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lines changed Original file line number Diff line number Diff line change @@ -1168,7 +1168,7 @@ $$ (eq:decoder102)
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Since $\Phi$ has $p$ linearly independent columns, the generalized inverse of $\Phi$ is
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$$
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- \Phi^{\dagger } = (\Phi^T \Phi)^{-1} \Phi^T
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+ \Phi^{+ } = (\Phi^T \Phi)^{-1} \Phi^T
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$$
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and so
@@ -1177,12 +1177,12 @@ $$
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\check b = (\Phi^T \Phi)^{-1} \Phi^T X
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$$ (eq:checkbform)
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- $\check b$ is recognizable as the matrix of least squares regression coefficients of the matrix
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+ The matrix $\check b$ is recognizable as the matrix of least squares regression coefficients of the matrix
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$X$ on the matrix $\Phi$ and
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$$
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\check X = \Phi \check b
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- $$
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+ $$ (eq:Xcheck_)
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is the least squares projection of $X$ on $\Phi$.
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@@ -1195,6 +1195,12 @@ we can represent $X$ as the sum of the projection $\check X$ of $X$ on $\Phi$ p
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To verify this, note that the least squares projection $\check X$ is related to $X$ by
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+ $$
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+ X = \check X + \epsilon
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+ $$
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+
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+ or
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+
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$$
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X = \Phi \check b + \epsilon
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$$
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(X - \Phi \check b)^T \Phi = 0_ {m \times p}
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$$ (eq:orthls)
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- Rearranging the orthogonality conditions {eq}`eq:orthls` gives $X^T \Phi = \check b \Phi^T \Phi$
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+ Rearranging the orthogonality conditions {eq}`eq:orthls` gives $X^T \Phi = \check b \Phi^T \Phi$,
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which implies formula {eq}`eq:checkbform`.
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