Uncertainty Propagation

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Myra Ford
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1 Setting: Uncertainty Propagation We assume that we have determined distributions for parameters e.g., Bayesian inference, prior experiments, expert opinion Ṫ 1 = 1 - d 1 T 1 - (1 - ")k 1 VT 1 Ṫ 2 = 2 - d 2 T 2 - (1 - f ")k 2 VT 2 Ṫ 1 =(1 - ")k 1 VT 1 - T 1 - m 1 ET 1 Ṫ 2 =(1 - f ")k 2 VT 2 - T 2 - m 2 ET 2 V = N T (T 1 + T 2 ) - cv - [(1 - ") 1 k 1 T 1 +(1 - f ") 2 k 2 T 2 ]V Ė = E + b E(T1 + T 2 ) T1 + T 2 + K E - d E(T1 + T 2 ) b T1 + T 2 + K E - E E d Goal: Construct statistics for quantities of interest e.g., Expected viral load in HIV patient with appropriate uncertainty intervals Note: Often involves moderate to highdimensional integration Z E[V (t)] = V (t, q) (q)dq R 6

2 Example: Uncertainty Propagation Yesterday, we computed mean responses, credible intervals, and prediction intervals for the SIR model ds dt = N S IS, S(0) = S 0 di dt = IS (r + )I, I(0) = I 0 dr dt = ri R, R(0) = R 0 Susceptible Infectious Recovered Goal: Discuss sampling-based techniques to compute statistics and intervals.

3 Issues: Uncertainty Propagation Uncertainty propagation and computation of statistical quantities of interest much more difficult for PDE models. e.g. Sinko-Streifer model: u(t, x): Number of fish of size x at @x = -µu g(t, x)u(t, x) x=x0 = u(0, x) = (x) Z x1 Random Field Representation: px g(x) = q j j (x) j=1 Quantity of Interest: Z E[u(t, x)] = u(t, x, q) (q)dq R p x 0 k(t, )u(t, )d Issues: How do we efficiently propagate input uncertainties through models? Surrogate models. How do we approximately integrate in moderate to high dimensions; e.g., p = ? Monte Carlo sampling, sparse grid quadrature

Forward Uncertainty Propagation: Linear Models Linear Models: Analytic mean and variance relations Example: Linear stress-strain relation i = Ee i + E 2 e 3 i + " i,i=1,.

4 Forward Uncertainty Propagation: Linear Models Linear Models: Analytic mean and variance relations Example: Linear stress-strain relation i = Ee i + E 2 e 3 i + " i,i=1,...,n Model Statistics: Let E,E 2 and var(e), var(e 2 ) denote parameter means and variance. Then E[Ee i + E 2 e 3 i ]=Ee i + E 2 e 3 i var[ee i + E 2 e 3 i ]=e 2 i var(e)+e 6 i var(e 2 )+2e 4 i cov(e,e 2 ) Response Statistics: Assume measurement errors uncorrelated from model response. E[ i ]=Ee i + E 2 e 3 i var[ i ]=e 2 i var(e)+e 6 i var(e 2 )+2e 4 i cov(e,e 2 ) + var(" i ) Problem: Models are almost always nonlinearly parameterized

5 Forward Uncertainty Propagation: Sampling Methods Strategy: Randomly sample from parameter and measurement error distributions and propagate through model to quantify response uncertainty. Advantages: Applicable to nonlinear models. Parameters can be correlated and non-gaussian. Straight-forward to apply and convergence rate is independent of number of parameters. Can directly incorporate both parameter and measurement uncertainties. Disadvantages: Very slow convergence rate: O (1/ p M) where M is the number of samples. 100-fold more evaluations required to gain additional place of accuracy. This motivates numerical analysis techniques: Next talk by C. Webster

6 Uncertainty Propagation Sampling-Based Approaches: Quadrature: Monte Carlo, Latin hypercube, Sobol Interval definitions and construction Prediction intervals for HIV model via DRAM algorithm Numerical Analysis-Based Approaches: Next presentation by C. Webster

7 Numerical Quadrature Motivation: Computation of expected values requires approximation of integrals Z E[u(t, x)] = u(t, x, q) (q)dq R p Numerical Quadrature: Z RX f (q) (q)dq f (q r )w r R p Questions: r=1 How do we choose the quadrature points and weights? E.g., Newton-Cotes; e.g., trapezoid rule Z b a f (q)dq h 2 " f (a)+f (b)+2 q r = a + hr, h = b - a R - 1 XR-2 r=1 f (q r ) # a b

8 Numerical Quadrature Motivation: Computation of expected values requires approximation of integrals Z E[u(t, x)] = u(t, x, q) (q)dq R p Numerical Quadrature: Z RX f (q) (q)dq f (q r )w r R p r=1 Questions: How do we choose the quadrature points and weights? E.g., Newton-Cotes, Gaussian algorithms a b a xx x j 1 xx x j b

9 Numerical Quadrature: Z RX f (q) (q)dq f (q r )w r R p r=1 Numerical Quadrature Questions: Can we construct nested algorithms to improve efficiency? E.g., employ Clenshaw-Curtis points Level

10 Questions: Numerical Quadrature How do we reduce required number of points while maintaining accuracy? Tensored Grids: Exponential growth Sparse Grids: Same accuracy p R` Sparse Grid R Tensored Grid R =(R`) p , > ,901 > ,353,801 >

11 Numerical Quadrature Problem: Accuracy of methods diminishes as parameter dimension p increases Suppose f 2 C ([0, 1] p ) Tensor products: Take R` points in each dimension so R =(R`) p total points Quadrature errors: Newton-Cotes: E O (R - ` )=O (R - /p ) Gaussian: E O (e - R`) =O e - pp R Sparse Grid: E O R - log (R) (p-1)( +1)

12 Numerical Quadrature Problem: Accuracy of methods diminishes as parameter dimension p increases Suppose f 2 C ([0, 1] p ) Tensor products: Take R` points in each dimension so R =(R`) p total points Quadrature errors: Newton-Cotes: E O (R - ` )=O (R - /p ) Gaussian: E O (e - R`) =O e - pp R Sparse Grid: E O R - log (R) (p-1)( +1) Alternative: Monte Carlo quadrature Z f (q) (q)dq 1 R R p RX f (q r ), E r=1 1 pr Advantage: Errors independent of dimension p Disadvantage: Convergence is very slow!

13 Numerical Quadrature Problem: Accuracy of methods diminishes as parameter dimension p increases Suppose f 2 C ([0, 1] p ) Tensor products: Take R` points in each dimension so R =(R`) p total points Quadrature errors: Newton-Cotes: E O (R - ` )=O (R - /p ) Gaussian: E O (e - R`) =O e - pp R Sparse Grid: E O R - log (R) (p-1)( +1) Alternative: Monte Carlo quadrature Z R p f (q) (q)dq 1 R RX f (q r ), E r=1 1 pr Advantage: Errors independent of dimension p Disadvantage: Convergence is very slow! Conclusion: For high enough dimension p, monkeys throwing darts will beat Gaussian and sparse grid techniques!

14 Monte Carlo Sampling Techniques Issues: Very low accuracy and slow convergence Random sampling may not randomly cover space 1 Samples from Uniform Distribution

15 Monte Carlo Sampling Techniques Issues: Very low accuracy and slow convergence Random sampling may not randomly cover space 1 Samples from Uniform Distribution 1 Sobol Points Sobol Sequence: Use a base of two to form successively finer uniform partitions of unit interval and reorder coordinates in each dimension.

16 Quasi-Monte Carlo Sampling Techniques Issues: Very low accuracy and slow convergence Random sampling may not randomly cover space 1 Samples from Uniform Distribution 1 Sobol Points Z R p f (q) (q)dq 1 R RX f (q r ), E r=1 1 pr E O (log R) p R

Monte Carlo Sampling Techniques Example: Use Monte Carlo sampling to approximate area of circle A c A s = r 2 4r 2 = 4 ) A c = 4 A s Strategy: Randomly sample N points in square )

17 Monte Carlo Sampling Techniques Example: Use Monte Carlo sampling to approximate area of circle A c A s = r 2 4r 2 = 4 ) A c = 4 A s Strategy: Randomly sample N points in square ) approximately N 4 Count M points in circle in circle ) 4M N Later Tutorial: Run code and see if we see convergence rate 1 p N Quasi-Monte Carlo: Talk with M. Hyman SAMSI Program on QMC in

18 Confidence, Credible and Prediction Intervals Note: We now know how to compute the mean response for the QoI. How do we compute appropriate intervals? SIR Model: ds dt = N S IS, S(0) = S 0 di dt = IS (r + )I, I(0) = I 0 dr dt = ri R, R(0) = R 0 Susceptible Infectious Recovered

19 Confidence, Credible and Prediction Intervals Data: =[ 1,, n ] of iid random observations Confidence Interval (Frequentist): A 100 (1 )% confidence interval for a fixed, unknown parameter q 0 is a random interval [L c ( ),U c ( )], having probability at least 1 of covering q 0 under the joint distribution of. Credible Interval (Bayesian): A 100 (1 ) % credible interval is that having probability at least 1 of containing q. Strategy: Sample out of parameter density Q (q) 90% Confidence Intervals 90% Credible Interval

),U c ( )] having probability at least 1 of of containing n+1 under the joint distribution of (, n+1

20 Confidence, Credible and Prediction Intervals Data: =[ 1,, n ] of iid random observations Prediction Interval: A 100 (1 )% prediction interval for a future observable n+1 is a random interval [L c ( ),U c ( )] having probability at least 1 of of containing n+1 under the joint distribution of (, n+1 ). Example: Consider linear model i = q 0 + q 1 x i + " i,i=1,,n Prediction interval Credible interval

21 Example: HIV Model Model: Ṫ 1 = 1 - d 1 T 1 - (1 - ")k 1 VT 1 Ṫ 2 = 2 - d 2 T 2 - (1 - f ")k 2 VT 2 Ṫ 1 =(1 - ")k 1 VT 1 - T 1 - m 1 ET 1 Ṫ 2 =(1 - f ")k 2 VT 2 - T 2 - m 2 ET 2 V = N T (T 1 + T 2 ) - cv - [(1 - ") 1 k 1 T 1 +(1 - f ") 2 k 2 T 2 ]V Ė = E + b E(T1 + T 2 ) T1 + T 2 + K E - d E(T1 + T 2 ) b T1 + T 2 + K E - E E d Parameter Chains and Densities: q =[b E,, d 1, k 2, 1, K b ]

22 Propagation of Uncertainty HIV Example Parameter Densities: Techniques: Sample from parameter and observation error densities to construct mean response, credible intervals, and prediction intervals for QoI. p Slow convergence rate O (1/ M ) Samples from Chain Data, Credible Intervals and Prediction Intervals Non-Gaussian Credible and Prediction Intervals

23 Monte Carlo Quadrature: MATLAB Tutorial Run rand_points.m to observe uniformly sampled and Sobol points. Run pi_approx.m with different values of N to see if you observe convergence rate of 1/ p N SIR Model: Note: ds dt = N S IS, S(0) = S 0 di dt = IS (r + )I, I(0) = I 0 dr dt = ri R, R(0) = R 0 Susceptible Infectious Recovered Run either the posted code for the 4 parameter model or your modification for the 3 parameter model and compute the prediction intervals.

Parameter Selection Techniques and Surrogate Models

Parameter Selection Techniques and Surrogate Models Model Reduction: Will discuss two forms Parameter space reduction Surrogate models to reduce model complexity Input Representation Local Sensitivity