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Unformatted text preview: Geometry of the GaussMarkov Linear Model Reminder from the last section of the notes: y = X + We saw two possible X matrices for the ttest. This section focuses on the Q: does it matter which X we use? Important pieces of information for what follows: I X is sometimes referred to as the design matrix . It is an n p matrix of constants with columns corresponding to explanatory variables. I is an unknown parameter vector in IR p . Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 32 The Column Space of the Design Matrix I X is a linear combination of the columns of X : X = [ x 1 , .. . , x p ] 1 . . . p = 1 x 1 + + p x p . I The set of all possible linear combinations of the columns of X is called the column space of X and is denoted by C ( X ) = { Xa : a IR p } . I The GaussMarkov linear model says y is a random vector whose: mean is in the column space of X and whose variance is 2 I for some positive real number 2 , i.e., E ( y ) C ( X ) and Var ( y ) = 2 I , 2 IR + . Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 32 An Example Column Space X = 1 1 = C ( X ) = { Xa : a IR } = 1 1 a 1 : a 1 IR = a 1 a 1 : a 1 IR What does this column space look like? Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 321.00.5 0.0 0.5 1.01.00.5 0.0 0.5 1.0 X1 X2 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 32 Another Example Column Space X = 1 1 1 1 = C ( X ) = 1 1 1 1 a 1 a 2 : a IR 2 = a 1 1 1 + a 2 1 1 : a 1 , a 2 IR = a 1 a 1 a 2 a 2 : a 1 , a 2 IR What is this column space? A plane living in IR 4 . Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 5 / 32 A Third Column Space Example X 2 = 1 1 0 1 1 0 1 0 1 1 0 1 x C ( X 2 ) = x = X 2 a for some a IR 3 = x = a 1 1 1 1 1 + a 2 1 1 + a 3 1 1 for some a IR 3 = x = a 1 + a 2 a 1 + a 2 a 1 + a 3 a 1 + a 3 = b 1 b 1 b 2 b 2 for some b 1 , b 2 IR This is also a plane in R 4 . Is it the same plane? Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 6 / 32 Proving that two column spaces, C ( X 1 ) and C ( X 2 ) , are the same Concept: I If you can start with x C ( X 1 ) and derive x = X 2 b , this implies x C ( X 2 ) and C ( X 1 ) C ( X 2 ) ....
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 Spring '08
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