Difference: AVFLogA012LeastSquares (11 vs. 12)

Revision 122015-03-25 - AlexanderFedotov

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 META TOPICPARENT name="AVFedotovLogA"
-- AlexanderFedotov - 2015-03-03
Line: 313 to 313
Summing up (4.2.10) and (4.2.11) gives
Changed:
<
<
\beta^{(k)}
>
>
\boxed { \beta^{(k)} }
& = & ( I_{g_k} + I_{ \bar{g_k} } ) \beta^{(k)} = \ & = & (X_{g_k}^T W X_{g_k})^{-1}_E X^T W y + [I - (X_{g_k}^T W X_{g_k})^{-1}_E X_T W X ] I_{ \bar{g_k} } \beta^{(k-1)} =\
Changed:
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<
& = & A_{g_k} X^T W y + B_{g_k} \beta^{(k-1)}
>
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& = & \boxed {A_{g_k} X^T W y + B_{g_k} \beta^{(k-1)} }
\begin{array}{lllrr} \boxed{ A_{g_k}} & = & (X_{g_k}^T W X_{g_k})^{-1}_E & \qquad , & \qquad \qquad (4.2.13) \
Changed:
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<
\boxed{ B_{g_k}} & = & (I - A_{g_k} X_T W X ) I_{ \bar{g_k} } & \qquad . & \qquad \qquad (4.2.14)
>
>
\boxed{ B_{g_k}} & = & (I - A_{g_k} X^T W X ) I_{ \bar{g_k} } & \qquad . & \qquad \qquad (4.2.14)
\end{array}
Line: 334 to 334

4.3.1 as a function of

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<
Let us define matrices and as
>
>
Let us define matrices and as

Changed:
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<
\boxed{ D_k } = \left\{
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\boxed{ A_k } = \left\{
\begin{array}{lll} 0 & , & k=0 \
Changed:
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A_{g_k} + B_{g_k} D_{k-1} & , & k > 0 .
>
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A_{g_k} + B_{g_k} A_{k-1} & , & k > 0 .

Line: 351 to 351
Then, from (4.2.12),
Changed:
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<
\boxed{ \beta^{(k)} = D_k X^T W y + B_k \beta^{(0)} }
>
>
\boxed{ \beta^{(k)} = A_k X^T W y + B_k \beta^{(0)} }
• eq.(4.3.3) holds for :
Changed:
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<
\beta^{(1)} = A_{g_1} X^T W y + B_{g_1} \beta^{(0)} = D_1 X^T W y + B_1 \beta^{(0)}
>
>
\beta^{(1)} = A_{g_1} X^T W y + B_{g_1} \beta^{(0)} = A_1 X^T W y + B_1 \beta^{(0)}
• and assuming it is true for , leads to
\beta^{(p+1)} & = & A_{g_{p+1}} X^T W y + B_{g_{p+1}} \beta^{(p)} = \
Changed:
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<
& = & A_{g_{p+1}} X^T W y + B_{g_{p+1}} ( D_p X^T W y + B_p \beta^{(0)} ) = \ & = & ( A_{g_{p+1}} + B_{g_{p+1}} D_p ) X^T W y + B_{g_{p+1}} B_p \beta^{(0)} = \ & = & D_{p+1} X^T W y + B_{p+1} \beta^{(0)}
>
>
& = & A_{g_{p+1}} X^T W y + B_{g_{p+1}} ( A_p X^T W y + B_p \beta^{(0)} ) = \ & = & ( A_{g_{p+1}} + B_{g_{p+1}} A_p ) X^T W y + B_{g_{p+1}} B_p \beta^{(0)} = \ & = & A_{p+1} X^T W y + B_{p+1} \beta^{(0)}

In particular, the eq.(4.3.3) is valid for :

Changed:
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\boxed{ \beta^{(N)} = D_N X^T W y + B_N \beta^{(0)} }
>
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\boxed{ \beta^{(N)} = A_N X^T W y + B_N \beta^{(0)} }
Line: 384 to 384
eq.(4.3.3) gives rise to
M_{ \beta^{(k)} }
Changed:
<
<
& = & D_k X^T W M_y (D_k X^T W)^T = D_k X^T W M_y W^T X D_k^T = \ & = & D_k X^T W X D_k^T
>
>
& = & A_k X^T W M_y (A_k X^T W)^T = A_k X^T W M_y W^T X A_k^T = \ & = & A_k X^T W X A_k^T
\qquad . \qquad \qquad (4.3.7) Comparing this with the covariance matrix of the exact solution, eq.(2.6), yields
Changed:
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\boxed{ \lim_{k \to \infty} D_k X^T W X D_k^T = M_{\hat{\beta}} = (X^T W X ) ^{-1} }
>
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\boxed{ \lim_{k \to \infty} A_k X^T W X A_k^T = M_{\hat{\beta}} = (X^T W X ) ^{-1} }
Line: 402 to 402
Changed:
<
<
we can build the expression for the case in one step as
>
>
one can build the expression for the case in one step as

Line: 410 to 410
Substituting (4.3.6) into (4.4.2) gives
\beta^{(2N)}
Changed:
<
<
& = & D_N X^T W y + B_N ( D_N X^T W y + B_N \beta^{(0)} ) =\ & = & (I + B_N)D_N X^T W y + B_N^2 \beta^{(0)}
>
>
& = & A_N X^T W y + B_N ( A_N X^T W y + B_N \beta^{(0)} ) =\ & = & (I + B_N)A_N X^T W y + B_N^2 \beta^{(0)}
Changed:
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<
\beta^{(4N)} = (I + B_N^2)(I + B_N)D_N X^T W y + B_N^4 \beta^{(0)}
>
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\beta^{(4N)} = (I + B_N^2)(I + B_N)A_N X^T W y + B_N^4 \beta^{(0)}
Changed:
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<
\beta^{(8N)} = (I + B_N^4)(I + B_N^2)(I + B_N)D_N X^T W y + B_N^8 \beta^{(0)}
>
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\beta^{(8N)} = (I + B_N^4)(I + B_N^2)(I + B_N)A_N X^T W y + B_N^8 \beta^{(0)}
Line: 429 to 429
This generalizes to
\boxed{
Changed:
<
<
\beta^{(2^p N)} = \left\lgroup \prod_{i=p-1}^0 (I + B_N^{2^i}) \right\rgroup D_N X^T W y
>
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\beta^{(2^p N)} = \left\lgroup \prod_{i=p-1}^0 (I + B_N^{2^i}) \right\rgroup A_N X^T W y