Task 6Statistical Approaches

Scope of Work

Bob Youngs

NGA Workshop #5

March 25, 2003

Working Group 6

• Norm Abrahamson

• David Brillinger

• Brian Chiou

• Bob Youngs

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

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

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

Treatment of Censored/Truncated Response Data

Standard Statistical Model

recordediii

recordediii

recordediiiN

iii

xySS

xyL

xyfL

xy

2

222

),()ln(

sdifference squared of sum theminimizingby or

2/),()ln(2/)ln()ln(

hood)log(Likeli themaximizingby Solved

),(

data observed of Likelihood

),()ln(

β

β

β

β

Censored Data

• Known number of recordings where value of yi < Zcensor and value of xi is known

(McLaughlin, 1991)0.001

0.01

0.1

1

10

1 10 100 1000

Distance

PG

A

Zcensor

Censored Data Statistical Model

censoredjjcensorN

recordediii

recordedi censoredjjcensorNiiN

xZF

xyL

xZFxyfL

),(ln

2/),()ln(2/)ln()ln(

hood)log(Likeli themaximizingby Solved

),(),(

data observed of Likelihood

222

β

β

ββ

Truncated Data

• Unknown number of recordings where value of yi < Ztrunc , value of xi is unknown

(Toro, 1981)

0.001

0.01

0.1

1

10

1 10 100 1000

Distance

PG

A

Ztrunc

Truncated Data Statistical Model

recordediitruncN

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xZF

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),(1ln

2/),()ln(2/)ln()ln(

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data observed of Likelihood

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β

β

ββ

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

0.001

0.01

0.1

1

10

0.1 1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

< 0.03g

Generating function

Fit to all data

Fit to Censored/Truncated Data Ignoring Effect

0.001

0.01

0.1

1

10

0.1 1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

Generating function

Fit to all data

Fit to data > 0.03

Fit Using Censored Data Model

0.001

0.01

0.1

1

10

0.1 1 10 100 1000

Distance

Acc

eler

atio

n

> 0.03g

< 0.03g

Generating function

Fit to all data

Censored fit

Censored x's

Fit Using Truncated Data Model

0.001

0.01

0.1

1

10

0.1 1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

Generating function

Fit to all data

Truncated fit

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

0.001

0.01

0.1

1

10

1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

< 0.03g

Generating function

Fit to all data

Fit to Censored/Truncated Data Ignoring Effect

0.001

0.01

0.1

1

10

1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

Generating function

Fit to all data

Fit to data > 0.03g

Fit Using Censored Data Model

0.001

0.01

0.1

1

10

1 10 100 1000

Distance

Acc

eler

atio

n

> 0.03g

< 0.03g

Generating function

Fit to all data

Censored fit

censored x's

Fit Using Truncated Data Model

0.001

0.01

0.1

1

10

1 10 100 1000

Distance

Acc

eler

atio

n > 0.03g

Generating function

Fit to all data

Truncated fit

Example Model Parameters

Case Number of Records 1 2 3 4

Model 4.5 -1.6 20 -5.00E-03 0.5

Fit all data 1000 4.328 -1.549 20.1 -5.74E-03 0.502

Fit to data > 0.03 858 4.057 -1.547 16.8 0 0.500

Censored fit 858 + 142c 2.311 -1.012 13.5 -1.25E-02 0.507

Truncated fit 858 4.000 -1.470 18.9 -6.40E-03 0.511

Fit all data 20 0.889 -0.598 7.1 -1.59E-02 0.395

Fit to data > 0.03 16 2.391 -1.120 10.5 0 0.327

Censored fit 16+4c 0.268 -0.427 2.8 -1.68E-02 0.374

Truncated fit 16 0.486 -0.553 2.9 -9.07E-03 0.349

Minimum PGA versusDate of Earthquake in NGA Data Set

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

1938-1970

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

1971-1980

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

1981-1990

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

1991-2002

Minimum PGA versusNumber of Records/Earthquake in NGA

Data Set

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

1 to 5

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

6 to 10

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

11 to 50

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5 8

Magnitude

Min

imu

m P

GA

>50

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?

Treatment of Correlations in Response Data(Peak Motions)

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”

effectsitej

effectearthquakeiijij xy ),( β

Potential Data Correlations from Earthquake and Site Classifications

Number of Stations

Number of Recordings per Station

648 1235 2149 3119 495 5

145 617 7-10

Number of Earthquakes

Number of Recordings per

Earthquake56 121 216 35 49 527 6-1011 11-2120 22-836 118-420

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

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?

Treatment of Missing or Uncertain Predictor Variables

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?

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

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