2. Engineers often: Regress data to a model  Used for assessing theory  Used for predicting  Empirical or theoretical model Use the regression of others  Antoine Equation  DIPPR What are uncertainties of regression? Do the data fit the model? What are the errors in the prediction? What are the errors in the parameters? What are uncertainties of regression? Do the data fit the model? What are the errors in the prediction? What are the errors in the parameters?

3. Two classes for a model  Linear  Non-linear “Linear” refers to parameters, not the dependent variable (i.e. written in matrix form- linear algebra) You can use the Mathcad function “linfit” on linear equations = x 2 x 1 abcabc y parameters are linear x is not linear

4. 1. 2. 3. 4. 5. Which models can you use linear regression?

5. Regression- Straight line Y= b 0 + b 1 X Two parameters Practice Open Excel Regression example (1 st order Example) If necessary, add-in the Data Analysis ToolPak Regress the data in Columns B and C- select 95 % confidence and residual plot. You may include data labels but you must select label. Copy the regressed slope and intercept into m and b cells in column H The next slides will help us understand the output

6. residual (error) “X” data “Y” measured slope Intercept “Y” predicted

7. residual (error) sum squared error “X” data “Y” data 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 42.76608602 “Y” predicted b 0 = 0.923, b 1 = 0.516 Number of fitted parameters: 2 for a two-parameter model

8. 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. Average measured y value

12. intercept slope Called standard error in Excel Number of fitted parameters: 2 for a two-parameter model

13. Confidence Interval on the Prediction Expected Range of Data n  ∞ S Y  0 n  ∞ S Y  σ

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

15. 1.Complete columns D-H in 1 st order example- look at 95% confidence level of data. 2.Complete multiple regression Practice example a)Highlight all x columns for x input b)Note that x data could represent non-linear data such as T, 1/T, T 2, etc. c)Look at regressed parameter associated with x 1 and calculate the 95% confidence. Compare with Excel. d)Select residuals and residual plots. Note that there are two types of residual information: 1) each dependent variable has a residual plot where the x-axis is the data and the y-axis is the residual for the data; 2) the summary output gives a residual for y (actual Y – predicted Y). If desired, you can plot as a function of the # of data points.