1. 2014

2. Engineers often: Regress data  Analysis  Fit to theory  Data reduction Use the regression of others  Antoine Equation  DIPPR We need to be able to report uncertainties associated with regression. Do the data fit the model? What are the errors in the prediction? What are the errors in the parameters? We need to be able to report uncertainties associated with regression. Do the data fit the model? What are the errors in the prediction? What are the errors in the parameters?

3. There are two classes of regressions  Linear  Non-linear “Linear” refers to the parameters, not the functional dependence of the independent variable  You can use the Mathcad function “linfit” on linear equations

4. There are two classes of regressions  Linear  Non-linear “Linear” refers to the parameters, not the functional dependence of the independent variable  You can use the Mathcad function “linfit” on linear equations 1. 2. 3. 4. 5.

5. residual (error) “X” data “Y” Measured slope Intercept “Y” Predicted

6. residual (error) “X” data “Y” data intercept 0.92291455 slope 0.516173934 12.7490321781.4390884831.309943694 23.7199102242.3620030331.357907192 30.9259950173.284917582-2.35892257 42.6234826864.207832132-1.58434945 56.5397973425.1307466811.409050661 66.7799091776.0536612310.726247946 74.9461504016.976575781-2.03042538 89.6741780697.899490331.774687739 97.619598218.82240488-1.20280667 107.6500209969.745319429-2.09529843 1111.51410.668233980.845766021 1213.1828506811.591148531.591702152 1313.2817363512.514063080.767673275 1413.6044459213.436977630.16746829 1512.7953521814.35989218-1.56454 1617.8237477815.282806732.540941056 1714.5506837916.20572128-1.65503748 42.76608602SS E “Y” predicted sum squared error mean squared error Number of fitted parameters: 2 for a two-parameter model Number of fitted parameters: 2 for a two-parameter model

7. A useful statistic but not definitive Tells you how well the data fit the model. It does not tell you if the model is correct.

11. How well does the line fit the data at each point (not just the mean)? In what range should the data lie (are there outliers)? What are the confidence intervals on the slope and intercept?

12. Confidence Intervals from Example Green is confidence interval How well does the model fit the data? Red is the prediction band Where should the data fall? Can I throw out any points?

13. When you fit data to a straight line, the slope and intercept are only estimates of the true slope and intercept. (slope) (intercept)

14. Confidence Interval on the Prediction Prediction Band: Expected Range of Data

15. CONFIDENCE INTERVAL “Mean Prediction Band” Shows possible errors in where the line will go Interval narrows with increasing number of data points  See equation on previous slide PREDICTION BAND “Single (another) Point Prediction Band” Shows where the data should lie Interval does not narrow much with increasing number of data points  See equation on previous slide

16. Igor has the equations already programmed and will Plot the confidence intervals Plot the prediction bands Print the  confidence intervals on the slope and intercept Assignment

19. Linear regression can be written in matrix form. XY 21186 24214 32288 47425 50455 59539 68622 74675 62562 50453 41370 30274

20. Standard Error of b i is the square root of the i-th diagonal term of the matrix