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lec7 - Lecture 7 Linear Regression Linear Regression Please...

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Lecture 7: Linear Regression Please read Chapter 20.5 in Advanced Engineering Advanced Engineering Mathematics 1
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What is Regression? What is regression? Given n data points ) , ( , ... ), , ( ), , ( 2 2 1 1 n n y x y x y x best fit ) ( x f y to the data. The best fit is generally based on i i i i th f th f th id l S minimizing the sum of the square of the residuals, r Residual at a point is ) , ( n n y x . ) ( i i i x f y ) ( x f y n i i i r x f y S 1 2 )) ( ( ) , ( 1 1 y x Sum of the square of the residuals 2 Figure. Basic model for regression
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Linear Regression Criterion#1 Linear Regression-Criterion#1 ) ( ) ( ) ( Gi d t i t b t fit , , ... ), , ), , 2 2 1 1 n n y x y x y x Given n data points best fit x a a y 1 0 to the data. y n n y x , i i y x , i i i x a a y 1 0 2 2 , y x 3 3 , y x x i i i x a a y 1 0 1 1 , y x Figure. Linear regression of y vs. x data showing residuals at a typical point, x i . 3 Does minimizing n i i 1 work as a criterion, where ) ( 1 0 i i i x a a y i
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Example for Criterion#1 Example: Given the data points (2,4), (3,6), (2,6) and (3,8), best fit the data to a straight line using Criterion#1 Table. Data Points 8 10 x y 2.0 4.0 3.0 6.0 2 4 6 y 2.0 6.0 3.0 8.0 0 0 1 2 3 4 x 4 Figure. Data points for y vs. x data.
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