Environmental Engineering Reference
In-Depth Information
by observing the result of one single coin flipping? The
same question arises when estimating the statistical law
describing the spatial variability of a parameter such as,
for example, the ore concentration in a gold mine. There is
a strong dependence between neighbouring observations,
so the inference of a reasonable stochastic model of the
ore concentration requires a sufficiently diverse sample
covering the different scales at stake. The guarantee (if
any) for a successful statistical inference from a unique
realization of a spatial process leads to the difficult but
fundamental assumption of
ergodicity
(Matheron, 1969).
In the same line of argument, Matheron introduced the
notion of
operatory reconstruction
for spatial methods:
in order to reach objective conclusions, the randomness
attributed to a stochastic model should potentially be
reconstructed from the unique observable reality.
phenomenon presents some variability. Furthermore, the
fact that a radionuclide gets older does not make it more
amenable to undergo decay. This phenomenon is the so-
called absence of memory. Yet, despite the unpredictable
nature of decay, it is still possible to define a mean lifetime
of the radionuclide. This mean lifetime is the expectation
E[
X
] of the random variable
X
and it corresponds to its
first statistical moment (see also Chapter 3). Whenever it
exists, the expectation of
X
is defined as the sum of all
possible values of the variable weighted by their corre-
sponding probability of occurrence, or in the continuous
case as:
+∞
E
[
X
]
=
xf
(
x
)
dx
,
(8.4)
−∞
where
f
(
x
) denotes the probability density function (pdf
hereinafter) of
X
. Back to the radionuclide, it is not
possible to calculate the theoretical expected lifetime of
Equation (8.4) if
f
(
x
) is not known (which is always the
case in practice). Instead, a statistical estimation based
on an available sample of observed radionuclide lifetimes
{
8.3 Tools and methods
In the following, we will distinguish between (1)
statistical
models
, based on statistical concepts only, (2)
determinis-
tic models
, yielding a
'single best solution'
and (3)
stochastic
models
, yielding a manifold of equally likely solutions.
However, the reader should bear in mind that this classi-
fication is not unique but just aimed at clarifying concepts.
For instance, both deterministic and stochastic models
make use of statistical concepts. Stochastic models are
another counterexample breaking the classification. They
are often formulated by a stochastic partial differential
equation. Yet, they can also make use of a determin-
istic equation and solve it a number of times using
different parameters or initial conditions drawn from a
prior statistical model (i.e. a probability-density function
designed from available observations). This section is
aimed at describing the strengths and weaknesses of these
model types.
x
i
,
i
∈
[1,
N
]
}
makes more sense. The average, defined as:
N
1
N
=
x
x
i
(8.5)
i
=
1
is a natural estimate of
E
[
X
], whose accuracy depends
largely on the number of measurements,
N
. The life
expectancy
E
[
X
] is, however, not sufficient to describe
precisely the way the radionuclides decay. Since the decay
may vary significantly from one radionuclide to the
other, it is important to have a second statistic describing
the variability of the lifetime around the expectation.
This is what the variance (second statistical moment of
the random variable
X
) does. The variance is defined as
the expected value of the squared variation of
X
below
and above around [
X
]:
+∞
x
[
X
]
E
[
X
])
2
]
E
[
X
])
2
f
(
x
)
dx
,
σ
=
E
[(
X
−
=
(
x
−
8.3.1 Statisticalmodels
−∞
(8.6)
When a large set of field observations or measurements
is available, the first step is to figure out their statistical
distribution, which allows us to quantify the degree of
variability of the variable under study and to investigate
whether it can be summarized by a simple statistical dis-
tribution. In this perspective, the variable of interest,
X
,
is modelled as a
random variable
. For example,
X
can
be the lifetime of a radionuclide. It is well known that
not all radionuclides of the same family will decay in the
same manner and exactly at the same time. Indeed, this
Equation (8.6) holds for random variables of contin-
uous nature. Note that, like in the case of
E
[
X
], the
observations of
x
i
can be used as a basis to construct a
statistical estimate of the variance in a discrete manner.
Overall, the expectation and the variance (most
often, their estimates) play an important role in
descriptive statistics. They allow us to summarize the
basic properties of potentially large data sets with just two
numbers. Higher order statistical moments such as the
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