inferring release characteristics from an atmospheric ... · diablo canyon nuclear power plant 3...
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1
Inferring Release Characteristics From an Atmospheric Dispersion Model
Bruno Sansó www.ams.ucsc.edu/~brunoDepartment of Applied Mathematics and Statistics University of California Santa Cruz
The CastThe work presented in this talk was done in collaboration with
2
Devin Francom, LANLVera Bulaevskaya, TCC
Donald Lucas, LLNL
Matthew Simpson, LLNL
Diablo Canyon Nuclear Power Plant
3
35.0
35.1
35.2
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−120.8 −120.6 −120.4longitude
latitude
101102103104105
Background SF6
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local time
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Time Series
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SF6
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cent
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gm
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8 10 12 14 16 18
101
102
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local time
site 317
site 317
site 342
site 342
site 507site 507
Diablo Canyon Nuclear Power Plant
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146 Kilos of SF6 where released from the Diablo Canyon plant on Sept. 4, 1986 for 8 hours, starting at 8:00.
35.0
35.1
35.2
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−120.8 −120.6 −120.4longitude
latitude
101102103104105
Background SF6
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local time
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Time Series
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local time
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SF6
con
cent
ratio
n (n
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3 )
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8 10 12 14 16 18
101
102
103
104
local time
site 317
site 317
site 342
site 342
site 507site 507
Diablo Canyon Nuclear Power Plant
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146 Kilos of SF6 where released from the Diablo Canyon plant on Sept. 4, 1986 for 8 hours, starting at 8:00.
Air samples were obtained from 7:00 to 18:00 at 150 sites. 24% are missing. 35.0
35.1
35.2
35.3
−120.8 −120.6 −120.4longitude
latitude
101102103104105
Background SF6
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local time
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102
103
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Time Series
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local time
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102
103
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SF6
con
cent
ratio
n (n
gm
3 )
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8 10 12 14 16 18
101
102
103
104
local time
site 317
site 317
site 342
site 342
site 507site 507
FLEXPART Simulations
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The FLEXPART simulator is used to explore the dispersion of the release.
35.0
35.1
35.2
35.3
−120.8 −120.6 −120.4longitude
latitude
101102103104105
site 317
site 317
site 342
site 342
site 507 site 507
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local time
101
102
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Time Series
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local time
101
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SF6
con
cent
ratio
n (n
gm
3 )
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10 15 20
local time
101
102
103
104
105
local time
FLEXPART Simulations
4
The FLEXPART simulator is used to explore the dispersion of the release.
35.0
35.1
35.2
35.3
−120.8 −120.6 −120.4longitude
latitude
101102103104105
site 317
site 317
site 342
site 342
site 507 site 507
●●●●
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local time
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102
103
104
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Time Series
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local time
101
102
103
104
105
SF6
con
cent
ratio
n (n
gm
3 )
●●●●●●●
●
●●
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10 15 20
local time
101
102
103
104
105
local time
18,000 different combinations of the 11 input parameters of FLEXPART are sampled from a latin hypercube. These result in 18,000 plumes varying in space and time.
Input Parameters
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Continuous Input Parameters
Input Lower Bound Upper Bound True ValueLatitude 35.1977 35.2250 35.2111Longitude -120.87 -120.83 -120.8543Altitude 1 10 2
Start Time 7:00 9:00 8:00Duration 6 10 8Amount 10 1000 146.016
Categorical Inputs
Input Number of values
Pre-release Initialization time 2Boundary Layer
Model 3Nudging 3
Reanalysis 3Land Model 3
There are five nested domains for WRF models. Each combination of the five categorical variables produces a different wind field at 300 meters resolution.
EmulatorWe build an emulator for the computer output corresponding to location s, time t and input values x, by using the representation on empirical orthogonal functions
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The EOFs are calculated from the model runs
yc(s, t, x) =kX
i=1
Ki(s, t)wi(x) + u(s, t)
EmulatorWe build an emulator for the computer output corresponding to location s, time t and input values x, by using the representation on empirical orthogonal functions
6
The EOFs are calculated from the model runs
yc(s, t, x) =kX
i=1
Ki(s, t)wi(x) + u(s, t)
For the truncation we take a non-Gaussian error
This is important in order to propagate truncation uncertaintyj = 1, . . . , nx]
u(s, t) ⇠ Unif [yc(s, t, xj)�kX
i=1
Ki(s, t)wi(xj),
Estimating the EOF coefficientsTo estimate the coefficients in the EOF we set:
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Regression Splines
Somehand-pickedknots
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0.0 0.2 0.4 0.6 0.8 1.0
05
10
Spline Fit
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Basis Functions
x
h(x)
wi(x) = ⌘i(x) + ✏iwhere
⌘(x) = a0 +MX
m=1
amBm(x)
is a representation on adaptive spline basis composed of M (unknown) terms, that uses products of hockey sticks with varying signs, number of interactions, and unknown knots.
Categorical an Continuous InputsOur application requires the emulator to handle continuous and categorical inputs. Assume that x1 and x2 are continuous, and x3 and x4 are categorical, then
8
B(x) = [s1(x1 � t1)]↵+[s2(x2 � t2)]
↵+1x32C31x42C4
where t1 and t2 are the knots andsi = ±1
and Ci corresponds to one or more categories of the i-th variable.
Categorical an Continuous InputsOur application requires the emulator to handle continuous and categorical inputs. Assume that x1 and x2 are continuous, and x3 and x4 are categorical, then
8
• Allows for basis functions specific to some categories• Allows for basis functions common to all categories• Learn from the data the categorical variables in each basis function, if any
B(x) = [s1(x1 � t1)]↵+[s2(x2 � t2)]
↵+1x32C31x42C4
where t1 and t2 are the knots andsi = ±1
and Ci corresponds to one or more categories of the i-th variable.
Emulation Performance
Predictions at 15 of the of the 137 locations considered for a held out configuration of the input parameters.
9
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Morro Bay
101
102
103
104
simulatorsurrogate
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
San Luis Obisbo
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
1 mile from plant
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
4 miles from plant
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Whalers Island
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Avila Beach
101
102
103
104
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
coastline extent of plume
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
midway up See Canyon
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
mouth of See Canyon
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
1 mile inland along Hwy 101
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
inland from coastline extent
101
102
103
104
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Pismo Beach
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Edna05
:00
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Arroyo Grande
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.si
m[m
at[ll
, ], h
oldo
ut.s
im[Y
Y.us
e]]
Santa Maria
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
time
conc
entra
tion
Emulation Performance
10
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Morro Bay
101
102
103
104
simulatorsurrogate
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
San Luis Obisbo
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
1 mile from plant
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
4 miles from plant
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Whalers Island
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Avila Beach
101
102
103
104
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
coastline extent of plume
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
midway up See Canyon
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
mouth of See Canyon
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
1 mile inland along Hwy 101
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
inland from coastline extent
101
102
103
104
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Pismo Beach
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Edna
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Arroyo Grande
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Index
Y.sim
[mat
[ll, ],
hol
dout
.sim
[YY.
use]
]
Santa Maria
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
time
conc
entra
tion
Predictions at 15 of the of the 137 locations considered for another held out configuration of the input parameters.
Global Sensitivity
Analytic expressions are available for the time and space-varying Sobol coefficients for the different inputs and interactions. 11
Morro Bay
0.0
0.2
0.4
0.6
other (>3rd)other (2nd/3rd)7x111x61x21110765421
11 − reanalysis10 − nudge9 − LSM8 − PBL7 − init time6 − amount5 − duration4 − start time3 − altitude2 − longitude1 − latitude
San Luis Obisbo 1 mile from plant 4 miles from plant Whalers Island
Avila Beach
0.0
0.2
0.4
0.6
coastline extent of plume midway up See Canyon mouth of See Canyon 1 mile inland along Hwy 101
inland from coastline extent
0.0
0.2
0.4
0.6
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Pismo Beach
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0Edna
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Arroyo Grande
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
Santa Maria
05:0
0
07:0
0
09:0
0
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
time
parti
tione
d va
rianc
e
Observation Equation with Gaussian error
12
Observationsz }| {yF (s, t) = ⇣(s, t)| {z }
True System
+
Oberv. Errorz }| {v(s, t)
⇣(s, t) = yP (s, t, ✓)| {z }Best Estimate
+
Discrepancyz }| {��(s, t)
System Equation with additive discrepancy with U[0,2] multiplication factor
p(✓|Y F , Y C)Calibration:
Observation Equation with Gaussian error
12
Observationsz }| {yF (s, t) = ⇣(s, t)| {z }
True System
+
Oberv. Errorz }| {v(s, t)
⇣(s, t) = yP (s, t, ✓)| {z }Best Estimate
+
Discrepancyz }| {��(s, t)
System Equation with additive discrepancy with U[0,2] multiplication factor
We estimate the discrepancy by fitting the emulator at the prior mean of the inputs. We then fit the discrepancy using adaptive splines (BASS). 𝛾 provides information about the relevance of the discrepancy.
p(✓|Y F , Y C)Calibration:
Posteriors for Continuous Inputs
13
No discrepancy Discrepancy
Posteriors for Categorical Inputs
14
Index
1
init time
9 15
0.0
0.2
0.4
0.6
0.8
1.0
prob
abilit
y
Index
1
PBL
YSU
MYJ
MYN
N
Index
1
LSM
TD
Noa
h
RUC
Index
1
nudge
none low
high
Index
1
reanalysis
NAR
R
ECM
WF
CFS
R
No discrepancy
DiscrepancyIndex
1
init time
9 15
0.0
0.2
0.4
0.6
0.8
1.0
prob
abilit
y
Index
1
PBL
YSU
MYJ
MYN
N
Index
1
LSMTD
Noa
h
RUC
Index
1
nudge
none low
high
Index1
reanalysis
NAR
R
ECM
WF
CFS
R
Calibrated Predictions
15
No discrepancy
Discrepancy 𝛾∈[0.58,1.02]
Index
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●
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●
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315
101
102
103
104
Index
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316
Index
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●
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317
Index
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● ●
318
Index
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319
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102
103
104
Index
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●
320
Index
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321
Index
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●
●
322
Index
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●
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●
●
323
101
102
103
104
06:0
0
08:0
0
10:0
0
12:0
0
14:0
0
16:0
0
18:0
0
20:0
0
22:0
0
Index
●
●●
●
●
●
●
●
● ●
324
06:0
0
08:0
0
10:0
0
12:0
0
14:0
0
16:0
0
18:0
0
20:0
0
22:0
0
Index
● ●
●
●
●
●
●
●
●
325
06:0
0
08:0
0
10:0
0
12:0
0
14:0
0
16:0
0
18:0
0
20:0
0
22:0
0
time
conc
entra
tion
● observationsmean predictions
Index
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315
101
102
103
104
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104
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320
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322
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●
323
101
102
103
104
06:00
08:00
10:00
12:00
14:00
16:00
18:00
20:00
22:00
Index
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●
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●
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● ●
324
06:00
08:00
10:00
12:00
14:00
16:00
18:00
20:00
22:00
Index
● ●
●
●
●
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●
325
06:00
08:00
10:00
12:00
14:00
16:00
18:00
20:00
22:00
time
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1 2 3 4
12
34
Without Discrepancy
observed
predicted
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1 2 3 4
12
34
With Discrepancy
observed
predicted
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1 2 3 4
12
34
Without Discrepancy
observed
predicted
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1 2 3 41
23
4
With Discrepancy
observed
predicted
Calibrated Release Location
16
4000
8000
12000 16000 20000
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reported release locationcorrected release locationmeasurement sitesprior rangeposterior contours
The release location was originally mis-reported. Our posterior distribution reveals that a second source of information corresponds to a much more probable location
Analysis of Discrepancies
17
Clusters of discrepancy curves identifying clear location patterns.
10 15 20
−10
12
Cluster 1
time
disc
repa
ncy
10 15 20
−10
12
Cluster 2
timedi
scre
panc
y
10 15 20
−10
12
Cluster 3
time
disc
repa
ncy
10 15 20
−10
12
Cluster 4
time
disc
repa
ncy
10 15 20
−10
12
Cluster 5
time
disc
repa
ncy
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−120.9 −120.8 −120.7 −120.6 −120.5
35.0
35.1
35.2
35.3
LongitudeLa
titud
e
● cluster 1cluster 2cluster 3cluster 4cluster 5
Conclusions
18
Conclusions
18
• Adaptive regression splines have been effectively used to tackle an emulation and calibration problem for a massive computer experiment with large spatio-temporal output.
Conclusions
18
• Adaptive regression splines have been effectively used to tackle an emulation and calibration problem for a massive computer experiment with large spatio-temporal output.
• Our method scales well to large amounts of data, providing accurate emulation and prediction.
Conclusions
18
• Adaptive regression splines have been effectively used to tackle an emulation and calibration problem for a massive computer experiment with large spatio-temporal output.
• Our method scales well to large amounts of data, providing accurate emulation and prediction.
• Our method can handle continuous and categorical inputs.
Conclusions
18
• Adaptive regression splines have been effectively used to tackle an emulation and calibration problem for a massive computer experiment with large spatio-temporal output.
• Our method scales well to large amounts of data, providing accurate emulation and prediction.
• Our method can handle continuous and categorical inputs.• We able to perform a time and space-varying sensitivity analysis of the inputs based on accurate analytic expressions for the global sensitivity coefficients.
Conclusions
18
• Adaptive regression splines have been effectively used to tackle an emulation and calibration problem for a massive computer experiment with large spatio-temporal output.
• Our method scales well to large amounts of data, providing accurate emulation and prediction.
• Our method can handle continuous and categorical inputs.• We able to perform a time and space-varying sensitivity analysis of the inputs based on accurate analytic expressions for the global sensitivity coefficients.
• The method uses a fully probabilistic approach that allows to account for all sources of variability and provides a coherent quantification of the uncertainty.
References
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysisof Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysisof Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
• Francom, D., Sansó B., Kupresanin, A., and Johannesson, G. (2018), “Sensitivity AnalysisandEmulation for Functional Data using Bayesian Adaptive Splines,” Statistica Sinica, 28, 791-816.
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysisof Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
• Francom, D., Sansó B., Kupresanin, A., and Johannesson, G. (2018), “Sensitivity AnalysisandEmulation for Functional Data using Bayesian Adaptive Splines,” Statistica Sinica, 28, 791-816.
• Francom, D., Sansó B., Bulaevskaya, V., Lucas, D., Simpson, M. (2018), “Inferring AtmosphericRelease Characteristics in a Large Computer Experiment using Bayesian Adaptive Splines”,submitted
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysisof Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
• Francom, D., Sansó B., Kupresanin, A., and Johannesson, G. (2018), “Sensitivity AnalysisandEmulation for Functional Data using Bayesian Adaptive Splines,” Statistica Sinica, 28, 791-816.
• Francom, D., Sansó B., Bulaevskaya, V., Lucas, D., Simpson, M. (2018), “Inferring AtmosphericRelease Characteristics in a Large Computer Experiment using Bayesian Adaptive Splines”,submitted
• Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008), “Computer model calibration usinghigh-dimensional output,” Journal of the American Statistical Association, 103.
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J.,Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shockexperiments with measurement error,” Journal of the American StatisticalAssociation, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysisof Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
• Francom, D., Sansó B., Kupresanin, A., and Johannesson, G. (2018), “Sensitivity AnalysisandEmulation for Functional Data using Bayesian Adaptive Splines,” Statistica Sinica, 28, 791-816.
• Francom, D., Sansó B., Bulaevskaya, V., Lucas, D., Simpson, M. (2018), “Inferring AtmosphericRelease Characteristics in a Large Computer Experiment using Bayesian Adaptive Splines”,submitted
• Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008), “Computer model calibration usinghigh-dimensional output,” Journal of the American Statistical Association, 103.
• Kennedy, M. C., and O’Hagan, A. (2001), “Bayesian calibration of computer models,” Journal of theRoyal Statistical Society. Series B, Statistical Methodology, 425–464.
19
References• Chakraborty, A., Mallick, B. K., Mcclarren, R. G., Kuranz, C. C., Bingham, D., Grosskopf, M. J., Rutter, E. M., Stripling, H. F., and Drake, R. P. (2013), “Spline-based emulators for radiative shock experiments with measurement error,” Journal of the American Statistical Association, 108, 411–428.
• Francom, D., Sansó, B. (2018), “BASS: An R Package for Fitting and Performing Sensitivity Analysis of Bayesian Adaptive Spline Surfaces”, Journal of Statistical Software, to appear.
• Francom, D., Sansó B., Kupresanin, A., and Johannesson, G. (2018), “Sensitivity Analysis and Emulation for Functional Data using Bayesian Adaptive Splines,” Statistica Sinica, 28, 791-816.
• Francom, D., Sansó B., Bulaevskaya, V., Lucas, D., Simpson, M. (2018), “Inferring Atmospheric Release Characteristics in a Large Computer Experiment using Bayesian Adaptive Splines”, submitted
• Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008), “Computer model calibration using high-dimensional output,” Journal of the American Statistical Association, 103.
• Kennedy, M. C., and O’Hagan, A. (2001), “Bayesian calibration of computer models,” Journal of the Royal Statistical Society. Series B, Statistical Methodology, 425–464.
19