![]() \(x_1,x_2, \ldots ,x_n\) are called independent variables. In a multiple linear regression we are fitting a function of the form: If the above assumptions are not valid, then the regression model is unacceptable. \(\epsilon\) corresponds to the error term which is assumed to have zero mean, constant variance, uncorrelated, and normally distributed. In simple linear regression we are fitting a function of the form: Mathematical Foundations of Linear Regression If the shiny app is not working, you can repeat the above exercise using the excel workbook Reg.xls. Excel and R have functions which will automatically calculate the values of the slope and the intercept which minimizes the Residual Sum of Squares. The goal in linear regression is to choose the slope and intercept such that the Residual Sum of Squares is as small as possible. It does not matter if they are above or below the line. Observations which are far off from the line are poorly predicted. Why do we square the residuals? This is to prevent negative residuals and positive residuals from canceling each other out. This can be thought of as a measure of error. Square the residuals and sum them up across all values of \(x\).For each value of \(x\), calculate the residual which is the difference between the observed and the predicted value.What is the Residual Sum of Squares?įor a specific value of slope and intercept: Now observe how the Residual Sum of Squares changes with intercept and slope. The residual is the difference between the actual and predicted value. ![]() The predicted value of \(y\) is \(14 12 \times 5 = 74\) and the actual value of \(y\) is 79. Corresponding to \(x=5\) there are two values of \(y\), the actual observed value and the predicted value. Let us say the best value we got was 14 for the intercept and 12 for the slope. Visually try to find the intercept and slope which best represents the data. Change the values of intercept and slope. ![]() This section is inspired by the example in Chapter 4 of Barreto and Howland (2006).
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