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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“Childrens view of the world and their capacity to understand keep expanding as they mature, and they need to ask the same questions over and over, fitting the information into their new level of understanding.”
—Joanna Cole (20th century)
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