least squares_slides - 1 Ordinary Least-Squares Ordinary...

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Unformatted text preview: 1 Ordinary Least-Squares Ordinary Least-Squares Ordinary Least-Squares Outline Linear regression Geometry of least-squares Discussion of the Gauss-Markov theorem Ordinary Least-Squares One-dimensional regression a b Ordinary Least-Squares One-dimensional regression ax b = Find a line that represent the best linear relationship: a b Ordinary Least-Squares One-dimensional regression x a b e i i i- = Problem: the data does not go through a line x a b i i- a b Ordinary Least-Squares One-dimensional regression x a b e i i i- = Problem: the data does not go through a line Find the line that minimizes the sum: 2 ) ( - i i i x a b x a b i i- a b 2 Ordinary Least-Squares One-dimensional regression x x a b e i i i- = 2 ) ( ) ( - = i i i x a b x e Problem: the data does not go through a line Find the line that minimizes the sum: We are looking for that minimizes 2 ) ( - i i i x a b x a b i i- a b Ordinary Least-Squares Matrix notation Using the following notations and = n a a : 1 a = n b b : 1 b Ordinary Least-Squares Matrix notation Using the following notations and we can rewrite the error function using linear algebra as: = n a a : 1 a = n b b : 1 b 2 2 ) ( ) ( ) ( ) ( ) ( a b a b a b x x e x x x a b x e T i i i- =-- =- = Ordinary Least-Squares Matrix notation Using the following notations and we can rewrite the error function using linear algebra as: = n a a : 1 a = n b b : 1 b 2 2 ) ( ) ( ) ( ) ( ) ( a b a b a b x x e x x x a b x e T i i i- =-- =- = Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters = + + = j j j m m x a x a x a b ......
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This note was uploaded on 08/22/2010 for the course CAP 6701 taught by Professor Staff during the Spring '10 term at University of Central Florida.

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least squares_slides - 1 Ordinary Least-Squares Ordinary...

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