Questions from lectures The figure compares several approximations. Define each one, and interpret the message of the figure

Questions posed in lecture slides The true function is y=x. We fitted noisy data at 10 points. The data at x=10, the last point was y10=11. The fit was y=1.06x. Provide the values of 10, e10, and the surrogate error at x=10. The pairs (1,1), (2,1), (3,2.25) represent strain (millistrains) and stress (ksi) measurements. For the material tested Estimate k using regression. Calculate the error in k using cross validation both from the definition and from the formula.

Questions from lecture slides (continued) The following 20 numbers were produced from by x=rand(1,20) 0.81 0.90 0.12 0.91 0.63 0.09 0.27 0.54 0.95 0.96 0.15 0.97 0.95 0.48 0.80 0.14 0.42 0.91 0.79 0.95 What estimate of the probability of X<0.25 would you estimate from the sample? What uncertainty would you estimate for your estimate? How does it compare to the error in the estimate?

Meaning of basic terminology Two random variables X and Y are sampled as X: 1,1,1 and Y: 0,1,2. Define standard deviation, and estimate the standard deviations of X and Y. If you know that the exact values of these standard deviations are 0.1 and 1, respectively, estimate the correlation coefficients of X and Y. Is the regression on the previous slide linear regression or nonlinear regression? Why? What is D-optimal design? Given the function y=bx, which design is better from D-optimality: Design 1: x=0,1,4. Design 2: x=1,2,3

Additional examples of terminology Intervening variables, reciprocal approximation, local-global approximation. Coefficient of multiple determination, prediction variance Universal kriging, sparse data, adaptive sampling, EGO Design of experiments, space filling DOEs, Monte Carlo sampling Aleatory and epistemic uncertainties, PDF, CDF, histogram Risk allocation, probabilistic design.