Chiou and Youngs PEER-NGA Empirical Ground Motion Model for ...

Figure 27: Example of data truncation. Left hand plot shows broad-band peak acceleration data for
three small southern California earthquakes and right hand plot shows PEER-NGA data for the same
events. Curves show truncated regression fits to the combined data with the truncation level indicated
by the horizontal dashed line.
The sample likelihood and log likelihood functions for a regression model without random
effects in which the responses have been truncated below a level ytruncation are given by:
ln( L)
=
L =
∏
ϕ N ( yi
xi
, β)
1−
Φ ( Z x , β)
i N trunc i
2
2 2
[ − ln( σ ) / 2 − { ln( y ) − μ(
x , β)
} / 2σ
] − ln[
1−
Φ ( y x , β)
]
∑ i
i ∑
i i
truncation
In Equation (17) φN and ΦN are the normal density and cumulative normal functions, xi is the
vector of predictor variables for the i th observation and β is the vector of model coefficients.
Application of this to the analysis of ground motion data has been performed by Toro (1981)
and more recently by Bragato (2004). We have extended the concept of truncation to the
mixed effects model. Using the formulation of Brillinger and Preisler (1984), the sample
likelihood function is:
L
⎡
1
2
∏∫ ⎢ exp{
− zi
/ 2}
⋅∏
⎢
⎣
⎛
⎜ 1 exp
⎜
⎝
σ 2π
2 2
{ − ( y − μ ( θ ) −τ
⋅ z ) / 2σ
}
ij ij
i
[ 1−
Φ(
y μ ( θ ) + τ ⋅ z , σ ) ]
=
i 2π
j truncation ij
i
C&Y2006 Page 35
N
i
⎞⎤
⎟⎥dz
⎟
⎠
⎥
⎦
In Equation (18), the index i refers to the individual earthquakes and the index j refers to the
separate observations for the i th earthquake. The term [ 1− Φ(
ytruncation μij ( θ ) + τ ⋅ zi,
σ ) ]
normalizes the probability for the yij th observation by dividing by 1 minus the probability
distribution tail below the truncation limit ytruncation. The random effect for the i th earthquake,
τ·zi, appears in both the error term for the ij th observation and in the density normalization
(17)
(18)