Least Squares Proof

Least Squares Proof - The Calculus Way Using calculus, a...

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The Calculus Way Using calculus, a function has its minimum where the derivative is 0 . Since we need to adjust both m and b, we take the derivative of E with respect to m, and separately with respect to b, and set both to 0: Each equation then gets divided by the common factor 2, and the terms not involving m or b are moved to the other side. With a little thought you can recognize the result as two simultaneous equations in m and b, namely: The summation expressions are all just numbers, the result of summing x and y in various combinations. (By the way, how do we know that these will give us a minimum and not a maximum or inflection point? Because each second derivative is 2 for all values of m and b, and if the first derivative is 0 and the second derivative is positive you have a minimum.) These simultaneous equations can be solved like any others: by substitution or by linear combination. Let’s try substitution. The second equation looks easy to solve for b: Substitute that in the other equation and you eventually come up with

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This note was uploaded on 04/02/2008 for the course MATH 283 taught by Professor Brown during the Spring '08 term at NMT.

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Least Squares Proof - The Calculus Way Using calculus, a...

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