1. Task 6 Statistical Approaches
Bob Youngs
NGA Workshop #6
July 19, 2004
July 19, 2004
Peer-NGA Project

2. Truncated Data
Unknown number of recordings where value of yi < Ztrunc , value of xi is unknown
(Toro, 1981)
July 19, 2004
Peer-NGA Project

3. Truncated Data Statistical Model
July 19, 2004
Peer-NGA Project

4. Fit to Truncated Data Ignoring Effect
July 19, 2004
Peer-NGA Project

5. Fit Using Truncated Data Model
July 19, 2004
Peer-NGA Project

6. Fit to Simulated Data
July 19, 2004
Peer-NGA Project

7. Fit to Truncated Simulated Data
July 19, 2004
Peer-NGA Project

8. Uncertain and Missing Predictor Variables
Uncertain predictors
Magnitude
Distance/Rupture Geometry
Site parameters (discrete and continuous)
Missing predictors
Rupture Geometry (for smaller events)
July 19, 2004
Peer-NGA Project

9. 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

10. Magnitude Uncertainty (Rhoades, BSSA, 1997)
Start with random (mixed) effects model
Reported magnitude, , contains error δi [N(0,si2)]
Revised mixed effects model
Solution obtained using “standard” approaches, including analytical inversion of variance matrix
July 19, 2004
Peer-NGA Project

11. Magnitude Uncertainty for NGA
Models likely to be non-linear in magnitude
Reported magnitude, , contains error δi [N(0,si2)]
Revised mixed effects model
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

12. 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
July 19, 2004
Peer-NGA Project

13. Simulation Extrapolation (continued)
Define a functional relationship for
Extrapolate back to λ = -1
Coefficients -1 1 2
July 19, 2004
Peer-NGA Project

14. 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
July 19, 2004
Peer-NGA Project

15. Assess the Effect of Magnitude Uncertainty
Start with a “True” Model
Simulate PGA values from “True” model using NGA M-R disribution
Calculate mean of model parameters from simulated data sets (parametric bootstrap)
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

16. Simulated Data
July 19, 2004
Peer-NGA Project

17. July 19, 2004
Peer-NGA Project

18. July 19, 2004
Peer-NGA Project

19. July 19, 2004
Peer-NGA Project

20. July 19, 2004
Peer-NGA Project

21. July 19, 2004
Peer-NGA Project

22. 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

23. 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

24. Pattern of Missing Predictors
Univariate
Monotone
Special
Random
July 19, 2004
Peer-NGA Project

25. 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

26. Missing Data Methods Imputation Multiple 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

27. Missing Data Methods Maximum Likelihood Bayesian Simulation Methods
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

28. 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