Parameter Selection, Model Calibration, and Uncertainty Propagation for Physical Models

Size: px

Start display at page:

Download "Parameter Selection, Model Calibration, and Uncertainty Propagation for Physical Models"

Jody McLaughlin
5 years ago
Views:

1 Parameter Selection, Model Calibration, and Uncertainty Propagation for Physical Models Ralph C. Smith Department of Mathematics North Carolina State University Experimental Setup! d 2 T s 2(a + b) = dx2 ab Heat Model! h k [T s(x) T amb ] dt s dx (0) = Φ k, dt s dx (L) =h k [T amb T s (L)]

2 Experimental Setup and Data:! Heat Model Example Aluminum Rod Data 80 Temperature ( o C) Steady State Model:! d 2 T s 2(a + b) = dx2 ab dt s dx (0) = Φ k h k [T s(x) T amb ] dt s dx (L) = h k [T amb T s (L)] Location (cm) Objectives: Employ Bayesian analysis for! Model calibration! Uncertainty propagation! Experimental design! Note:! Parameter set q =[h, k, Φ] is not identifiable

3 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.

4 Bayes Theorem: P (A B) = Example: Coin Flip Likelihood: π(υ q) = P (B A)P (A) P (B) Υ i (ω) = N q υ i (1 q) 1 υ i i=1 Bayesian Model Calibration 0, ω = T 1, ω = H = q υ i (1 q) N υ i = q N 1 (1 q) N 0 Posterior with Noninformative Prior: π 0 (q) =1 Bayesian Model Calibration: Parameters assumed to be random variables π(q υ) = π(υ q)π 0 (q) R p π(υ q)π 0 (q)dq π(q υ) = qn 1 (1 q) N qn 1 (1 q) N 0dq = (N + 1)! N 0!N 1! qn 1 (1 q) N 0

5 Bayesian Model Calibration: Parameters considered to be random variables with associated densities. Bayesian Model Calibration π(q υ) = π(υ q)π 0 (q) R p π(υ q)π 0 (q)dq 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

6 Markov Chain Techniques Markov Chain: Sequence of events where current state depends only on last value. Baseball: States are S = {win,lose}. Initial state is p 0 =[0.8, 0.2]. 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 win lose 0.6 Percentage of teams who win/lose next game given by p 1 =[0.8, 0.2] =[0.64, 0.36] Question: does the following limit exist? p n =[0.8, 0.2] n

7 Markov Chain Techniques Baseball Example: Solve constrained relation π = πp, πi =1 to obtain [π win, π lose ] π =[0.5714, ] =[πwin, π lose ], π win + π lose =1

8 Markov Chain Techniques Baseball Example: Solve constrained relation π = πp, πi =1 to obtain [π win, π lose ] π =[0.5714, ] =[πwin, π lose ], π win + π lose =1 Alternative: Iterate to compute solution Notes: n p n n p n n p n 0 [0.8000, ] 4 [0.5733, ] 8 [0.5714, ] 1 [0.6400, ] 5 [0.5720, ] 9 [0.5714, ] 2 [0.5920, ] 6 [0.5716, ] 10 [0.5714, ] 3 [0.5776, ] 7 [0.5715, ] Forms basis for Markov Chain Monte Carlo (MCMC) techniques Goal: construct chains whose stationary distribution is the posterior density

9 Markov Chain Monte Carlo Methods Strategy: Markov chain simulation used when it is impossible, or computationally prohibitive, to sample q directly from π(q υ) = π(υ q)π 0 (q) R p π(υ q)π 0 (q)dq Note: Create a Markov process whose stationary distribution is π(q υ). In Markov chain theory, we are given a Markov chain P, and we construct its equilibrium distribution. In MCMC theory, we are given a distribution and we want to construct a Markov chain that is reversible with respect to it.

10 Model Calibration Problem Assumption: Assume that measurement errors are iid and ε i N(0, σ 2 ) Likelihood: π(υ q) =L(q, σ υ) = 1 2 (2πσ 2 ) n/2 e SS q/2σ where n SS q = [υ i f i (q)] 2 i=1 is the sum of squares error.

11 Markov Chain Monte Carlo Methods General Strategy: Current value: X k 1 = q k 1 Propose candidate q J(q q k 1 ) from proposal (jumping) distribution With probability α(q,q k 1 ), accept q ; i.e., X k = q Otherwise, stay where you are: X k = q k 1 Intuition: Recall that π(υ q)π 0 (q) π(q υ) = π(υ q)π R p 0 (q)dq π(υ q) =! q) 1 n (2πσ 2 e i=1 [υ i f i (q)] 2 /2σ 2 = ) n/2 SSq 1 2 (2πσ 2 ) n/2 e SS q/2σ q q * q k 1 q* q k 1 q

12 Intuition: Markov Chain Monte Carlo Methods! q) SSq q q * q k 1 q* q k 1 q Consider r(q q k 1 )= π(q υ) π(q k 1 υ) = π(υ q )π 0 (q ) π(υ q k 1 )π 0 (q k 1 ) If r<1 π(υ q ) < π(υ q k 1 ), accept with probability α = r If r>1, accept with probability α =1 Note: Narrower proposal distribution yields higher probability of acceptance.

13 Markov Chain Monte Carlo Methods Note: Narrower proposal distribution yields higher probability of acceptance. q * q 1 =q* q 2 q =q =q* 3 2 q0 J( q* q k 1 )! q) q0 q* q* q 1 =q* q 3 =q 1 q 2 =q 1 J( q* q k 1 )! q) Parameter Value Chain Iteration Parameter Value Chain Iteration

14 Proposal Distribution Proposal Distribution: Significantly affects mixing Too wide: Too many points rejected and chain stays still for long periods; Too narrow: Acceptance ratio is high but algorithm is slow to explore parameter space Ideally, it should have similar shape to posterior distribution. (q! q 2 J ( q* q k 1 ) (q! J q* q 2 ( q k 1 ) q 1 q 1 (a) (b) Problem: Anisotropic posterior, isotropic proposal; Efficiency nonuniform for different parameters Result: Recovers efficiency of univariate case

15 Proposal Distribution Proposal Distribution: Two basic approaches Choose a fixed proposal function o Independent Metropolis Random walk (local Metropolis) q = q k 1 + Rz o Two (of several) choices: (i) R = ci q N(q k 1,cI) (ii) R = chol(v ) q N(q k 1,V) where Sensitivity Matrix V = σ 2 OLS X T (q OLS )X (q OLS ) 1 σ 2 OLS = 1 n [υ i f i (q OLS )] 2 n p i=1 (q! q 2 J X ik (q OLS )= f i(q OLS ) q k ( q* q k 1 ) (q! J q* q 2 ( q k 1 ) q 1 q 1 (a) (b)

16 Random Walk Metropolis Algorithm for Parameter Estimation 1. Set number of chain elements M and design parameters n s, σ s 2. Determine q 0 = arg min q N i=1 [υ i f i (q)] 2 3. Set SS q 0 = N i=1 [υ i f i (q 0 )] 2 4. Compute initial variance estimate: s 2 0 = SS q 0 n p 5. Construct covariance estimate V = s 2 0[X T (q 0 )X (q 0 )] 1 and R = chol(v ) 6. For k =1,,M (a) Sample z k N(0, 1) (b) Construct candidate q = q k 1 + Rz k (c) Sample u α U(0, 1) (d) Compute SS q (e) Compute α(q q k 1 )=min = N i=1 [υ i f i (q )] 2 1,e [SS q SS q k 1 ]/2s 2 k 1 (f) If u α < α, Set q k = q,ss q k = SS q else Set q k = q k 1,SS q k = SS q k 1 endif (g) Update s k Inv-gamma(a val,b val ) where a val =0.5(n s + n),b val =0.5(n s σ 2 s + SS q k) Parameter Value Chain Iteration

17 Delayed Rejection Adaptive Metropolis (DRAM) Adaptive Metropolis: Update chain covariance matrix as chain values are accepted. Diminishing adaptation and bounded convergence required since no longer Markov chain. Employ recursive relations V k = s p cov(q 0,q 1,,q k 1 )+εi p q k = 1 k +1 k i=0 q i = k k +1 1 k k 1 i=0 V k+1 = k 1 k V k + s p k q i + 1 k +1 qk = k k +1 qk k +1 qk k q k 1 ( q k 1 ) T (k + 1) q k ( q k ) T + q k (q k ) T + εi p

18 Chain Convergence (Burn-In) Techniques: Visually check chains Statistical tests Often abused in the literature Parameter Value Chain Iteration Chain not converged Chain for nonidentifiable parameter

19 Delayed Rejection Adaptive Metropolis (DRAM) Websites Examples Examples on using the toolbox for some statistical problems.

20 Delayed Rejection Adaptive Metropolis (DRAM) We fit the Monod model y = θ 1 1 θ , N(0,Iσ2 ) to observations x (mg / L COD): y (1 / h): 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 = [ ]'; % x (mg / L COD) data.ydata = [ ]'; % y (1 / h) Construct model modelfun theta(1)*x./(theta(2)+x); ssfun sum((data.ydata-modelfun(data.xdata,theta)).^2); model.ssfun = ssfun; model.sigma2 = 0.01^2;

21 Delayed Rejection Adaptive Metropolis (DRAM) Input parameters params = { {'theta1', tmin(1), 0} {'theta2', tmin(2), 0} }; and set options options.nsimu = 4000; options.updatesigma = 1; options.qcov = tcov; Run code [res,chain,s2chain] = mcmcrun(model,data,params,options);

22 Delayed Rejection Adaptive Metropolis (DRAM) Plot results figure(2); clf mcmcplot(chain,[],res,'chainpanel'); figure(3); clf mcmcplot(chain,[],res,'pairs'); Examples: Several available in MCMC_EXAMPLES ODE solver illustrated in algae example

Delayed Rejection Adaptive Metropolis (DRAM) Construct credible and prediction intervals figure(5); clf out = mcmcpred(res,chain,[],x,modelfun); mcmcpredplot(out);

23 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')

24 Steady State Model:! d 2 T s 2(a + b) = dx2 ab dt s dx (0) = Φ k DRAM for Heat Example h k [T s(x) T amb ] dt s dx (L) = h k [T amb T s (L)] Note:! Parameter set q =[Φ,h,k] is not identifiable Website Temperature ( o C) Aluminum Rod Data Location (cm)

25 DRAM for Heat Model with 3 Parameters: Results Note:! Parameter set q =[Φ,h,k] is not identifiable Notes:! Cond(V) = 1e+35! Data and model are not informing priors!

26 2 Parameter Heat Model:! Notes:! d 2 T s 2(a + b) = dx2 ab dt s dx (0) = Φ k Assignment:! Assignment h k [T s(x) T amb ] dt s dx (L) = h k [T amb T s (L)] Set k = 2.37 W/cm C, which is the physical value for aluminum Parameter set q =[h, Φ] is now identifiable 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.!

Parameter Selection and Model Calibration for an SIR Model

Parameter Selection and Model Calibration for an SIR Model Ralph C. Smith Department of Mathematics North Carolina State Universit SIR Model ds = N - S - kis, S(0) =S 0 di = kis - (r + )I, I(0) =I 0 dr