Uncertainties in ground motion prediction in probabilistic seismic hazard analysis (PSHA) of civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 48

check that certain terms within the functional form are not strongly cor-
related with one another. However, recently it has been highlighted that
this information can also be used in order to defi ne a component of model-
specifi c epistemic uncertainty.
In a routine regression analysis one would perform the regression and
then look at the model coeffi cients and decide whether the observed trends
are statistically signifi cant or not. In the case that they are found to be not
signifi cant then the coeffi cient is commonly removed from the functional
form. The basis for deciding whether or not a term is signifi cant is to look
at its p- value. The p -value can be interpreted as being the probability that
a coeffi cient would take on its computed value purely by chance (as a result
of the random noise in the data). That is, if a coeffi cient has a very low
p- value then there is a very good chance that it is indicating a real depen-
dence within the data. Ordinarily a derived coeffi cient is regarded as being
signifi cant if its p -value is less than 0.05 (corresponding to a 95% confi dence
limit). To summarise this process: a regression analysis is conducted and if
a coeffi cient is found to have a p- value greater than 0.05 then it is often
discarded, while those terms with values below 0.05 are retained. Coeffi -
cients that are near this limit, but that are retained, are clearly not as well
constrained as other coeffi cients with much lower p -values. The implication
is that the terms of the model that relate to these coeffi cients will be less
well constrained that other parts. When one considers the model-specifi c
epistemic uncertainty this varying degree of constraint should obviously be
considered.
Often the reason why a particular coeffi cient is not well constrained is
because the choice of the functional form is not ideal. In other cases, the
lack of constraint can arise because there is little data corresponding to the
effect that is trying to be modelled. In order to refl ect both of these effects,
Arroyo & Ordaz (2011) presented the concept of prediction intervals for
ground-motion models. This concept is not at all new within regression
analysis and can be found in most textbooks on the subject. However, it
had (surprisingly) not previously been used in the context of ground-motion
prediction. While Arroyo & Ordaz (2011) only considered the case of linear
models, the standard framework can be extended to be described more
generally by the following equation:
ˆ

+ ∇ (

) ( ) ∇ ( )

T

σσ

2

=

2

μ X θ COV θ μ X θ [2.16]

ln Sa ln Sa

are the estimated model
coeffi cients, COV represents the covariance, and the vector X is a set of
input parameters that defi ne a future earthquake scenario (the magnitude,
distance, etc.). From Equation (2.16) it is clear that the variance for a pre-
diction associated with a ground-motion model,

In Equation (2.16), ∇ is the gradient operator, θ

ˆ

σ

2

ln Sa , is larger than the

2
ln Sa . The extent to which

variance obtained from the regression analysis, σ