From fb0906d974a101644dd290548e25078d8d461026 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Mon, 11 Nov 2019 04:47:44 +0100 Subject: [PATCH] cleaning more --- .../html/._DecisionTrees-bs048.html | 4 ++-- .../html/._DecisionTrees-bs056.html | 2 +- .../html/DecisionTrees-reveal.html | 6 +++--- .../html/DecisionTrees-solarized.html | 6 +++--- doc/pub/DecisionTrees/html/DecisionTrees.html | 6 +++--- .../DecisionTrees/ipynb/DecisionTrees.ipynb | 6 +++--- .../ipynb/ipynb-DecisionTrees-src.tar.gz | Bin 294061 -> 294061 bytes .../pdf/DecisionTrees-minted.pdf | Bin 542561 -> 542776 bytes doc/src/DecisionTrees/DecisionTrees.do.txt | 6 +++--- 9 files changed, 18 insertions(+), 18 deletions(-) diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs048.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs048.html index ada510754..dac77df12 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs048.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs048.html @@ -293,7 +293,7 @@ $$ \frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. $$ -We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma x_i) \) with \( \boldsymbol{e} \) being the unit vector) +We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector) $$ \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, $$ @@ -305,7 +305,7 @@ $$

which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting -for \( \beta \) gives us an equation for \( \gamma \). +for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.

The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html index 4a6f767cb..ce76f15ab 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs056.html @@ -269,7 +269,7 @@ MathJax.Hub.Config({

Gradient Boosting, algorithm

-Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard square-error function +Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function $$ C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html index 59ee06a8b..9b9b3510e 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html @@ -2098,7 +2098,7 @@ $$ $$

 
-We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma x_i) \) with \( \boldsymbol{e} \) being the unit vector) +We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector)

 
$$ \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, @@ -2114,7 +2114,7 @@ $$

which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting -for \( \beta \) gives us an equation for \( \gamma \). +for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.

The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as @@ -2376,7 +2376,7 @@ See discussion during lecture November 8.

Gradient Boosting, algorithm

-Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard square-error function +Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function

 
$$ C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html index 27309b5be..d0156f2db 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html @@ -2094,7 +2094,7 @@ $$ \frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. $$ -We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma x_i) \) with \( \boldsymbol{e} \) being the unit vector) +We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector) $$ \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, $$ @@ -2106,7 +2106,7 @@ $$

which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting -for \( \beta \) gives us an equation for \( \gamma \). +for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.

The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as @@ -2337,7 +2337,7 @@ See discussion during lecture November 8.

Gradient Boosting, algorithm

-Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard square-error function +Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function $$ C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ diff --git a/doc/pub/DecisionTrees/html/DecisionTrees.html b/doc/pub/DecisionTrees/html/DecisionTrees.html index f596aab80..941aaf5f3 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees.html @@ -2099,7 +2099,7 @@ $$ \frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. $$ -We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma x_i) \) with \( \boldsymbol{e} \) being the unit vector) +We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector) $$ \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, $$ @@ -2111,7 +2111,7 @@ $$

which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting -for \( \beta \) gives us an equation for \( \gamma \). +for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.

The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as @@ -2342,7 +2342,7 @@ See discussion during lecture November 8.

Gradient Boosting, algorithm

-Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard square-error function +Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function $$ C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$ diff --git a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb index 4bfc45466..2aeaf36de 100644 --- a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb +++ b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb @@ -2172,7 +2172,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma x_i)$ with $\\boldsymbol{e}$ being the unit vector)" + "We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma \\boldsymbol{x})$ with $\\boldsymbol{e}$ being the unit vector)" ] }, { @@ -2205,7 +2205,7 @@ "metadata": {}, "source": [ "which leads to $\\gamma =(\\boldsymbol{x}^T\\boldsymbol{y}-\\beta\\boldsymbol{x}^T\\boldsymbol{e})/(\\beta\\boldsymbol{x}^T\\boldsymbol{x})$. Inserting\n", - "for $\\beta$ gives us an equation for $\\gamma$.\n", + "for $\\beta$ gives us an equation for $\\gamma$. This is a non-linear equation in the unknown $\\gamma$ and has to be solved numerically. \n", "\n", "The solution to these two equations gives us in turn $\\beta_1$ and $\\gamma_1$ leading to the new expression for $f_1(x)$ as\n", "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. \n", @@ -2569,7 +2569,7 @@ "\n", "## Gradient Boosting, algorithm\n", "\n", - "Suppose we have a cost function $C(f)=\\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard square-error function" + "Suppose we have a cost function $C(f)=\\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. 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