1. Chapter 7 Regression and Correlation Analyses Instructor: Prof. Wilson Tang Instructor: Prof. Wilson Tang CIVL 181 Modelling Systems with Uncertainties

2. Soil Strength Example Dam Soil 6 24 Histogram 0 0.5 1.01.5 2.0

3. Previous Model Alternate Model

4. A General Formulation Y is the r.v. of interest where x is the independent variable. (x i, y i )  +  x i xixi x y yiyi y  +  x

5. Method of Least Square

8. E 7.1 r 2 = % reduction in uncertainty by regression line 0 to 100 % Completely random Straight line relationship

9. Assume Y is Normal at given x  e.g. at 24’  E(Y  x = 24) = 0.018 + 0.0516  24 = 1.26 Var(Y  x = 24) = 0.0368   = 0.19  N(1.26, 0.19) Read E 7.2

10. Advanced topics 1. Non-constant variance, Var(Y  x) 2. Multiple linear regression 3. Non-linear regression

11. x (GNP) y (per capita energy consumption) P 7.5

12. In general, there are 2 different lines except when X and Y are perfectly dependent. The angle between 2 lines depend on scatter of data.  A measure of scatter. Application of Regression Analysis in Engineering 1.Determining empirical relationship from observed data 2.Checking or verifying proposed model 3.Economics of indirect measurements 4.Obtaining preliminary information (to save time)

13. Correlation Analysis To estimate , the correlation coefficient between X and Y

14. % reduction in variance through regression  dependent  complete reduction of uncertainty

15. Confidence interval on regression line Due to possible variations in both  (intercept) and  (slope) more accurate around the middle. x y