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02Geometry(1)

# 02Geometry(1) - Geometry of the Gauss-Markov Linear Model X...

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Geometry of the Gauss-Markov Linear Model Reminder from the last section of the notes: y = X β + We saw two possible X matrices for the t-test. This section focuses on the Q: does it matter which X we use? Important pieces of information for what follows: X is sometimes referred to as the design matrix . It is an n × p matrix of constants with columns corresponding to explanatory variables. β 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 X β is a linear combination of the columns of X : X β = [ x 1 , . . . , x p ] β 1 . . . β p = β 1 x 1 + · · · + β p x p . 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 } . The Gauss-Markov linear model says y is a random vector whose: mean is in the column space of X and whose variance is σ 2 I for some positive real number σ 2 , i.e., E ( y ) ∈ C ( X ) and Var ( y ) = σ 2 I , σ 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 / 32 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 X1 X2 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 32

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Another Example Column Space X = 1 0 1 0 0 1 0 1 = ⇒ C ( X ) = 1 0 1 0 0 1 0 1 a 1 a 2 : a IR 2 = a 1 1 1 0 0 + a 2 0 0 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 0 0 + a 3 0 0 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: 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 ) . N.B. Not (at least yet) C ( X 1 ) = C ( X 2 ) because there may be some b for which X 2 b is not in C ( X 1 ) .
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02Geometry(1) - Geometry of the Gauss-Markov Linear Model X...

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