Global predictors of regression fidelity A single number to characterize the overall quality of the surrogate. Equivalence measures –Coefficient of multiple determination –Adjusted coefficient of multiple determination Prediction accuracy measures –Model independent: Cross validation error –Model dependent: Standard error

Linear Regression

Coefficient of multiple determination Equivalence of surrogate with data is often measured by how much of the variance in the data is captured by the surrogate. Coefficient of multiple determination and adjusted version

R 2 does not reflect accuracy Compare y1=x to y2=0.1x plus same noise (normally distributed with zero mean and standard deviation of 1. Estimate the average errors between the function (red) and surrogate (blue). R 2 = R 2 =0.3016

Cross validation Validation consists of checking the surrogate at a set of validation points. This may be considered wasteful because we do not use all the points for fitting the best possible surrogate. Cross validation divides data into n g groups. Fit the approximation to n g -1 groups, and use last group to estimate error. Repeat for each group. When each group consists of one point, error often called PRESS (prediction error sum of squares) Calculate error at each point and then present r.m.s error For linear regression can be shown that

Model based error for linear regression The common assumptions for linear regression –Surrogate is in functional form of true function –The data is contaminated with normally distributed error with the same standard deviation at every point. –The errors at different points are not correlated. Under these assumptions, the noise standard deviation (called standard error) is estimated as. Similarly, the standard error in the coefficients is

Comparison of errors

Top hat question We sample the function y=x with noise at x=0, 1, 2 to get 0.5, 0.5, 2.5. Assume that the linear regression fit is y=0.8x. What are the noise (epsilon), the discrepancy (e), the cross-validation error, and the actual error at x=2.

Prediction variance Linear regression model Define then With some algebra Standard error

Example of prediction variance For a linear polynomial RS y=b 1 +b 2 x 1 +b 3 x 2 find the prediction variance in the region (a) For data at three vertices (omitting (1,1))

Interpolation vs. Extrapolation At origin. At 3 vertices. At (1,1)

Standard error contours

Data at four vertices Now And Error at vertices At the origin minimum is How can we reduce error without adding points?

Graphical Comparison of Standard Errors Three pointsFour points

Problems The pairs (0,0), (1,1), (2,1) represent strain (millistrains) and stress (ksi) measurements. –Estimate Young’s modulus using regression. –Calculate the error in Young modulus using cross validation both from the definition and from the formula on Slide 5. Repeat the example of y=x, using only data at x=3,6,9,…,30. Use the same noise values as given for these points in the notes for Slide 4.