lec7 - Lecture 7: Linear Regression Linear Regression...

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Lecture 7: inear Regression Linear Regression Please read Chapter 20.5 in dvanced 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 minimizing the sum of the square of the residuals, r S Residual at a point is ) , ( n n y x . ) ( i i i x f y ) ( x f y n i i r x f y S 2 )) ( ( ) , ( 1 1 y x Sum of the square of the residuals 2 i 1 Figure. Basic model for regression
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inear Regression riterion#1 Linear Regression-Criterion#1 ) , ( , ... ), , ( ), , ( 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 . 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 10 x y 2.0 4.0 3.0 6.0 4 6 8 y 2.0 6.0 3.0 8.0 0 2 01234 x 4 Figure. Data points for y vs. x data.
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Linear Regression-Criteria#1 able. esiduals at each point for Using y=4x-4 as the regression curve xy y predicted ε = y - y predicted 2.0 4.0 4.0 0.0 Table. Residuals at each point for regression model y = 4x – 4. 6 8 10 3.0 6.0 8.0 -2.0 2.0 6.0 4.0 2.0 3.0 8.0 8.0 0.0 2 4 y 0 4 1 i i igure egression curve for y=4x yv sxd a t a 0 01234 x 5 Figure. Regression curve for y=4x-4, y vs. x data
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Linear Regression-Criteria#1 able. esiduals at each point for y=6 Using y=6 as a regression curve xy y predicted ε = y - y predicted 2.0 4.0 6.0 -2.0 3.0 6.0 6.0 0.0 6 8 10 y Table. Residuals at each point for y6 2.0 6.0 6.0 0.0 3.0 8.0 6.0 2.0 4 0 2 4 0 1 i i 01234 x Figure. Regression curve for y=6, y vs. x data 6 g gy , y
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Linear Regression – Criterion #1 0 4 1 i i for both regression models of y=4x-4 and y=6.
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lec7 - Lecture 7: Linear Regression Linear Regression...

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