Parameter Selection, Model Calibration, and Uncertainty Propagation for Physical Models Ralph C. Smith Department of Mathematics North Carolina State University Heat Model Experimental Setup d 2 T s dx 2 = 2(a + b) ab h k [T s (x) − T amb ] dT s dx (0) = Φ k , dT s dx (L)= h k [T amb − T s (L)]
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Parameter Selection, Model Calibration, and … Selection, Model Calibration, and Uncertainty Propagation for Physical Models Ralph C. Smith Department of Mathematics North Carolina
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Parameter Selection, Model Calibration, and Uncertainty Propagation for Physical Models
Ralph C. Smith Department of Mathematics
North Carolina State University
Heat Model!
Experimental Setup!
d2Ts
dx2=
2(a+ b)
ab
h
k[Ts(x)− Tamb]
dTs
dx(0) =
Φ
k,dTs
dx(L) =
h
k[Tamb − Ts(L)]
Heat Model Example Experimental Setup and Data:!
Note:!
10 20 30 40 50 60 7020
30
40
50
60
70
80
90
100
Location (cm)
Tem
pera
ture
(o C)
Aluminum Rod Data
Steady State Model:!
Objectives: Employ Bayesian analysis for!• Model calibration!
• Uncertainty propagation!• Experimental design!
d2Ts
dx2=
2(a+ b)
ab
h
k[Ts(x)− Tamb]
dTs
dx(0) =
Φ
kdTsdx (L) = h
k [Tamb − Ts(L)]
Parameter set q = [h, k,Φ] is not identifiable
Statistical Inference
Goal: The goal in statistical inference is to make conclusions about a phenomenon based on observed data.
Frequentist: Observations made in the past are analyzed with a specified model. Result is regarded as confidence about state of real world.
• Probabilities defined as frequencies with which an event occurs if experiment is repeated several times.
• Parameter Estimation:
o Relies on estimators derived from different data sets and a specific sampling distribution.
o Parameters may be unknown but are fixed and deterministic.
Bayesian: Interpretation of probability is subjective and can be updated with new data.
• Parameter Estimation: Parameters are considered to be random variables having associated densities.
Bayesian Model Calibration Bayesian Model Calibration:
• Parameters assumed to be random variables
Bayes’ Theorem:
P (A|B) =P (B|A)P (A)
P (B)
Example: Coin Flip
π(q|υ) = π(υ|q)π0(q)�Rp π(υ|q)π0(q)dq
Υi(ω) =
�0 , ω = T
1 , ω = H
Likelihood:
π(υ|q) =N�
i=1
qυi(1− q)1−υi
= q�
υi(1− q)N−�
υi
= qN1(1− q)N0
Posterior with Noninformative Prior:
π(q|υ) = qN1(1− q)N0
� 10 qN1(1− q)N0dq
=(N + 1)!
N0!N1!qN1(1− q)N0
π0(q) = 1
Bayesian Model Calibration
Bayesian Model Calibration:
• Parameters considered to be random variables with associated densities.
Problem:
• Often requires high dimensional integration;
o e.g., p = 18 for MFC model
o p = thousands to millions for some models
Strategies:
• Sampling methods
• Sparse grid quadrature techniques
π(q|υ) = π(υ|q)π0(q)�Rp π(υ|q)π0(q)dq
Markov Chain Techniques Markov Chain: Sequence of events where current state depends only on last value.
Baseball:
• Assume that team which won last game has 70% chance of winning next game and 30% chance of losing next game.
• Assume losing team wins 40% and loses 60% of next games.
• Percentage of teams who win/lose next game given by
• Question: does the following limit exist?
States are S = {win,lose}. Initial state is p0 = [0.8, 0.2].
First clear some variables from possible previous runs. clear data model options
Next, create a data structure for the observations and control variables. Typically one could make a structure data that contains fields xdata and ydata. data.xdata = [28 55 83 110 138 225 375]'; % x (mg / L COD)
Delayed Rejection Adaptive Metropolis (DRAM) Construct credible and prediction intervals figure(5); clf out = mcmcpred(res,chain,[],x,modelfun); mcmcpredplot(out); hold on plot(data.xdata,data.ydata,'s'); % add data points to the plot xlabel('x [mg/L COD]'); ylabel('y [1/h]'); hold off title('Predictive envelopes of the model')
DRAM for Heat Example
Note:!
Website
• http://helios.fmi.fi/~lainema/mcmc/
• http://www4.ncsu.edu/~rsmith/
Steady State Model:!
10 20 30 40 50 60 7020
30
40
50
60
70
80
90
100
Location (cm)
Tem
pera
ture
(o C)
Aluminum Rod Data
d2Ts
dx2=
2(a+ b)
ab
h
k[Ts(x)− Tamb]
dTs
dx(0) =
Φ
kdTsdx (L) = h
k [Tamb − Ts(L)]
Parameter set q = [Φ, h, k] is not identifiable
DRAM for Heat Model with 3 Parameters: Results
Note:!
Notes:!• Cond(V) = 1e+35!• Data and model are not informing priors!
Parameter set q = [Φ, h, k] is not identifiable
Assignment 2 Parameter Heat Model:!
Notes:!
Assignment:!• Modify the posted 3 parameter code for the 2 parameter model. How do
your chains and results compare?!• Consider various chain lengths to establish burn-in.!
d2Ts
dx2=
2(a+ b)
ab
h
k[Ts(x)− Tamb]
dTs
dx(0) =
Φ
kdTsdx (L) = h
k [Tamb − Ts(L)]
• Parameter set q = [h,Φ] is now identifiable• Set k = 2.37 W/cm C, which is the physical value for aluminum