Agriculture Reference
In-Depth Information
analysis (Saltelli
et al
., 2000, 2004) provide
a much more accurate assessment of sensi-
tivity to model inputs or parameter estimates.
While UnS analysis assesses sensitivity in
only one dimension, global sensitivity (GS)
analysis offers a measure of sensitivity that
is multi-dimensionally averaged, meaning
that GS coefficients provide a measure of
variance contribution for an input variable
X
i
that is averaged over all possible values
of the remaining variables
X
j≠i
. One such GS
method that has gained wide acceptance is
the Fourrier amplitude sensitivity test (FAST).
Software to implement FAST is now avail-
able and relatively easy to implement (Salt-
elli
et al
., 2004; Giglioli and Saltelli, 2013).
For the total quality cost model of forage,
the FAST analysis showed that 55.9% of total
cost sensitivity is attributable to
N
c
, whereas
19.1% of sampling interval sensitivity is
linked to
N
c
. This is because 85.0% of the
cost sensitivity to
N
c
is linear compared to
only 20.2% for sampling interval.
parameters to function
f
1
;
Z
j
are variables
affecting model predictions; and γ
m
are
unknown parameters to function
f
2
.
Ultimately, deciding which method (i.e.
observations vs model) to call
O
and
P
should be totally incidental. The conclusion
being reached regarding the two systems
should be totally invariant to what system
constitutes the observations and which sys-
tem is deemed the predictor. That is, we
should reach the same conclusion regardless
of whether we ask 'Are the predictions close
to the observations?' or 'Are the observations
close to the predictions?'
Validation methods for stochastic models
can be classified into three groups: deviance
analysis, concordance analysis, and linear
functional relationship.
Deviance Analysis
The expectation of the square of the diffe-
rence between two random variables
Y
1
and
Y
2
is (van Belle, 2002):
E (
Y
1
−
Y
2
)
2
= (μ
1
- μ
2
)
2
+ (σ
1
− σ
2
)
2
+
2
(
1
− ρ)σ
1
σ
2
Model Validation
(5.11)
Model validation is as much an issue with
stochastic models as with deterministic
ones. However, the conceptual framework is
vastly different. In the deterministic realm,
model outcomes are fixed and errorless.
Differences between observed and predicted
values are entirely due to errors in the obser-
vations. In the stochastic realm, both obser-
vations and model predictions contain errors.
The validation of such models is really an
attempt at answering the question: are model
outcomes from a system similar to those from
physical observations. This is a concept
similar to that of inter-observer agreement.
Observations and predictions play a symmet-
rical role because they are both transform-
ations of other (hidden) variables. That is:
Where E stands for the mathematical ex-
pectation; μ
1
is the mean of the
i
th
variable;
σ
2
is the standard deviation of the
i
th
vari-
able; and ρ is the correlation between the
two variables.
If the two variables agreed perfectly, then
they would fall perfectly on a 45° line through
the origin. The average of the square of the
distances of each pair from the 45° line
is equal to 2E (
Y
1
- Y
2
)
2
.
Thus Eqn 5.11 offers
potential to assess the agreement between
two random variables such as observed values
and output from a stochastic model. Equation
5.11 can be rewritten as:
2
2
EY Y
(
-
)
(
mm
ss
ss
ss
-
)
1
2
1
2
=
2
ss
2
12
12
O
=
f
1
(
X
1
,
X
2
, ...,
X
i
; φ
1
, φ
2
, …, φ
k
)
P
=
f
2
(
Z
1
,
Z
2
, ...,
Z
j
; γ
1
, γ
2
, …, γ
m
) (5.10)
(
-
)
2
+
1
2
+
-
(
1
r (5.12)
)
2
12
Where
O
is a vector of observed values;
P
is
a vector of predicted values;
f
1
and
f
2
are
some unknown functions;
X
i
are variables
affecting observations; φ
k
are unknown
In Eqn 5.12, the left-hand side is known as
the deviance, and the three terms on the right-
hand side are known as bias, scale difference
and imprecision, respectively. We can easily
