Least squares method:
Least squares method:
Objective: Find a line [pic] which best fits the data [pic].
[pic]
The famous method , proposed by Gauss and used to find the best fitted line, is called least squares methods. This method is to minimize the sum of squares of the distance between the data points [pic] and the points [pic] in the fitted line. That is, find the estimates of a and b minimizing
[pic].
After some calculus techniques, it turns out that the estimates of a and b can be obtained by solving the following linear system,
[pic],
where
[pic] and [pic].
Example:
Find the least square line for the points [pic] and [pic].
[solution:]
[pic] and [pic].
[pic] [pic].
[pic]
Thus, the estimates of a and b can be obtained by solving the following linear system:
[pic].
The solution is
[pic]The fitted line is [pic].
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