Task 6
Statistical 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 crossclassification 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
ln( yi )   ( xi , β)   i
Likelihood of observed data
L
f
N
i  recorded
( yi xi , β)
Solved by maximizing the log(Likeli hood)
ln( L) 
  ln( 
i  recorded
2
) / 2  ln( yi )   ( xi , β) / 2 2
2
or by minimizing the sum of squared difference s
SS 
 ln( y )   ( x , β)
2
i  recorded
i
i

Censored Data
• Known number of
recordings where
value of yi < Zcensor
and value of xi is
known
10
PGA
1
0.1
Zcensor
0.01
0.001
1
10
100
Distance
1000
(McLaughlin, 1991)
Censored Data Statistical Model
Likelihood of observed data
L

f N ( yi xi , β)
i  recorded
F
N
j  censored
( Z censor x j , β)
Solved by maximizing the log(Likeli hood)
ln( L) 
  ln( 
i  recorded

2
) / 2  ln( yi )   ( xi , β) / 2 2
2
 ln F
j  censored
N
( Z censor x j , β)


Truncated Data
• Unknown number of
recordings where
value of yi < Ztrunc ,
value of xi is unknown
10
PGA
1
0.1
Ztrunc
0.01
0.001
1
10
100
Distance
1000
(Toro, 1981)
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
N
( Z trunc xi , β)

Example Large Synthetic Data Set (1000)
ln(y)=1 + 2ln(r + 3) + 4r
10
Acceleration
1
> 0.03g
< 0.03g
0.1
Generating function
Fit to all data
0.01
0.001
0.1
1
10
Distance
100
1000
Fit to Censored/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
Distance
100
1000
Fit Using Censored Data Model
10
Acceleration
1
> 0.03g
< 0.03g
Generating function
0.1
Fit to all data
Censored fit
Censored x's
0.01
0.001
0.1
1
10
Distance
100
1000
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
Distance
100
1000
Example Small Synthetic Data Set (20)
ln(y)=1 + 2ln(r + 3) + 4r
10
Acceleration
1
> 0.03g
< 0.03g
0.1
Generating function
Fit to all data
0.01
0.001
1
10
100
Distance
1000
Fit to Censored/Truncated Data
Ignoring Effect
10
Acceleration
1
> 0.03g
Generating function
0.1
Fit to all data
Fit to data > 0.03g
0.01
0.001
1
10
100
Distance
1000
Fit Using Censored Data Model
10
Acceleration
1
> 0.03g
< 0.03g
Generating function
0.1
Fit to all data
Censored fit
censored x's
0.01
0.001
1
10
100
Distance
1000
Fit Using Truncated Data Model
10
Acceleration
1
> 0.03g
Generating function
0.1
Fit to all data
Truncated fit
0.01
0.001
1
10
100
Distance
1000
Example Model Parameters
Case
Number of
Records
Model
1
2
3
4

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 versus
Date of Earthquake in NGA Data Set
10
10
1938-1970
1981-1990
1
Minimum PGA
Minimum PGA
1
0.1
0.01
0.001
0.1
0.01
0.001
0.0001
0.0001
4
4.5
5
5.5
6
6.5
7
7.5
8
4
Magnitude
10
5
5.5
6
6.5
7
7.5
8
6.5
7
7.5
8
Magnitude
1991-2002
1971-1980
1
Minimum PGA
1
Minimum PGA
4.5
10
0.1
0.01
0.1
0.01
0.001
0.001
0.0001
0.0001
4
4.5
5
5.5
6
Magnitude
6.5
7
7.5
8
4
4.5
5
5.5
6
Magnitude
Minimum PGA versus
Number of Records/Earthquake in NGA
Data Set
10
10
1 to 5
11 to 50
1
Minimum PGA
Minimum PGA
1
0.1
0.01
0.001
0.1
0.01
0.001
0.0001
0.0001
4
4.5
5
5.5
6
6.5
7
7.5
8
4
Magnitude
10
5
10
5.5
6
6.5
7
7.5
8
6.5
7
7.5
8
Magnitude
>50
6 to 10
1
Minimum PGA
1
Minimum PGA
4.5
0.1
0.01
0.1
0.01
0.001
0.001
0.0001
0.0001
4
4.5
5
5.5
6
Magnitude
6.5
7
7.5
8
4
4.5
5
5.5
6
Magnitude
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
yij   ( xij , β)  
earthquake effect
i

site effect
j

• 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”
Potential Data Correlations from
Earthquake and Site Classifications
Number of
Number of Recordings per
Earthquakes
Earthquake
56
1
21
2
16
3
5
4
9
5
27
6-10
11
11-21
20
22-83
6
118-420
Number of
Stations
648
235
149
119
95
145
17
Number of
Recordings
per Station
1
2
3
4
5
6
7-10
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