Civil Engineering Reference
In-Depth Information
2.4.2 Infl uence of uncertainty in independent variables
Previously it has been asserted that one is better off working with a smaller
high-quality dataset than a larger dataset in which the metadata is not as
well constrained. The reason why this is the case is that any uncertainty that
is associated with the independent variables gets mapped into apparent
aleatory variability during the regression analysis. For example, if one
derived a deterministic expression that precisely related two variables to
each other via some constant, such as
E
mc
2
, then if one knew the exact
value of the mass,
m
, in this case, then one would also know the exact value
of the energy,
E
. However, if the mass was not known precisely, but was
rather a normally distributed random variable with variance
=
m
, then the
energy would also be a normally distributed random variable with a vari-
ance of
σ
2
m
. If one attempted to derive a relationship between
observed energy quantities that were exactly known and estimated mass
values using the expression
E
i
σ
E
2
=
c
4
σ
2
∼
α
m
i
+
ε
i
, then one would observe two
effects. Firstly, the regression coeffi cient
would most likely not be equal
to its true value of
c
2
and secondly one would fi nd that the residual error
denoted by
α
i
would have a non-zero variance, i.e., it would be required in
order to explain the observations. The point here for the context of empiri-
cal ground-motion models is that when an independent variable is not
known precisely then the computed variance of the model will be an over-
estimate and the derived coeffi cients may also be biased.
While it is clear that treating independent variables as though they are
exactly known, when in fact they are not, will infl uence the obtained model
for both the median predictions and the standard deviation, this effect is
very rarely accounted for. Rhoades (1997) demonstrated that the effect of
magnitude uncertainties is not particularly large when a reasonable con-
straint is available on the catalogue magnitudes. More recently, Gehl
et al
.
(2011) have also applied a similar approach in order to account for the
uncertainties in site characterisation using the average shear-wave velocity.
The basic idea in both of these studies is to partition either the event-
specifi c residual (in the case of Rhoades) or the site-specifi c residual (in the
case of Gehl
et al
.) into two components, one of which represents the dif-
ference between the true value of the independent variable (and its infl u-
ence upon the predicted motions) while the other represents the true
residual. For example, if one began with a regression model of the form of
Equation (2.10) one could replace the terms
ε
δ
E
,
i
and
δ
S
,
k
with the new terms
given in Equations (2.11) and (2.12).
(
)
∂
μθ
Z
ijk
δ
′
=
Δ
M
+
δ
[2.11]
Ei
,
w
,
i
∂
M
w
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