Methodology
2.1 ANOVA sensitivity analysis techniques
In order to use the same terminology to present each sensitivity
technique, a generalized model is defined as:
Where represent the
independent variable (such as
model parameters, or model structure) and Y represents the response
(such as the model performance).
Variance-based
methods use a variance ratio to estimate the importance of
parameters. According to the ANOVA theory, the total sum of the squares
(SST) can be divided into the sum of squares due to individual model
parameters and their interactions as follows
(Saltelli et al., 2008.,
Saltelli et al., 2010).
where represents the
squares
due to the individual effect of
and to represent the squares due to interactions among k factors. In
this model, we summarize all interaction terms into the term.
Then, for each effect, the variance fractions are derived as follows:
where:
The symbol “o” indicates averaging over the particular index. Values
of 0 and 1 for the variance fraction correspond to a contribution of an
effect to the total ensemble variance (uncertainty) of 0% and 100%,
respectively. Obviously:
Subsampling
To diminish the effect of the sample size on contribution quantification
in ANOVA, (Bosshard et al., 2013) proposed
a subsampling scheme.
Assume
that there are elements for each
parameter,
the vector can be represented as. In each subsampling iteration, two
elements
are selected out of the
total
Ti elements which results in a total of (specify that C
is the combination symbol) possible element pairs for. Therefore, for
element, the is replaced by, which is a matrix as Formula (9). Here h
means the row number and j means the column number. The total number of
columns is defined as J. Therefore, h=1 or 2 and j=1, 2,
3,……,J. For more details of subsampling scheme, please
refer to the literature (Bosshard et al.,
2013).
Single-subsampling ANOVA
Single-subsampling ANOVA means that only one parameter from the
parameter vector ( ) is subsampled before the ANOVA. Assuming that the
is subsampled, which mean the two elements selected from vector are used
for . As for all other parameter, there are still elements for each. We
estimate the terms in equations (2) and (3) using the subsampling
procedure introduced in section 2.2 as follows.
For :
For :
The symbol indicates averaging over the particular index. Then, for each
effect, the variance fraction is derived as follows:
Multiple-Subsampling
ANOVA
The single-subsampling ANOVA may lead to biased results if different
parameters are chosen for subsampling. As an extension of the
single-subsampling ANOVA, a multiple-subsampling scheme is introduced to
ANOVA, leading to a multiple-subsampling ANOVA approach.
Multiple-subsampling ANOVA means that more than one parameter from the
parameter vector () are going to be subsampled at the same time before
the ANOVA is calculated. Assume that are subsampled then are replaced by
respectively. We estimate the terms in equations (2) and (3) using the
subsampling procedure as follows:
For :
For :
Then, for each effect, the variance fraction is derived as follows:
Full-Subsampling ANOVA
Moreover, a full-subsampling approach can be formulated when all
parameters are going to be subsampled. In detail, the full-subsampling
ANOVA means that all parameters are subsampled before ANOVA is
calculated. Consequently, are replaced by respectively. We estimate the
terms in equations (2) and (3) using the subsampling procedure as
follows:
Then, for each effect, the variance fraction is derived as follows:
To evaluate the performance of the above different subsampling ANOVA
approaches, two test cases are applied in the following.