Lecture_Statistics_Spring_2013b

Vi linear regression variance of the regression s2

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Unformatted text preview: s we need to minimize the function S, which is the sum of the squared deviations between the N observed y values, yi, and the best fit line y values, ŷi. VI. Linear Regression For simple linear regression, y = mx + b 2 N S = ∑ (y i − b − mxi) i =1 k=1, where b=β0 and m=β1 N 2 ˆ S = ∑ (y i − y i ) i =1 N ȹ k ȹ ȹ ȹ = ∑ ȹ y i − ȹ β 0 + ∑ β i X i ȹ ȹ ȹ ȹ ȹ ȹ i =1ȹ i =1 ȹ Ⱥ Ⱥ 2 VI. Linear Regression N 2 S = ∑ (y i − b − mxi) i =1 Note: xi and yi are simply values Partial differentiation with respect to b (the intercept), and then with respect to m (the slope) 2 ȹ 2 ȹ ∂ ȹ N ȹ ∂S ȹ ∂ ȹ N ȹ ∂S ȹ ȹ ∑ (y − b − mx ) ȹ = 0 ȹ ∑ (y − b − mx ) ȹ = 0 ȹ ȹ = ȹ ȹ = i i ȹ i i ȹ ∂b Ⱥ m ∂b ȹ i =1 ∂m Ⱥ b ∂m ȹ i =1 ȹ ȹ ȹ Ⱥ ȹ Ⱥ 2 ȹ 2 ȹ ȹ N ∂ ȹ N ∂ ȹ ∑ ȹ ∑ (y − b − mx ) ȹ = 0 (y i − b − mx i ) ȹ = 0 i i ȹ ȹ i =1∂m ȹ ȹ i =1∂b ȹ Ⱥ ȹ Ⱥ ȹ N ȹ ȹ N ȹ ȹ ∑ 2(y i − b − mx i )(− 1)ȹ = 0 ȹ ∑ 2(y i − b − mx i )(− x i )ȹ = 0 ȹ ȹ ȹ ȹ ȹ i =1 Ⱥ ȹ i =1 Ⱥ N N N i =1 i =1 i =1 N N ȹ N ȹ Nb + ȹ ∑ x i ȹm = ∑ y i ȹ ȹ i...
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This note was uploaded on 01/26/2014 for the course CHEM 3625 taught by Professor Mrjohnson during the Spring '08 term at Virginia Tech.

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