Page 6 - Volume 4, Issue 1, January 2011

© Benaki Phytopathological Institute
Makowski 4
used for parameter estimation (Vaclavik and
Meentemeyer, 2009). Sensitivity was also
used to identify the most important factors
influencing the predicted efficiencies of dif-
ferent management options (Stansbury et
al., 2002; Roura-Pascual et al ., 2010).
3. Main steps for uncertainty and
sensitivity analysis
Uncertainty analysis typically compris-
es three main steps: (i) definition of uncer-
tainty ranges and/or of probability distribu-
tions for uncertain model input factors, (ii)
generation of values for the uncertain input
factors, (iii) model output computation and
description of model output distribution.
Sensitivity analysis includes another step to
compute sensitivity indices (step iv). Finally,
when several model equations are available
for predicting a given quantity of interest, a
further step is to analyse uncertainty about
model equations using specific techniques.
All these steps are detailed below.
3.1. Step (i). Uncertainty ranges and
probability distributions for uncer-
tain input factors
Uncertainty in an input factor can be
described in different ways. It is often de-
scribed by the most likely factor value plus
or minus a given percentage (e.g. Koch et
al., 2009) or it is specified through a discrete
or continuous probability distribution over
a range of possible values. Among prob-
ability distributions, the uniform distribu-
tion, which gives equal weight to each value
within the uncertainty range, is commonly
used in uncertainty and sensitivity analysis
when the main objective is to understand
model behaviour.
More flexible probability distributions
are sometimes needed to represent the in-
put uncertainty. When the input corre-
sponds to a discrete variable (e.g. number
of imported consignments, number of suc-
cessful entries, etc.), discrete probability dis-
tribution (e.g. Poisson distribution) is often
appropriate (e.g. Yen et al., 2010). Among
continuous distributions, the well-known
Gaussian distribution is often convenient
since it requires only the specification of a
mean value and a standard deviation. It is
often replaced by the truncated Gaussian
distribution, triangular, or by beta distribu-
tions, which give upper and lower bounds
to the possible values (e.g. Peterson et al.,
2009; Yen et al., 2010). When the distribution
should be asymmetric, for example when
input factors are likely to be near zero, log-
normal, triangular, or beta distributions of-
fer a large range of possibilities (e.g. Peter-
son et al., 2009).
Probability distributions can be derived
from expert knowledge and/or from exper-
imental data. Bayesian statistics now offer
a variety of methods and algorithms to de-
rive probability distributions by combining
expert knowledge and data (e.g. Gelman et
al., 2004).
3.2.Step (ii). Generation of values of un-
certain factors
Monte Carlo sampling is a popular meth-
od for generating representative samples
from uncertain factor distributions. In Mon-
te Carlo sampling, the samples are drawn in-
dependently, and this approach provides
unbiased estimates of the expectation and
variance of each output variable. Other al-
ternative sampling techniques like Latin Hy-
percube can be used. It is also possible to
generate combinations of values of uncer-
tain factors by using experimental designs
like, for example, complete factorial designs.
This approach was used by EFSA (2008b) to
combine minimum, maximum, and most
likely values of several uncertain input fac-
tors.
3.3. Step (iii). Model output computa-
tion and description of the model
output distribution
This step may be difficult to carry out
when computation of model output is time-
consuming. With some very complex mod-
els, the number of samples generated at the
previous step must be set equal to a small
value due to computation time constraint.