-If we then compute the expectation value +If we then compute the expectation value (note the \( 1/n \) factor instead of \( 1/(n-1) \))
$$
-\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix}
+\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\frac{1}{n}\begin{bmatrix}
x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
\end{bmatrix},
@@ -2675,7 +2675,7 @@ $$
where we wrote
$$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$
-
to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
+
to indicate that this is the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html index bbcada960..02bae429b 100644 --- a/doc/pub/week35/html/week35-solarized.html +++ b/doc/pub/week35/html/week35-solarized.html @@ -2666,9 +2666,9 @@ x_{10} & x_{11}\\ $$
-If we then compute the expectation value +If we then compute the expectation value (note the \( 1/n \) factor instead of \( 1/(n-1) \)) $$ -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} +\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\frac{1}{n}\begin{bmatrix} x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ \end{bmatrix}, @@ -2681,7 +2681,7 @@ $$ \end{bmatrix}, $$ -where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \). +where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this is the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html index 3efd21e35..2fbf430b4 100644 --- a/doc/pub/week35/html/week35.html +++ b/doc/pub/week35/html/week35.html @@ -2671,9 +2671,9 @@ x_{10} & x_{11}\\ $$
-If we then compute the expectation value +If we then compute the expectation value (note the \( 1/n \) factor instead of \( 1/(n-1) \)) $$ -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} +\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\frac{1}{n}\begin{bmatrix} x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ \end{bmatrix}, @@ -2686,7 +2686,7 @@ $$ \end{bmatrix}, $$ -where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \). +where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this is the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz index 76b4c6ce1..f1916750f 100644 Binary files a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz and b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz differ diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb index 86a83a788..55177c1d4 100644 --- a/doc/pub/week35/ipynb/week35.ipynb +++ b/doc/pub/week35/ipynb/week35.ipynb @@ -3402,7 +3402,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "If we then compute the expectation value" + "If we then compute the expectation value (note the $1/n$ factor instead of $1/(n-1)$)" ] }, { @@ -3410,7 +3410,7 @@ "metadata": {}, "source": [ "$$\n", - "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", + "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\frac{1}{n}\\begin{bmatrix}\n", "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", "\\end{bmatrix},\n", @@ -3439,7 +3439,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", + "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this is the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", "\n", "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", "\n", diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt index 0038d9465..5d31e1f9a 100644 --- a/doc/src/week35/week35.do.txt +++ b/doc/src/week35/week35.do.txt @@ -2147,10 +2147,10 @@ x_{10} & x_{11}\\ \] !et -If we then compute the expectation value +If we then compute the expectation value (note the $1/n$ factor instead of $1/(n-1)$) !bt \[ -\mathbb{E}[\bm{X}^T\bm{X}] = \frac{1}{n}\bm{X}^T\bm{X}=\begin{bmatrix} +\mathbb{E}[\bm{X}^T\bm{X}] = \frac{1}{n}\bm{X}^T\bm{X}=\frac{1}{n}\begin{bmatrix} x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ \end{bmatrix}, @@ -2164,7 +2164,7 @@ which is just \end{bmatrix}, \] !et -where we wrote $$\bm{C}[\bm{x}_0,\bm{x}_1] = \bm{C}[\bm{x}]$$ to indicate that this the covariance of the vectors $\bm{x}$ of the design/feature matrix $\bm{X}$. +where we wrote $$\bm{C}[\bm{x}_0,\bm{x}_1] = \bm{C}[\bm{x}]$$ to indicate that this is the covariance of the vectors $\bm{x}$ of the design/feature matrix $\bm{X}$. It is easy to generalize this to a matrix $\bm{X}\in {\mathbb{R}}^{n\times p}$.