APPENDIX A: PROPOSED REGRESSION MODELS
We propose 30 regression models for each of the
“
collapse
”
and
“
unrepairable
”
response
variables. We define the probability of failure (either
“
collapse
”
or
“
unrepairable
”
) given a
transformed intensity measure,
x
,as
EQ-TARGET;temp:intralink-;ea1;62;587
p
ð
failure
j
x
Þ¼
π
ð
x
Þ
(A1)
where
π
ð
x
Þ
is a monotonically increasing function on the interval (0, 1). The logistic, hazard,
and cumulative normal models are three commonly used forms for
π
(Burnham and
Anderson 2002, pp. 195
–
196). These functions are
EQ-TARGET;temp:intralink-;x1;62;530
π
ð
x
Þ¼
1
1
þ
exp
ð
x
Þ
ð
logistic
Þ
π
ð
x
Þ¼
1
exp
½
exp
ð
x
Þð
hazard
Þ
π
ð
x
Þ¼
Φ
ð
x
Þð
cumulative normal
Þ
where
Φ
ð
·
Þ
represents the standard normal cumulative probability distribution.
Based on the discussion in the Results Section, we propose seven transformed intensity
measures from the two vector intensity measures. These seven are
EQ-TARGET;temp:intralink-;x1;62;427
x
1
¼
β
0
þ
β
1
log
10
S
a
þ
β
2
ε
þ
β
3
ð
log
10
S
a
Þ
ε
þ
β
4
ð
log
10
S
a
Þ
2
þ
β
5
ε
2
x
2
¼
β
0
þ
β
1
log
10
r
þ
β
2
θ
þ
β
3
ð
log
10
r
Þ
θ
þ
β
4
ð
log
10
r
Þ
2
þ
β
5
θ
2
x
3
¼
β
0
þ
β
1
log
10
S
a
þ
β
2
ε
þ
β
3
ð
log
10
S
a
Þ
ε
x
4
¼
β
0
þ
β
1
log
10
r
þ
β
2
θ
þ
β
3
ð
log
10
r
Þ
θ
x
5
¼
β
0
þ
β
1
log
10
S
a
x
6
¼
β
0
þ
β
1
log
10
PGD
x
7
¼
β
0
þ
β
1
log
10
PGV
In these equations,
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PGD
2
þ
PGV
2
p
and
θ
¼
tan
1
ð
PGV
∕
PGD
Þ
. We form 21
regression models by combining the three common models for
π
with these seven trans-
formed intensity measures. Table
A1
summarizes the proposed regression models; the
ones so far described are denoted Models 1
–
7, 11
–
17, and 21
–
27.
The remaining nine regression models are formed as products. We propose an additional
four intensity measure transformations:
EQ-TARGET;temp:intralink-;x1;62;218
x
8
¼
β
2
þ
β
3
ε
x
9
¼
β
0
þ
β
1
log
10
r
x
10
¼
β
2
þ
β
3
θ
x
11
¼
β
4
þ
β
5
ð
θ
Þ
For Models 8, 9, 18, 19, 28, and 29, we define the probability of failure as
EQ-TARGET;temp:intralink-;de2;62;127
p
ð
failure
j
x
i
;
x
j
Þ¼
π
ð
x
i
Þ
π
ð
x
j
Þ
(A2)
where
x
i
¼
x
5
and
x
j
¼
x
8
(Models 8, 18, and 28) or
x
i
¼
x
6
and
x
j
¼
x
7
(Models 9, 19, and
29; note that the parameter subscripts in the equation for
x
7
must be incremented to
ES
‐
1
distinguish the four parameters in these models). The contours of equal probability of failure
for these six models approximate
L
-shaped curves in the
S
a
‐
ε
and PGD-PGV planes. For
Models 10, 20, and 30, we define the probability of failure as
EQ-TARGET;temp:intralink-;da3;41;378
p
ð
failure
j
x
9
;
x
10
;
x
11
Þ¼
π
ð
x
9
Þ
π
ð
x
10
Þ
π
ð
x
11
Þ
(A3)
The contours of equal probability of failure for these three models approximate
V
-or
U
-shaped curves in the PGD-PGV plane.
We propose seven regression models to predict the IDR given an intensity measure value
and given that the response is also
“
repairable.
”
These models are
EQ-TARGET;temp:intralink-;da4;41;301
log
10
IDR
¼
β
0
þ
β
1
log
10
S
a
þ
β
2
ε
þ
β
3
ð
log
10
S
a
Þ
ε
þ
β
4
ð
log
10
S
a
Þ
2
þ
β
5
ε
2
þ
ε
1
(A4)
EQ-TARGET;temp:intralink-;da5;41;269
log
10
IDR
¼
β
0
þ
β
1
log
10
S
a
þ
β
2
ε
þ
β
3
ð
log
10
S
a
Þ
ε
þ
ε
2
(A5)
EQ-TARGET;temp:intralink-;da6;41;243
log
10
IDR
¼
β
0
þ
β
1
log
10
S
a
þ
ε
3
(A6)
EQ-TARGET;temp:intralink-;da7;41;216
log
10
IDR
¼
β
0
þ
β
1
log
10
PGD
þ
β
2
log
10
PGV
þ
β
3
ð
log
10
PGD
Þð
log
10
PGV
Þ
:::
þ
β
4
ð
log
10
PGD
Þ
2
þ
β
5
ð
log
10
PGV
Þ
2
þ
ε
4
ð
A7
Þ
EQ-TARGET;temp:intralink-;da8;41;170
log
10
IDR
¼
β
0
þ
β
1
log
10
PGD
þ
β
2
log
10
PGV
þ
β
3
ð
log
10
PGD
Þð
log
10
PGV
Þþ
ε
5
(A8)
EQ-TARGET;temp:intralink-;da9;41;143
log
10
IDR
¼
β
0
þ
β
1
log
10
PGD
þ
ε
6
(A9)
EQ-TARGET;temp:intralink-;da10;41;117
log
10
IDR
¼
β
0
þ
β
1
log
10
PGV
þ
ε
7
(A10)
where
ε
i
(
i
¼
1
;
2
;
3
:::
7
) is normally distributed with mean zero and variance
σ
2
i
.
Table A1.
Proposed regression models for the probability of
“
collapse
”
or
“
unrepairable,
”
denoted
π
Model
Transformed
intensity
measure
π
ð
x
) Model
Transformed
intensity
measure
π
ð
x
) Model
Transformed
intensity
measure
π
ð
x
)
1
x
1
Logistic 11
x
1
Hazard 21
x
1
Cumulative normal
2
x
2
Logistic 12
x
2
Hazard 22
x
2
Cumulative normal
3
x
3
Logistic 13
x
3
Hazard 23
x
3
Cumulative normal
4
x
4
Logistic 14
x
4
Hazard 24
x
4
Cumulative normal
5
x
5
Logistic 15
x
5
Hazard 25
x
5
Cumulative normal
6
x
6
Logistic 16
x
6
Hazard 26
x
6
Cumulative normal
7
x
7
Logistic 17
x
7
Hazard 27
x
7
Cumulative normal
8
x
5
,
x
8
Logistic 18
x
5
,
x
8
Hazard 28
x
5
,
x
8
Cumulative normal
9
x
6
,
x
7
Logistic 19
x
6
,
x
7
Hazard 29
x
6
,
x
7
Cumulative normal
10
x
9
,
x
10
,
x
11
Logistic 20
x
9
,
x
10
,
x
11
Hazard 30
x
9
,
x
10
,
x
11
Cumulative normal
Note: A regression model is composed of one or more transformed intensity measure(s),
x
, and a functional form for
π
.
ES-2