Computer Model Validation via Dynamic Linear Modelfei/slides/poster_dlm.pdf · Computer Model...
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Computer Model Validation via Dynamic LinearModel
Fei Liu1 Liang Zhang1 Mike West1
1Institute of Statistics and Decision SciencesDuke University
Kickoff Workshop, SAMSI, September 10-14, 2006
Fei Liu, Liang Zhang, Mike West CMV-DLM
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Data
0 500 1000 1500 2000 2500 3000 3500
u = 0.5
u = 0.25
u = 0.35
u = 0.45
u = 0.55
u = 0.65
u = 0.75
Computer Model Ouptuts Data
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Statistical Modeling of theSpatially Correlated Computer Outputs
Xt(u1)Xt(u2)
. . .Xt(un)
=
Xt−1(u1) Xt−2(u1) . . . Xt−p(u1)Xt−1(u2) Xt−2(u2) . . . Xt−p(u2)
......
. . ....
Xt−1(un) Xt−2(un) . . . Xt−p(un)
φt,1φt,2
...φt,p
+
εt(u1)εt(u2)
...εt(un)
Random walk for the TVAR coefficients,
Φt =
φt ,1φt ,2
...φt ,p
; Φt = Φt−1 + wt
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Gaussian Stochastic Process For the TVARinnovations
Gaussian Random Field for εt :
εt(·) ∼ GP(0, vtc(·, ·))
For finite observations specifically,εt(u1)εt(u2)
...εt(un)
∼ MVN(0, Vt × Σ(u1, . . . , un))
Power Exponential Family of spatial correlation:
c(u, u′) = exp(−β | u − u
′ |),Σi,j = c(ui , uj)
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Discounting Variances
Discount factor δ1 for vt to allow its stochastic changes,
v−1t | Dt−1 ∼ G(δ1nt−1/2, δ1dt−1/2)
choose n0 = 1, d0 = var(X ).δ2 for Ct ,
wt ∼ MVN(mt , Ct)
Ct | Dt−1 = (1 − δ2)Ct−1/δ2
Ct−1 = Cov(Φt−1 | Dt−1)
choose m0 = 0, C0 = 10Ip×p
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DLM Representation
(F , G, V , W )t = (Ft , Gt , Vt , Wt)
F′t =
Xt−1(u1) Xt−2(u1) . . . Xt−p(u1)Xt−1(u2) Xt−2(u2) . . . Xt−p(u2)
......
. . ....
Xt−1(un) Xt−2(un) . . . Xt−p(un)
Gt = Ip×p
Vt = vtΣ(u1, . . . , un)
Finally, vt and Wt are sequentially specified.
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Backward Sampling
Gibbs Sampler
Interested in: ({v1, . . . vT}; {Φ1, . . . ,ΦT}; {β} | DT )
Sample (β | DT , v1:T ,Φ1:T ).
Sample (v1:T ,Φ1:T | DT , β).
Sample (v1:T | DT , β).
Sample (Φ1:T | v1:T , DT , β).
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Sample (Vt , t = 1, . . . , T | DT , β):
(a.) Forward filtering with unknown variances.(b.) Sample (V−1
T | DT , β) ∼ G(nT /2, dT /2).
(c.) Recursively sample vt , t = T − 1, . . . , 1 from,
v−1t = δ1v−1
t+1 + G ((1 − δ1)nt/2, dt/2)
Sample (Φ1:T | DT , v1:T , β):
(a.) Forward filtering again with known v1:T .
(b.) Sample (ΦT | DT , v1:T ) ∼ MVN(mT , CT ).
(c.) Recursively sample Φt , t = T − 1, . . . , 1 from,
(Φt | DT ,Φt+1, V1:T ) ∼ MVN ((1 − δ2)mt + δ2Φt+1, (1 − δ2)Ct)
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Posterior Distribution of β
0 200 400 600 800 1000 1200 1400 1600 1800 20001.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85Trace Plot of the MCMC Samples −− beta
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Lag
Auto
corre
latio
n
ACF of MCMC Samples −− beta
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.850
100
200
300
400
500
600Histogram of the Posterior Samples −− beta
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5Prior Density of beta
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Posterior Distribution of vt
0 500 1000 1500 2000 2500 3000 35007.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5Posterior Mean of the Standard Deviation
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Posterior Distribution of φ
0 500 1000 1500 2000 2500 3000 3500−3
−2
−1
0
1
2
3Posterior Mean of the AR Coefficients
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Spatial Interpolation —-Predict Output of a computer model with new input
At new input u, we can predict (approximate) the computermodel output its from the posterior draws.
(xt(u) | xt−1:t−p(u), Data,Φ1:T v1:T , β
)∼ N(µt(u), σ2
t (u))
µt(u) =∑
j
xt−j(u)φt ,j + ρt(u, u1:n)Σ−1(u1:n, β)
εt(u1)εt(u2)
...εt(un)
σ2
t (u) = Vt(1 − ρt(u, u1:n)Σ−1(u1:n, β)ρ(u, u1:n))
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Predictive Curve with Posterior Quantiles
1100 1120 1140 1160 1180 1200 1220 1240 1260 1280 1300−250
−200
−150
−100
−50
0
50
100
150
2001100 −− 1300 Sectional of Data with Confidence Bands
Posterior Predictive CurveTrue Data90% Confidence Bands90% Confidence Bands
2700 2720 2740 2760 2780 2800 2820 2840 2860 2880 2900−250
−200
−150
−100
−50
0
50
100
1502700 −− 2900 Sectional of Data with Confidence Bands
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Decomposition – Posterior Mean
0 500 1000 1500 2000 2500 3000−1
0
1
2
3
4
Data
Posterior Mean
Time
comp
comp
comp
comp
Decomposition Of the Posterior Mean
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Wave Lengths – Posterior Mean
0 500 1000 1500 2000 2500 30000
20
40
60
80
100
120Wavelength of the top 5 components
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Moduli – Posterior Mean
0 500 1000 1500 2000 2500 30000
0.2
0.4
0.6
0.8
1
1.2
1.4Moduli of the first 5 components
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