70 chapter 10 using data ence 627 assakkaf the

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Unformatted text preview: n ∑ ( yi − yi ) ˆ ˆ yi = the ith predicted value of Y yi = the ith measured value of Y ei = the ith error The objective function for the principle of least squares is to minimize the sum of the squares of the errors 35 Slide No. 70 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Procedure Solution for the Bivariate model ˆ Y = b0 + b1 X The objective function in this case is n ( ˆ F = min ∑ Yi − yi i =1 ) 2 n = min ∑ (b0 + b1 xi − yi ) 2 i =1 The derivatives of the sum of the squares of the errors with respect to the unknowns b0 and b1 are as follows: Slide No. 71 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach n ∂F = 2∑ (b0 + b1 xi − yi ) = 0 ∂b0 i =1 n ∂F = 2∑ (b0 + b1 xi − yi )xi = 0 ∂b1 i =1 Dividing each equation by 2, separating the terms in the summation, and rearranging yields the set of normal equations: n n n ∑b + ∑b x − ∑ y i =1 0 n 1i i =1 n i =1 n i i =1 n ∑b x + ∑b x − ∑ x y i =1 0i i =1 2 1i i =1 n i =1 = 0 = nb0 + b1 ∑ xi − ∑ yi i n i n n i =1 i =1 i =1 = 0 = b0 ∑ xi + b1 ∑ xi2 − ∑ xi yi 36 Slide No. 72 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach n n nb0 + b1 ∑ xi = ∑ yi i =1 i =1 n n n i =1 i =1 i =1 b0 ∑ xi + b1 ∑ xi2 = ∑ xi yi From which, n b1 = ∑x y i i =1 i − n 1n ∑ xi ∑ yi n i =1 i =1 1 n ∑ xi2 − 2 ∑ xi i =1 i =1 n 2 and b0 = Y − b1 X = 1n bn yi − 1 ∑ xi ∑ n i=1 n i =1 Slide No. 73 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Bivariate Model n ˆ Y = b0 + b1 X b1 = ∑ xi yi − i =1 n 1n xi ∑ yi ∑ n i =1 i =1 1 n ∑ x − n ∑ xi i =1 i =1 n 2 2 i b0 = Y − b1 X = 1n bn yi − 1 ∑ xi ∑ n i=1 n i =1 37 Slide No. 74 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Example: Bivariate Model Given the following pairs of observations, compute the regression coefficients for the bivariate (linear) model using the principle of least squares. X 0.8 1.6 3.1 4.4 6.3 7.9 9.2 Y 2.8 4.9 6.5 8.1 8.8 9.1 8.9 Slide No. 75 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Example: Bivariate Model 10 9 8 7 6 Y5 4 3 2 1 0 0 2 4 6 8 10 X 38 Slide No. 76 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Example: Bivariate Model Σ xi 0.8 1.6 3.1 4.4 6.3 7.9 9.2 33.3 yi 2.8 4.9 6.5 8.1 8.8 9.1 8.9 49.1 n b1 = ∑x y i i =1 i − xi2 0.64 2.56 9.61 19.36 39.69 62.41 84.64 218.91 n 1n ∑ xi ∑ yi n i =1 i =1 1 n x − ∑ xi ∑ n i=1 i =1 n 2 i 2 xi y i 2.24 7.84 20.15 35.64 55.44 71.89 81.88 275.08 = 1n bn ∑ yi − n1 ∑ xi n i =1 i =1 1 0.68605 (33.3) = (49.1) − 7 7 = 3.75063 b0 = 1 (33.3)(49.1) 7 = 0.68605 (33.3)2 (218.91) − 7 275.08 − Slide No. 77 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach Example: Bivariate Model 12 10 ˆ Y = 3.751 + 0.686 X 8 Y6 yi 4 2 0 0 2 4 6 8 10 X 39...
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This note was uploaded on 10/24/2012 for the course ESI 6385 taught by Professor Mansoorehmollaghasemi during the Fall '12 term at University of Central Florida.

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