For Lasso we need now, keeping a constraint on \( \vert\beta_0\vert+\vert\beta_1\vert=1 \), to take the derivative of the absolute values of \( \beta_0 \) -and \( beta_1 \). This gives us the following derivatives of the cost function +and \( \beta_1 \). This gives us the following derivatives of the cost function
$$
C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert),
@@ -1176,7 +1176,7 @@ $$
which leads to
$$
-\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}!
+\hat{\boldsymbol{\beta}}_{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}!
$$
diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html
index c31e6c599..cbb0689ab 100644
--- a/doc/pub/week36/html/week36-solarized.html
+++ b/doc/pub/week36/html/week36-solarized.html
@@ -808,7 +808,7 @@ which gives \( \lambda=4.571 \) and \( \beta_0=0.933 \) and \( \beta_1=0.359 \).
For Lasso we need now, keeping a constraint on \( \vert\beta_0\vert+\vert\beta_1\vert=1 \), to take the derivative of the absolute values of \( \beta_0 \) -and \( beta_1 \). This gives us the following derivatives of the cost function +and \( \beta_1 \). This gives us the following derivatives of the cost function $$ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert), $$ @@ -1150,7 +1150,7 @@ $$ which leads to $$ -\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}! +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}! $$
diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html index 828d35f4c..0a067dd6e 100644 --- a/doc/pub/week36/html/week36.html +++ b/doc/pub/week36/html/week36.html @@ -813,7 +813,7 @@ which gives \( \lambda=4.571 \) and \( \beta_0=0.933 \) and \( \beta_1=0.359 \).
For Lasso we need now, keeping a constraint on \( \vert\beta_0\vert+\vert\beta_1\vert=1 \), to take the derivative of the absolute values of \( \beta_0 \) -and \( beta_1 \). This gives us the following derivatives of the cost function +and \( \beta_1 \). This gives us the following derivatives of the cost function $$ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert), $$ @@ -1155,7 +1155,7 @@ $$ which leads to $$ -\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}! +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}! $$
diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz index e45110d14..45b8b91ed 100644 Binary files a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz and b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz differ diff --git a/doc/pub/week36/ipynb/week36.ipynb b/doc/pub/week36/ipynb/week36.ipynb index 919e6242a..3f50ff006 100644 --- a/doc/pub/week36/ipynb/week36.ipynb +++ b/doc/pub/week36/ipynb/week36.ipynb @@ -970,7 +970,7 @@ "## Lasso case\n", "\n", "For Lasso we need now, keeping a constraint on $\\vert\\beta_0\\vert+\\vert\\beta_1\\vert=1$, to take the derivative of the absolute values of $\\beta_0$\n", - "and $beta_1$. This gives us the following derivatives of the cost function" + "and $\\beta_1$. This gives us the following derivatives of the cost function" ] }, { @@ -1511,7 +1511,7 @@ "metadata": {}, "source": [ "$$\n", - "\\hat{\\boldsymbol{\\beta}}_{mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n", "$$" ] }, diff --git a/doc/src/week36/week36.do.txt b/doc/src/week36/week36.do.txt index 77f964392..047fe5215 100644 --- a/doc/src/week36/week36.do.txt +++ b/doc/src/week36/week36.do.txt @@ -531,7 +531,7 @@ which gives $\lambda=4.571$ and $\beta_0=0.933$ and $\beta_1=0.359$. ===== Lasso case ===== For Lasso we need now, keeping a constraint on $\vert\beta_0\vert+\vert\beta_1\vert=1$, to take the derivative of the absolute values of $\beta_0$ -and $beta_1$. This gives us the following derivatives of the cost function +and $\beta_1$. This gives us the following derivatives of the cost function !bt \[ C(\bm{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert), @@ -856,7 +856,7 @@ Taking the derivative of the *new* cost function with respect to the parameters which leads to !bt \[ -\hat{\bm{\beta}}_{mathrm{OLS}}=\left(\bm{X}^T\bm{X}\right^{-1}\bm{X}^T\bm{y}! +\hat{\bm{\beta}}_{\mathrm{OLS}}=\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}! \] !et