Linear Regression Y i =  0 +  1 x i +  i Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland)

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 2 Regression Analysis Aim: To know to which extent a certain response (dependent) variable is related to a set of explanatory (independent) variables. Example: James David Forbes (Edinburgh ) Aim: To know to which extent a certain response (dependent) variable is related to a set of explanatory (independent) variables. Example: James David Forbes (Edinburgh ) ResponseObservations Professor in glaciology. He measured the water boiling points and atmospheric pressures at 17 different locations in the Swiss alps (Jungfrau) and in Scotland with the aim of using the boiling temperature of water to estimate altitude.

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 3 Regression Model Input data: vectors x and Y, where: x i → i-th observation Y i → i-th response, or measurement Model: Y =  0 +  1 x +  or Y i =  0 +  1 x i +  i Output data: → estimated values of  0 and  1 Input data: vectors x and Y, where: x i → i-th observation Y i → i-th response, or measurement Model: Y =  0 +  1 x +  or Y i =  0 +  1 x i +  i Output data: → estimated values of  0 and  1 Measurement Error Fundamental assumption: errors are mutually independent and normally distributed with mean zero and variance  2 :

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 4 Residuals  i

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 5 Estimation of the Parameters Least Square Method: Minimum of S: Least Square Method: Minimum of S: The objective function (S) expresses a measure of the closeness between the regression line and the observations  I want to find the minimum of S

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 6 Example

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 7 Example: Parameter Estimation Averages Estimation of  0 and  1

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 8 Example: Matlab Regression Routine  = confidence interval

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 9 Residuals Outlier

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 10 Removal of the Outlier

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 11 Analysis of Variance (ANOVA) Total Sum of Squares Sum of Squares due to Regression Sum of Squares due to Error Total Sum of Squares Sum of Squares due to Regression Sum of Squares due to Error Coefficient of Determination R 2 = 1   i = 0 R 2 = 0  regression does not explain variation of Y

Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 12 Regression Analysis with Matlab Regression Routine Interval of confidence Regression Routine Interval of confidence

Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 13 Regression Analysis with Matlab Residuals Confidence interval for the residuals

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Simple Linear Regressions – Page # 14 Multiple Linear Regression Approximate model: Residuals Least Squares Sum of Square Residuals (SSR)