Fitting The Regression Line
Suppose there are n data points {yi, xi}, where i = 1, 2, …, n. The goal is to find the equation of the straight line
which would provide a "best" fit for the data points. Here the "best" will be understood as in the least-squares approach: such a line that minimizes the sum of squared residuals of the linear regression model. In other words, numbers α and β solve the following minimization problem:
By using either calculus, the geometry of inner product spaces or simply expanding to get a quadratic in α and β, it can be shown that the values of α and β that minimize the objective function Q are
where rxy is the sample correlation coefficient between x and y, sx is the standard deviation of x, and sy is correspondingly the standard deviation of y. Horizontal bar over a variable means the sample average of that variable. For example:
Substituting the above expressions for and into
yields
This shows the role plays in the regression line of standardized data points.
Read more about this topic: Simple Linear Regression
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