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1 C 0 −1
in VT Orthogonal basis forVT P FV
-1 0 0 1 τ τˆ CC
-1 0 0 1 τ τˆ CC
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-1 0 0 1 τ τˆ CC
Some vectors 4
2 C 0 −1 4
3 C 0 −1 4
4 B -1 0 1 0 1 0 τ τˆ BB
5 B -1 0 1 0 1 0 τ τˆ BB
6 A 1 1 0 1 0 0 τ τˆ
7 B -1 0 1 0 1 0 τ τˆ
1 0 1 0 0 τ τˆ
1 0 1 0 0 τ τˆ
10 B 0 0 1 0 1 0 τ τˆ
11 A 1 1 0 1 0 0 τ τˆ
12 C 0 -1 -1 0 0 1 τ τˆ
Linear Model
The vector τ in the previous table is a vector of unknown parameters. We also know that τ ∈ VT . Therefore, it can be written as the linear combination of the vectors in the orthogonal basis of VT . That is
τ = τAuA +τBuB +τCuC
From our data we can compute τˆ , τˆ , τˆ . Then vector of estimates is
τˆ=τˆu +τˆu +τˆu AABBCC
Linear Model
AlsonotethatavectorvinV canbewrittenasthesumofx+z. Generally speaking, all vectors in V can be written as the linear combination of x ∈ W and z ∈ W⊥. Therefore, V is the direct sum1 of W and W⊥.
V = W ⊕ W⊥
1Let U, W be subspaces of V . Then V = U ⊕ W if and only if for every v ∈ V there exist unique vectors u ∈ U and w ∈ Wsuch that v = u + w.
Linear Model
Let us consider a fixed v in V and any w ∈ W. The squared difference ||v − w||2 is minimized when w = PW v. The object PW can be seen as a linear operator, which when applied on v gives the orthogonal projection of v on W. If dimension of W is t then
t v.ui PW v = ∑ u .u ui
The operator PW v can be written as dim(V ) × dim(V ) matrix.
Linear Model
Properties of the operator matrix PW
The matrix PW is symmetric. i.e. PW = PTW The matrix PW is idempotent. i.e. PW = P2W The rank of PW = traceP(W) = dim (W).
Linear Model
Theorem .1
Let the linear model is
Y = τ1u1 + τ2u2 + …….τtut + ε
Where E(εij) = 0 Cov(ε) = σ2I. Also let W is a subspace of V. Then
1 E(PWY) = PW(E(Y)) = PWτ
2 E(||PWY||2) = ||PWτ||2 +dim(W)σ2
Y = τ1u1 + τ2u2 + …….τtut + ε
E(εij) = 0 =⇒ E(Y) = τ ∈ VT and Cov(Y) = σ2I Hence, E(PWY) = PW(E(Y)) = PWτ
Now consider a random vector X. Then X.X = ||X||2 = ∑w Xw2
E(X.X)=E||X||2 =E∑Xw2 w
= ∑ E ( X w2 ) w
= ∑{Var(Xw) + (E(Xw))2} w
=trace(Cov(X)) + ||E(X)||2 Where trace(Cov(X)) = σ12 + σ2 + …….σN2
Now let X = PW Y Then
Cov(X) =Cov(PWY) =PW Cov(Y )PTW
=PWσ2IPW = σ 2 P 2W =σ2PW
E(||PWY||2) =trace(σ2PW) + ||PWτ||2 =σ2trace(PW ) + ||PW τ||2
=σ2dimW + ||PW τ||2
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