Task 6
Statistical Approaches
Bob Youngs
NGA Workshop #6
July 19, 2004
July 19, 2004
Peer-NGA Project
1
Truncated Data
• Unknown number of
recordings where
value of yi < Ztrunc ,
value of xi is unknown
10
PGA
1
0.1
Ztrunc
0.01
(Toro, 1981)
0.001
1
10
100
1000
Distance
July 19, 2004
Peer-NGA Project
2
Truncated Data Statistical Model
Likelihood of observed data
L
f
N
i  recorded
( yi xi , β) /1  FN ( Z trunc xi , β)
Solved by maximizing the log(Likeli hood)
ln( L) 
  ln( 
i  recorded

2
) / 2  ln( yi )   ( xi , β) / 2 2
2
 ln 1  F
i  recorded
July 19, 2004
Peer-NGA Project
N

( Z trunc xi , β)
3
Fit to Truncated Data Ignoring Effect
10
Acceleration
1
> 0.03g
Generating function
0.1
Fit to all data
Fit to data > 0.03
0.01
0.001
0.1
1
10
100
1000
Distance
July 19, 2004
Peer-NGA Project
4
Fit Using Truncated Data Model
10
Acceleration
1
> 0.03g
Generating function
0.1
Fit to all data
Truncated fit
0.01
0.001
0.1
1
10
100
1000
Distance
July 19, 2004
Peer-NGA Project
5
0.1
0.01
0.03
PGA
0.3
1
3
Fit to Simulated Data
0.001
0.003
rock model
soil model
fit to rock
fit to soil
simulated data
0.1
0.3
1
3
10
30
100
300
R
July 19, 2004
Peer-NGA Project
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Fit to Truncated Simulated Data
July 19, 2004
Peer-NGA Project
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Uncertain and Missing Predictor
Variables
• Uncertain predictors
– Magnitude
– Distance/Rupture Geometry
– Site parameters (discrete and continuous)
• Missing predictors
– Site parameters (discrete and continuous)
– Rupture Geometry (for smaller events)
July 19, 2004
Peer-NGA Project
8
Predictor Variable Uncertainty
• General model Y = f(X) + ε
– Observe W which is imprecisely related to X
• Two types of error processes
– Error Model W = f(X) + U
applies when one wants X, but cannot measure it
precisely – “classical” measurement error
– Regression Calibration Model X = f(W) + U
one can measure W precisely, but quantity of
interest X is variable – often applies to
laboratory studies
July 19, 2004
Peer-NGA Project
9
Magnitude Uncertainty
(Rhoades, BSSA, 1997)
• Start with random (mixed) effects model
yij    M i  f (rij , )  i   ij
• Reported magnitude, M̂ i , contains error δi [N(0,si2)]
M̂i  Mi   i
• Revised mixed effects model
yij    M̂ i  f (rij , )  i   ij
i  i  i
• Solution obtained using “standard” approaches,
including analytical inversion of variance matrix
July 19, 2004
Peer-NGA Project
10
Magnitude Uncertainty for NGA
• Models likely to be non-linear in magnitude
yij  f (M i , rij , )  i   ij
• Reported magnitude, M̂ i , contains error δi [N(0,si2)]
M̂i  Mi   i
• Revised mixed effects model
yij  f (M i , rij , )   ij   ij
 ij 
f (M i , rij , )
M
 i  i
• Variance matrix terms due to error in magnitude i now vary
over j, - as a result not analytically invertible
July 19, 2004
Peer-NGA Project
11
Simulation Extrapolation Approach
• Applied in cases where W=X+U with U
N(0,s2)
• Simulate a series of data sets with
increasingly large measurement error
Wb,i(λ)=Wi + λ½Ub,i where Ub,i are simulated
error terms with 0 mean and variance s2
• For each value of λ average the parameters
of the model Θ over many simulations to
obtain an average value ˆ ( )
July 19, 2004
Peer-NGA Project
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Simulation Extrapolation (continued)
• Extrapolate back to
λ = -1
Coefficients
• Define a functional
relationship for ˆ ( )
-1
July 19, 2004
Peer-NGA Project
0
1
2
13
Example Application of Simulation
Extrapolation Approach
• Applied in cases where W=X+U with U
N(0,s2)
• Simulate a series of data sets with
increasingly large measurement error
Wb,i(λ)=Wi + λ½Ub,i where Ub,i are simulated
error terms with 0 mean and variance s2
• For each value of λ average the parameters
of the model Θ over many simulations to
obtain an average value ˆ ( )
July 19, 2004
Peer-NGA Project
14
Assess the Effect of Magnitude
Uncertainty
•
•
Start with a “True” Model
Simulate PGA values from “True” model using
NGA M-R disribution
1. Calculate mean of model parameters from simulated
data sets (parametric bootstrap)
2. Obtain simulated data set where fitted parameters are
closest to “True” Model
•
Using data set from 2, increase sigma in M
using NGA M values. Obtain mean parameter
from 500 simulations of uncertain M
July 19, 2004
Peer-NGA Project
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Simulated Data
ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
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ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
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ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
18
ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
19
ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
20
ln( pgarock )  C1  C2 M  C3 ln( R  eC4 C5M ) )  C6 R
ln( pgasoil )  ln( pgarock )  SC C7  C8 ln( pgarock  0.05)
 pga  V1  V2M
July 19, 2004
Peer-NGA Project
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Missing Predictor Variables
• Site classification variables
– VS30, NEHRP Categories, Other Site Categories,
– Depth to VS of 1.5 km/sec
• Rupture geometry variables
– Directivity variables
– Hanging wall/footwall determinations
– Confined to smaller events/distant recordings where
effect is believed to be minimal?
July 19, 2004
Peer-NGA Project
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Reason for Missing Predictors
• Independent of all data
• Dependent on value of the missing predictor
• Dependent on the values of other predictors
July 19, 2004
Peer-NGA Project
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Pattern of Missing Predictors
Univariate
Monotone
Special
July 19, 2004
Random
Peer-NGA Project
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Missing Data Methods
• Complete-case analysis
– Easily implemented
– Valid inferences when missing predictors
depend upon data
– May lead to elimination of a lot of useful
information
– Useful starting result
July 19, 2004
Peer-NGA Project
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Missing Data Methods
• Imputation
– Missing X’s estimated from correlations with
other X’s or X’s and Y’s
– Typically down weight imputed observations
• Multiple Imputation
– Simulate multiple data sets incorporating
uncertainty in estimated missing X’s
– Provides method for incorporation effect of
uncertainty in imputation on estimation
July 19, 2004
Peer-NGA Project
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Missing Data Methods
• Maximum Likelihood
– Need a model for joint distribution of Y and X,
including missing X’s
– Random missing patterns will need iterative
approaches
• Bayesian Simulation Methods
– e.g. Gibbs sampler
– Computer intensive (multiple thousands of
simulations)
July 19, 2004
Peer-NGA Project
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Missing/Uncertain Data
• If missing X’s are estimated from an external
model (e.g. VS30– becomes an uncertain predictor
problem
• Simulation methods appear to be useful for both
problems
• Implement these methods at later stage of model
development to obtain final coefficients and their
uncertainty
• Develop an implementation of each developer’s
final model to quantify the effects of
missing/uncertain data and provide parameter
uncertainty
July 19, 2004
Peer-NGA Project
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