probability space to a space
H
of functions or sequences. For a particular
t
t
ξ
t
,
z
t
,
ξ
,
z
ξ
)
is a realization,
i.e.
, a known
function. If
t
is discrete a random function is a random sequence,
i.e.
, a sequence of, in general,
correlated random variables
=
(
ξ
)
is random variable and for a particular ξ
=
(
t
,
i
,
ξ
)
i
,
i
,
i
,
T
measurement), ϕ
1
(
is a real number, and ϕ
(
ξ
)=[
ϕ
1
(
ξ
) ...
ϕ
m
(
ξ
)]
is a
random vector. Then
i
,
ξ
)=[
i
,
ξ
) ...
i
,
ξ
)]
T
ϕ
(
ϕ
1
(
ϕ
m
(
(4.118)
ξ
is a vector of real numbers for a particular realization (measurement) ξ
=
at a
i
particular time
i
=
of (4.117).
Let
θ
ϕ
T
y
(
i
,
)=
(
,
)
ξ
i
ξ
(4.119)
and
ε
(
i
,
ξ
)=
y
(
i
,
ξ
)−
y
(
i
,
ξ
).
(4.120)
Let
X
(
ξ
)
be a random matrix (
i.e.
, its elements are random variables) formed by
stacking ϕ
T
(
i
,
ξ
)
, for given times 1
,
2
, ...,
k
, ...,
N
.Then
⎡
⎤
⎡
⎤
ϕ
T
ϕ
1
(
1
,
ξ
) ···
ϕ
m
(
1
,
ξ
)
(
1
,
ξ
)
⎣
.
.
⎦
⎣
.
⎦
ϕ
T
X
(
ξ
)=
ϕ
1
(
k
,
ξ
) ···
ϕ
m
(
k
,
ξ
)
=
(
1
,
ξ
)
.
(4.121)
.
.
.
ϕ
T
ϕ
1
(
N
,
ξ
) ···
ϕ
m
(
N
,
ξ
)
(
N
,
ξ
)
Similarly, let
)=
y
)
T
Y
(
ξ
(
1
,
ξ
) ···
y
(
k
,
ξ
) ···
y
(
N
,
ξ
,
(4.122)
)=
y
)
T
Y
(
ξ
(
1
,
ξ
) ···
y
(
k
,
ξ
) ···
y
(
N
,
ξ
,
(4.123)
)=
ε
)
T
ε
(
ξ
(
1
,
ξ
) ···
ε
(
k
,
ξ
) ···
ε
(
N
,
ξ
,
(4.124)
)=
w
)
T
(
,
) ···
(
,
) ···
(
,
W
(
ξ
1
ξ
w
k
ξ
w
N
ξ
,
(4.125)
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