Difference: AVFLogA012LeastSquares (11 vs. 12)

Revision 122015-03-25 - AlexanderFedotov

Line: 1 to 1
 
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:
<
<
& = & A_{g_k} X^T W y + B_{g_k} \beta^{(k-1)}
>
>
& = & \boxed {A_{g_k} X^T W y + B_{g_k} \beta^{(k-1)} }
  \qquad . \qquad \qquad (4.2.12) where we denoted
\begin{array}{lllrr} \boxed{ A_{g_k}} & = & (X_{g_k}^T W X_{g_k})^{-1}_E & \qquad , & \qquad \qquad (4.2.13) \
Changed:
<
<
\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

Changed:
<
<
Let us define matrices and as
>
>
Let us define matrices and as
 
Changed:
<
<
\boxed{ D_k } = \left\{
>
>
\boxed{ A_k } = \left\{
  \begin{array}{lll} 0 & , & k=0 \
Changed:
<
<
A_{g_k} + B_{g_k} D_{k-1} & , & k > 0 .
>
>
A_{g_k} + B_{g_k} A_{k-1} & , & k > 0 .
  \end{array} \right. \qquad , \qquad\qquad (4.3.1)

Line: 351 to 351
  Then, from (4.2.12),
Changed:
<
<
\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)} }
  \qquad . \qquad \qquad (4.3.3) Indeed, by induction:
  • eq.(4.3.3) holds for :
Changed:
<
<
\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)}
  \qquad , \qquad \qquad (4.3.4)
  • 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:
<
<
& = & 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)}
  \qquad , \qquad \qquad (4.3.5)

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

Changed:
<
<
\boxed{ \beta^{(N)} = D_N X^T W y + B_N \beta^{(0)} }
>
>
\boxed{ \beta^{(N)} = A_N X^T W y + B_N \beta^{(0)} }
  \qquad . \qquad \qquad (4.3.6)
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:
<
<
\boxed{ \lim_{k \to \infty} D_k X^T W X D_k^T = M_{\hat{\beta}} = (X^T W X ) ^{-1} }
>
>
\boxed{ \lim_{k \to \infty} A_k X^T W X A_k^T = M_{\hat{\beta}} = (X^T W X ) ^{-1} }
  \qquad . \qquad \qquad (4.3.8)
Line: 402 to 402
  \beta^{(N)} = F(\beta^{(0)}) \qquad . \qquad \qquad (4.4.1)
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
 
\beta^{(2N)} = F(\beta^{(N)}) = F(F(\beta^{(0)})) \qquad . \qquad \qquad (4.4.2)
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)}
  \qquad . \qquad \qquad (4.4.3) Transforming similarly the formula to the one, gives
Changed:
<
<
\beta^{(4N)} = (I + B_N^2)(I + B_N)D_N X^T W y + B_N^4 \beta^{(0)}
>
>
\beta^{(4N)} = (I + B_N^2)(I + B_N)A_N X^T W y + B_N^4 \beta^{(0)}
  \qquad . \qquad \qquad (4.4.4) Then for the case one has
Changed:
<
<
\beta^{(8N)} = (I + B_N^4)(I + B_N^2)(I + B_N)D_N X^T W y + B_N^8 \beta^{(0)}
>
>
\beta^{(8N)} = (I + B_N^4)(I + B_N^2)(I + B_N)A_N X^T W y + B_N^8 \beta^{(0)}
  \qquad , \qquad \qquad (4.4.4) and so on...
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
>
>
\beta^{(2^p N)} = \left\lgroup \prod_{i=p-1}^0 (I + B_N^{2^i}) \right\rgroup A_N X^T W y
  + B_N^{2^p} \beta^{(0)} \quad , \quad p = 1,2, \: ... \quad }
 
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