Skip to content

Commit 64cc76e

Browse files
JoramSochJoramSoch
committed
updated some pages
Several small mistakes/errors were corrected in several proofs/definitions.
1 parent ed0b7bc commit 64cc76e

9 files changed

Lines changed: 35 additions & 23 deletions

File tree

‎D/cfm.md‎

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
77
e_mail: "joram.soch@bccn-berlin.de"
88
date: 2021-10-21 17:01
99

10-
title: "General linear model"
10+
title: "Corresponding forward model"
1111
chapter: "Statistical Models"
1212
section: "Multivariate normal data"
1313
topic: "Inverse general linear model"

‎D/iglm.md‎

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
77
e_mail: "joram.soch@bccn-berlin.de"
88
date: 2021-10-21 15:31:00
99

10-
title: "General linear model"
10+
title: "Inverse general linear model"
1111
chapter: "Statistical Models"
1212
section: "Multivariate normal data"
1313
topic: "Inverse general linear model"
@@ -28,7 +28,7 @@ username: "JoramSoch"
2828
---
2929

3030

31-
**Definition:** Let there be a [general linear models](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ in terms of the [design matrix](/D/glm) $X \in \mathbb{R}^{n \times p}$:
31+
**Definition:** Let there be a [general linear model](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ in terms of the [design matrix](/D/glm) $X \in \mathbb{R}^{n \times p}$:
3232

3333
$$ \label{eq:glm}
3434
Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; .

‎D/tglm.md‎

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
77
e_mail: "joram.soch@bccn-berlin.de"
88
date: 2021-10-21 14:43:00
99

10-
title: "General linear model"
10+
title: "Transformed general linear model"
1111
chapter: "Statistical Models"
1212
section: "Multivariate normal data"
1313
topic: "Transformed general linear model"

‎P/cfm-exist.md‎

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
77
e_mail: "joram.soch@bccn-berlin.de"
88
date: 2021-10-21 17:43:00
99

10-
title: "Existence of the corresponding forward model"
10+
title: "Existence of a corresponding forward model"
1111
chapter: "Statistical Models"
1212
section: "Multivariate normal data"
1313
topic: "Inverse general linear model"
@@ -28,7 +28,7 @@ username: "JoramSoch"
2828
---
2929

3030

31-
**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
31+
**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
3232

3333
$$ \label{eq:bda}
3434
\hat{X} = Y W \; .

‎P/cfm-para.md‎

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -28,7 +28,7 @@ username: "JoramSoch"
2828
---
2929

3030

31-
**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
31+
**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
3232

3333
$$ \label{eq:bda}
3434
\hat{X} = Y W \; .
@@ -40,7 +40,7 @@ $$ \label{eq:cfm-para}
4040
A = \Sigma_y W \Sigma_x^{-1}
4141
$$
4242

43-
with the [sample covariance](/D/cov-samp)
43+
with the "[sample covariances](/D/cov-samp)"
4444

4545
$$ \label{eq:Sx-Sy}
4646
\begin{split}

‎P/iglm-blue.md‎

Lines changed: 17 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -73,7 +73,13 @@ $$ \label{eq:W-hat-dist}
7373
\tilde{W} = M X \sim \mathcal{MN}(M Y W, M V M^T, \Sigma_x) \;
7474
$$
7575

76-
which requires that $M Y = I_v$. This is fulfilled by any matrix $M = (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D$ where $D$ is a $v \times n$ matrix which satisfies $D Y = 0$.
76+
[which requires](/P/matn-mean) that $M Y = I_v$. This is fulfilled by any matrix
77+
78+
$$ \label{eq:M-D}
79+
M = (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D
80+
$$
81+
82+
where $D$ is a $v \times n$ matrix which satisfies $D Y = 0$.
7783

7884
<br>
7985
3) Third, the [best linear unbiased estimator](/D/blue) is the one with minimum [variance](/D/var), i.e. the one that minimizes the expected Frobenius norm
@@ -85,26 +91,32 @@ $$
8591
Using the [matrix-normal distribution](/D/matn) of $\tilde{W}$ from \eqref{eq:W-hat-dist}
8692

8793
$$ \label{eq:W-hat-W-dist}
88-
\left( \tilde{W} - W \right) \sim \mathcal{MN}(0, M V M^T, \Sigma_x)
94+
\left( \tilde{W} - W \right) \sim \mathcal{MN}(0, M V M^\mathrm{T}, \Sigma_x)
8995
$$
9096

9197
and the property of the [Wishart distribution](/D/wish)
9298

9399
$$ \label{eq:E-XX}
94-
X \sim \mathcal{MN}(0, U, V) \quad \Rightarrow \quad \left\langle X X^T \right\rangle = \mathrm{tr}(V) \, U \; ,
100+
X \sim \mathcal{MN}(0, U, V) \quad \Rightarrow \quad \left\langle X X^\mathrm{T} \right\rangle = \mathrm{tr}(V) \, U \; ,
95101
$$
96102

97103
this [variance](/D/var) can be evaluated as a function of $M$:
98104

99105
$$ \label{eq:Var-M}
100-
\mathrm{Var}\left[ \tilde{W}(M) \right] = \mathrm{tr}(\Sigma_x) \; \mathrm{tr}(M V M^T) \; .
106+
\begin{split}
107+
\mathrm{Var}\left[ \tilde{W}(M) \right] &\overset{\eqref{eq:Var-W}}{=} \left\langle \mathrm{tr}\left[ (\tilde{W} - W)^\mathrm{T} (\tilde{W} - W) \right] \right\rangle \\
108+
&= \left\langle \mathrm{tr}\left[ (\tilde{W} - W) (\tilde{W} - W)^\mathrm{T} \right] \right\rangle \\
109+
&= \mathrm{tr}\left[ \left\langle (\tilde{W} - W) (\tilde{W} - W)^\mathrm{T} \right\rangle \right] \\
110+
&\overset{\eqref{eq:E-XX}}{=} \mathrm{tr}\left[ \mathrm{tr}(\Sigma_x) \, M V M^\mathrm{T} \right] \\
111+
&= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}(M V M^\mathrm{T}) \; .
112+
\end{split}
101113
$$
102114

103115
As a function of $D$ and using $D Y = 0$, it becomes:
104116

105117
$$ \label{eq:Var-D}
106118
\begin{split}
107-
\mathrm{Var}\left[ \tilde{W}(D) \right] &= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right) V \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right)^\mathrm{T} \right] \\
119+
\mathrm{Var}\left[ \tilde{W}(D) \right] &\overset{\eqref{eq:M-D}}{=} \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right) V \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right)^\mathrm{T} \right] \\
108120
&= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ (Y^\mathrm{T} V^{-1} Y)^{-1} \, Y^\mathrm{T} V^{-1} V V^{-1} Y \; (Y^\mathrm{T} V^{-1} Y)^{-1} + \right. \\
109121
&\hphantom{=\mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[\right.} \left. \, (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} V D^\mathrm{T} + D V V^{-1} Y (Y^\mathrm{T} V^{-1} Y)^{-1} + D V D^\mathrm{T} \right] \\
110122
&= \mathrm{tr}(\Sigma_x) \left[ \mathrm{tr}\!\left( (Y^\mathrm{T} V^{-1} Y)^{-1} \right) + \mathrm{tr}\!\left( D V D^\mathrm{T} \right) \right] \; .

‎P/iglm-dist.md‎

Lines changed: 3 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -40,7 +40,7 @@ $$ \label{eq:iglm}
4040
X = Y W + N, \; N \sim \mathcal{MN}(0, V, \Sigma_x)
4141
$$
4242

43-
where $W \in \mathbb{R}^{v \times p}$ is a matrix, such that $B \, W = I_p$, and the covariance across columns is $\Sigma_x = W^\mathrm{T} \Sigma W$.
43+
where $W \in \mathbb{R}^{v \times p}$ is a matrix, such that $B \, W = I_p$, and the [covariance across columns](/D/matn) is $\Sigma_x = W^\mathrm{T} \Sigma W$.
4444

4545

4646
**Proof:** The [linear transformation theorem for the matrix-normal distribution](/P/matn-ltt) states:
@@ -49,13 +49,13 @@ $$ \label{eq:matn-ltt}
4949
X \sim \mathcal{MN}(M, U, V) \quad \Rightarrow \quad Y = AXB + C \sim \mathcal{MN}(AMB+C, AUA^\mathrm{T}, B^\mathrm{T}VB) \; .
5050
$$
5151

52-
The matrix $W$ exists, if the rows of $B \in \mathbb{R}^{p \times v}$ are linearly independent, such that $\mathrm{rk}(B) = p$. Then, right-multiplying the model \eqref{eq:glm} and applying \eqref{eq:matn-ltt} with $W$ yields
52+
The matrix $W$ exists, if the rows of $B \in \mathbb{R}^{p \times v}$ are linearly independent, such that $\mathrm{rk}(B) = p$. Then, right-multiplying the model \eqref{eq:glm} with $W$ and applying \eqref{eq:matn-ltt} yields
5353

5454
$$ \label{eq:iglm-s1}
5555
Y W = X B W + E W, \; E W \sim \mathcal{MN}(0, V, W^\mathrm{T} \Sigma W) \; .
5656
$$
5757

58-
Applying $B \, W = I_p$ and rearranging, we have
58+
Employing $B \, W = I_p$ and rearranging, we have
5959

6060
$$ \label{eq:iglm-s2}
6161
X = Y W - E W, \; E W \sim \mathcal{MN}(0, V, W^\mathrm{T} \Sigma W) \; .

‎P/tglm-dist.md‎

Lines changed: 5 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -50,7 +50,7 @@ $$ \label{eq:tglm}
5050
\hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)
5151
$$
5252

53-
where the covariance across rows is $U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}$.
53+
where the [covariance across rows](/D/matn) is $U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}$.
5454

5555

5656
**Proof:** The [linear transformation theorem for the matrix-normal distribution](/P/matn-ltt) states:
@@ -75,13 +75,13 @@ Combining \eqref{eq:glm2-wls} with \eqref{eq:Y-dist}, the distribution of $\hat{
7575

7676
$$ \label{eq:G-dist}
7777
\begin{split}
78-
\hat{\Gamma} &\sim \mathrm{MN}\left( \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] X B, \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] V \left[ V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \right], \Sigma \right) \\
79-
&\sim \mathrm{MN}\left( ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t \, T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \\
80-
&\sim \mathrm{MN}\left( T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \; .
78+
\hat{\Gamma} &\sim \mathcal{MN}\left( \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] X B, \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] V \left[ V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \right], \Sigma \right) \\
79+
&\sim \mathcal{MN}\left( ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t \, T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \\
80+
&\sim \mathcal{MN}\left( T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \; .
8181
\end{split}
8282
$$
8383

84-
This can also written as
84+
This can also be written as
8585

8686
$$ \label{eq:tglm-qed}
8787
\hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}\left( 0, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right)

‎P/tglm-para.md‎

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -40,7 +40,7 @@ $$ \label{eq:tglm}
4040
\hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)
4141
$$
4242

43-
which are linked to each other via
43+
which [are linked to each other](/P/tglm-dist) via
4444

4545
$$ \label{eq:glm2-wls}
4646
\hat{\Gamma} = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} Y
@@ -52,7 +52,7 @@ $$ \label{eq:X-Xt-T}
5252
X = X_t \, T \; .
5353
$$
5454

55-
Then, the parameter estimates from \eqref{eq:glm1} and \eqref{eq:tglm} are equivalent.
55+
Then, the parameter estimates for $B$ from \eqref{eq:glm1} and \eqref{eq:tglm} are equivalent.
5656

5757

5858
**Proof:** The [weighted least squares parameter estimates](/P/glm-wls) for \eqref{eq:glm1} are given by

0 commit comments

Comments
 (0)