Civil Engineering Reference
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the residual error
(that is, the error estimate between estimated and
recorded deaths) is minimized:
ε
⎡
⎤
N
N
1
1
∑
∑
(
)
2
[
(
)
]
2
ε =
ln
EO
−
+
ln
EO
[31.4]
⎢
⎥
i
i
i
i
N
N
⎣
⎦
i
=
1
i
=
1
We use a standard iterative search technique often called Nelder-Mead
optimization available in MATLAB® for minimizing the objective function
(Eq. 31.4) with the two free parameters of the distribution function,
θ
and
β
. The combined-norm approach is simple and suitable for countries with
at least three or more fatal earthquakes in the catalog, thus providing earth-
quake fatality rates for a large number of countries. Although the choice of
the two-parameter lognormal cumulative distribution function for defi ning
the fatality rate, and the selection of a specifi c minimization norm (shown
in Eq. 31.4) were based on limited analysis. Owing to the limited observa-
tion period of earthquakes (for damage and losses covering only post-1973
earthquakes) for any given country, there are only a few to several high-loss
earthquakes in a set dominated by frequent low-loss earthquakes. The pro-
posed norm provides reasonable balance for all historical losses in a given
country, without losing accuracy towards estimating the few high-loss earth-
quakes (Jaiswal
et al.
, 2009).
In order to estimate the uncertainty associated with the model's predic-
tion for future earthquakes, we defi ne
as a variable (a measure of the
variability or dispersion associated with actual deaths) representing the
standard deviation of the logarithm of actual deaths given the logarithm of
model-estimated deaths. If the dispersion is constant across areas of low
and high losses in a scattergram for a given country, we can estimate this
dispersion using the following expression:
ξ
1
N
=
∑
[
]
2
(
−
ζ
=
ln
O
i
μ
[31.5]
ln
O
ln
E
i
i
N
−
2
i
1
varies between low and high losses, the
regression would have to be obtained differently due to heteroscedasticity.
The expected value of ln(
O
i
) given ln(
E
i
) is assumed to be the same as
model-estimated loss ln(
E
i
) in the forward prediction as given by Eq. 31.3.
Note that we do not have all the information necessary to systematically
quantify the total variability associated with actual impacts from any given
earthquake, given the real-time computation involved in PAGER opera-
tions. We only use the model's predictability to hindcast past earthquake
losses as a measure to infer the uncertainty that may be associated with
future losses. The sampling uncertainties and the uncertainties associated
with heteroscedasticity or bias or due to the factors that are not fully
refl ected in the historical earthquake catalog are not considered in the
present calculations. Given the objective of the PAGER system to provide
If the conditional dispersion
ξ
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