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least squares_slides

# least squares_slides - Outline Linear regression Geometry...

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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

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2 Ordinary Least-Squares One-dimensional regression x ˆ 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 ... 1 1 Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters 2 a = + + = j j j m m x a x a x a b ...
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least squares_slides - Outline Linear regression Geometry...

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