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# lecture12 - Lecture 12 Projection and Least Square...

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Lecture 12 Projection and Least Square Approximation Shang-Hua Teng

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Line Fitting and Predication Input: Table of paired data values (x, y) Some connection between x and y. Example: height ------ weight Example: revenue ------ stock price Example: Yesterday’s temperature at Pittsburgh --------- today’s temperature at Boston Output: a and b that best predicates y from x: y = ax + b
Scatter Plot of Data Revenue Stock Price

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Regression Line y = ax+b Revenue Stock Price
Predication with Regression Line y = ax+b Revenue Stock Price

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When Life is Perfect y = ax+b Revenue Stock Price x 1 x 7 y 7 x 2 y 1 y 3 x 3 y 2 How do we find a and b
When Life is Perfect y = ax+b Revenue Stock Price x 1 x 7 y 7 x 2 y 1 y 3 x 3 y 2 = 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 1 1 1 1 1 1 y y y y y y y b a x x x x x x x

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How to Solve it? By Elimination What will happen? = 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 1 1 1 1 1 1 y y y y y y y b a x x x x x x x
Another Method: Try to Solve In general: if A x = b has a solution, then A T A x = A T b has the same solution = 7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y y y y y y y x x x x x x x b a x x x x x x x x x x x x x x = = = = = = 7 1 7 1 7 1 7 1 7 1 2 1 i i i i i i i i i i i y y x b a x x x

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When Life is not Perfect No perfect Regression Line y = ax+b Revenue Stock Price
When Life is not Perfect No perfect Regression Line y = ax+b Revenue Stock Price No solution!!!!!!

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lecture12 - Lecture 12 Projection and Least Square...

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