Task 6 Statistical Approaches Scope of Work

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Task 6Statistical ApproachesScope of Work

• Bob Youngs
• NGA Workshop 5
• March 25, 2003

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Working Group 6

• Norm Abrahamson
• David Brillinger
• Brian Chiou
• Bob Youngs

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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

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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

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Progress to Date (contd)

• 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

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Treatment of Censored/Truncated Response Data
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Standard Statistical Model
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Censored Data

• Known number of recordings where value of yi lt
  Zcensor and value of xi is known
• (McLaughlin, 1991)

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Censored Data Statistical Model
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Truncated Data

• Unknown number of recordings where value of yi lt
  Ztrunc , value of xi is unknown
• (Toro, 1981)

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Truncated Data Statistical Model
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Example Large Synthetic Data Set (1000)ln(y)?1
?2ln(r ?3) ?4r
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Fit to Censored/Truncated Data Ignoring Effect
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Fit Using Censored Data Model
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Fit Using Truncated Data Model
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Example Small Synthetic Data Set (20)ln(y)?1
?2ln(r ?3) ?4r
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Fit to Censored/Truncated Data Ignoring Effect
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Fit Using Censored Data Model
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Fit Using Truncated Data Model
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Example Model Parameters
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Minimum PGA versusDate of Earthquake in NGA Data
Set
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Minimum PGA versusNumber of Records/Earthquake
in NGA Data Set
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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?

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Treatment of Correlations in Response Data(Peak
Motions)
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Source and Site Data Correlations

• Earthquake effect correlation in peak motions
  from the ith earthquake
• presently incorporated by random effects and
  two-stage regression approaches
• Site effect correlations in peak motions
  recorded at the jth site.
• This effect is cross-classified with the
  earthquake effect eliminates block-diagonal
  variance matrix, requiring tricks

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Potential Data Correlations from Earthquake and
Site Classifications
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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

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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?

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Treatment of Missing or Uncertain Predictor
Variables
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Missing Predictor Variables

• Site classification variables
• VS30, NEHRP Categories, Other Site Categories,
• Depth to VS 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?

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Possible Approaches

• Estimation of variable by an external model
• Example correlation of VS30 with surficial
  geology
• Correlations with other variables in the NGA data
  set
• Technique used in multivariate normal models

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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