1. Task 6 Statistical Approaches Scope of Work Bob Youngs NGA Workshop #5 March 25, 2003

2. Working Group 6 Norm Abrahamson David Brillinger Brian Chiou Bob Youngs

3. Primary Objectives Identify regression techniques that address uncertain/missing predictor variables, multiple levels of overlapping correlation in the residuals, and censoring/truncation of response Assess the significance of these issues in developing ground motion models Provide statistical tools to the NGA developers to assist them in addressing these issues

4. Progress to Date Treatment of Data Censoring/Truncation –Have identified an approach and begun implementation Treatment of correlations due to cross- classification of data (earthquake terms and site terms) –Have identified one method for analysis, but may not be an important issue in NGA

5. Progress to Date (cont’d) Treatment of other correlations (spatial within a given earthquake, and between frequencies) –Have not determined extent of need for treatment in NGA Treatment of missing/uncertain predictor variables –Identifying potential approaches to be explored

6. Treatment of Censored/Truncated Response Data

7. Standard Statistical Model

8. Censored Data Known number of recordings where value of y i < Z censor and value of x i is known (McLaughlin, 1991)

9. Censored Data Statistical Model

10. Truncated Data Unknown number of recordings where value of y i < Z trunc, value of x i is unknown (Toro, 1981)

11. Truncated Data Statistical Model

12. Example Large Synthetic Data Set (1000) ln(y)=  1 +  2 ln(r +  3 ) +  4 r

13. Fit to Censored/Truncated Data Ignoring Effect

14. Fit Using Censored Data Model

15. Fit Using Truncated Data Model

16. Example Small Synthetic Data Set (20) ln(y)=  1 +  2 ln(r +  3 ) +  4 r

17. Fit to Censored/Truncated Data Ignoring Effect

18. Fit Using Censored Data Model

19. Fit Using Truncated Data Model

20. Example Model Parameters

21. Minimum PGA versus Date of Earthquake in NGA Data Set

22. Minimum PGA versus Number of Records/Earthquake in NGA Data Set

23. Addition Work to be Done Incorporate into random effects model Investigate stability of estimation algorithms – maximum likelihood appears to be primary approach Evaluate sensitivity to selection of truncation level – treat as uncertain?

24. Treatment of Correlations in Response Data (Peak Motions)

25. Source and Site Data Correlations Earthquake effect – correlation in peak motions from the i th earthquake –presently incorporated by random effects and two-stage regression approaches Site effect – correlations in peak motions recorded at the j th site. –This effect is cross-classified with the earthquake effect – eliminates block-diagonal variance matrix, requiring “tricks”

26. Potential Data Correlations from Earthquake and Site Classifications

27. Tentative Conclusions Earthquake effect already addressed by developers Cross-classification by site effect term not a significant issue because of limited number of sites with many recordings –Need to do some testing with simulated data sets to confirm this conclusion

28. Additional Correlations Spatial Correlation of adjacent sites –Readily handled as nested classifications provided one has the correlation model –Need to investigate the potential extent in NGA data Correlation between adjacent spectral frequencies in a “global” regression –Is this of interest to then developers?

29. Treatment of Missing or Uncertain Predictor Variables

30. Missing Predictor Variables Site classification variables – V S30, NEHRP Categories, Other Site Categories, –Depth to V S of 1.0 and 2.5 km/sec Rupture geometry variables –Directivity variables –Hanging wall/footwall determinations –Confined to smaller events/distant recordings where effect is believed to be minimal?

31. Possible Approaches Estimation of variable by an external model –Example: correlation of V S30 with surficial geology Correlations with other variables in the NGA data set –Technique used in multivariate normal models

32. Treatment of Uncertainty in Predictor Variables Magnitude uncertainty –partition of earthquake random effect into an magnitude error term and an event term (Rhodes, 1997) Propagation of variable uncertainty into resulting model parameter uncertainty –Formal errors in variable methods –Simulation methods