Environmental Engineering Reference
In-Depth Information
its PDF
f
ε
. Let us first consider the case that the measurement error is additive, that is, the
measured value is equal to the true value plus the error,
x
i
=
X
+ ε. It follows that ε =
x
i
−
X
.
The probability of observing a measurement
x
i
given that the true value is
X
=
x
is equal to ε
taking the value
x
i
−
X
. Therefore, the likelihood is
Lx
()
=
fx
(
−
x
).
(5.17)
i
ε
i
Accordingly, if the measurement error is multiplicative, it is
x
i
= X × ε, so that ε =
x
i
/
X
and
the likelihood function is
x
x
i
Lx
()
=
f
.
(5.18)
i
ε
5.3.3.2 Samples of a spatially variable parameter
If samples of a spatially variable parameter
X
are taken, it must be carefully considered how
this parameter is modeled probabilistically, as discussed in Section 5.3.1. If the parameter is
modeled spatially explicit by means of a random field {
X
}, then the sample taken at a loca-
tion
z
is actually a measurement of the random variable
X
(
z
). In this case, the likelihood
describing the sample is as presented in the previous paragraph, where
X
is replaced by
X
(
z
).
More commonly, the spatially variable parameter is not modeled explicitly, but through a
single random variable
X
with corresponding PDF
f
X
describing the population. In this case,
the samples are measurements of realizations of
X
, which are used to learn the parameters
θ
of the distribution of
f
X
. Hence, the likelihood function would be defined on θ. To make the
dependence of
f
X
on its parameters explicit, we write it as
f
X
(
x
|θ). The likelihood describing
a sample
x
i
of
X
is then
L
()
θ
=
f
(
x
|
θ
).
(5.19)
i
X
i
As an example, if the variability of
X
is modeled through a normal distribution, then the
parameters are θ = [μ
X
;σ
X
], the mean and standard deviation of
X
. In this case, the likeli-
hood function describing a sample
x
i
is
L
π / /
In Illustration 1, we considered this case, but with only the mean μ
X
being uncertain and σ
X
being known.
It is common to assume independence among multiple samples; thus, the joint likelihood
(,
µσ
)(
=
2
σ
)exp[ (
−
1
−
1 2
)((
x
−
µ σ
)) ].
2
i
X X
X
i
X
X
m
∏
1
L
()
θ
=
L
( .
θ
(5.20)
i
i
=
5.3.3.3 Measurement of site performance parameters
In many instances, measurements of performances of the geotechnical construction are
made, such as deformation measurements. These measurements are related to the model
parameters
X
by means of the geotechnical model. Therefore, the likelihood function must
include this model. As an example, if deformations at a site are measured, the model pre-
dictions of these deformations for given values of
X
are required. Let
h
i
(
X
) denote such a
model prediction. Furthermore, let
y
i
denote the corresponding observed deformation and
let ε
i
denote the deviation of the model prediction from the observation. This deviation is
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