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Figure 4.7
. An example where the classical FOSM fails to identify the correct
range of variation of the output variable. The upper bound
f
2
of the
linearised function
f
is larger than the upper bound
y
2
of the real
function
y
because of the extremum in the function
y
. The lower
bound
f
1
of
f
is lower than the lower bound
y
1
of
y
because of the
convex nature of the real function.
4.2.3
Principle of the improved method
One possible way to improve the accuracy of the approximation is to use higher-order
terms from the Taylor series expansion. Unfortunately, the algebraic complexity
increases rapidly with the inclusion of higher-order terms. Moreover, the inclusion of
higher order terms also requires information on higher-order central moments, such as
skewness and kurtosis (Haldar and Mahadevan, 2000a). In practice, these moments are
difficult to assess due to the lack of data.
The Improved First-Order Second Moment (IFOSM) method presented here aims at
correcting the undesirable behaviour of the FOSM method near extrema (which is the
result of the linearisation) without making the computation too complicated. The
principle of the IFOSM method is based on the following premises:
1. The input-output function
y
is approximated using a parabolic function, that
is, a second-degree approximation.
2. The unknown PDFs of the input variables are replaced by equivalent uniform or
triangular density functions, which are derived from the given means and
standard deviations of the input variables.
3. The uncertainty is evaluated for the whole range of the variable defined by its
equivalent PDF, instead of using only the point estimate about the mean value as
in the FOSM method.
4.2.4
Mathematical derivation
For simplicity, the mathematical derivation of the IFOSM method is carried out for the
function of a single input variable
X
. The total variance of the output variable
Y
due to
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