From 703c79a4f155aab4a90c6fa0d37d58ea76f80a0d Mon Sep 17 00:00:00 2001 From: mhjensen Date: Sun, 24 Nov 2019 22:43:10 +0100 Subject: [PATCH] Forgot to update CART also --- .../html/._DecisionTrees-bs000.html | 2 +- .../html/._DecisionTrees-bs017.html | 5 +- .../html/._DecisionTrees-bs018.html | 22 +++++ .../html/._DecisionTrees-bs019.html | 23 +++++ .../DecisionTrees/html/DecisionTrees-bs.html | 2 +- .../html/DecisionTrees-reveal.html | 60 +++++++++++- .../html/DecisionTrees-solarized.html | 52 +++++++++- doc/pub/DecisionTrees/html/DecisionTrees.html | 52 +++++++++- .../DecisionTrees/ipynb/DecisionTrees.ipynb | 89 +++++++++++++++++- .../ipynb/ipynb-DecisionTrees-src.tar.gz | Bin 294061 -> 294061 bytes .../pdf/DecisionTrees-minted.pdf | Bin 556703 -> 563541 bytes doc/src/DecisionTrees/DecisionTrees.do.txt | 47 ++++++++- 12 files changed, 342 insertions(+), 12 deletions(-) diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html index 054a16087..d84cbeccc 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs000.html @@ -297,7 +297,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 13, 2019

+

Nov 24, 2019


diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs017.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs017.html index 97a71b5e7..dcc2c563b 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs017.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs017.html @@ -288,7 +288,10 @@ Two algorithms stand out in the set up of decision trees:

  • The ID3 algorithm based on the computation of the information gain for classification
  • -We discuss both algorithms with applications here. The popular library Scikit-Learn uses the CART algorithm. For classification problems you can use either the gini index or the entropy to split a tree in two branches. +We discuss both algorithms with applications here. The popular library +Scikit-Learn uses the CART algorithm. For classification problems +you can use either the gini index or the entropy to split a tree +in two branches.

    diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs018.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs018.html index f106735bf..b8afc2173 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs018.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs018.html @@ -280,6 +280,28 @@ MathJax.Hub.Config({

    The CART algorithm for Classification

    +

    +For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). +This could be for example a threshold set by a number below a certain circumference of a malign tumor. + +

    +How do we find these two quantities? +We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). +The cost function it tries to minimize is then +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +$$ + +where \( G_{\mathrm{left\left}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \) + is the number of instances in the left/right subset + +

    +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the \( min\_samples\_split \), +\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \). +

    diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs019.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs019.html index 424959151..19dc10456 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs019.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs019.html @@ -280,6 +280,29 @@ MathJax.Hub.Config({

    The CART algorithm for Regression

    +

    +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +$$ + +Here the MSE for a specific node is defined as +$$ +\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, +$$ + +with +$$ +\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, +$$ + +the mean value of all observations in a specific node. + +

    +Without any regularization, the regression task for decision trees, +just like for classification tasks, is prone to overfitting. +

    diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html index 054a16087..d84cbeccc 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html @@ -297,7 +297,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 13, 2019

    +

    Nov 24, 2019


    diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html index c2b91f094..b65e88beb 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Nov 13, 2019

    +

    Nov 24, 2019


    @@ -708,17 +708,73 @@ Two algorithms stand out in the set up of decision trees:

    -We discuss both algorithms with applications here. The popular library Scikit-Learn uses the CART algorithm. For classification problems you can use either the gini index or the entropy to split a tree in two branches. +We discuss both algorithms with applications here. The popular library +Scikit-Learn uses the CART algorithm. For classification problems +you can use either the gini index or the entropy to split a tree +in two branches.

    The CART algorithm for Classification

    + +

    +For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). +This could be for example a threshold set by a number below a certain circumference of a malign tumor. + +

    +How do we find these two quantities? +We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). +The cost function it tries to minimize is then +

     
    +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +$$ +

     
    + +where \( G_{\mathrm{left\left}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \) + is the number of instances in the left/right subset + +

    +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the \( min\_samples\_split \), +\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \).

    The CART algorithm for Regression

    + +

    +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +

     
    +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +$$ +

     
    + +Here the MSE for a specific node is defined as +

     
    +$$ +\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, +$$ +

     
    + +with +

     
    +$$ +\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, +$$ +

     
    + +the mean value of all observations in a specific node. + +

    +Without any regularization, the regression task for decision trees, +just like for classification tasks, is prone to overfitting.

    diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html index 3285766a2..a28aaf946 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html @@ -223,7 +223,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 13, 2019

    +

    Nov 24, 2019












    @@ -755,18 +755,66 @@ Two algorithms stand out in the set up of decision trees:

  • The ID3 algorithm based on the computation of the information gain for classification
  • -We discuss both algorithms with applications here. The popular library Scikit-Learn uses the CART algorithm. For classification problems you can use either the gini index or the entropy to split a tree in two branches. +We discuss both algorithms with applications here. The popular library +Scikit-Learn uses the CART algorithm. For classification problems +you can use either the gini index or the entropy to split a tree +in two branches.











    The CART algorithm for Classification

    +

    +For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). +This could be for example a threshold set by a number below a certain circumference of a malign tumor. + +

    +How do we find these two quantities? +We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). +The cost function it tries to minimize is then +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +$$ + +where \( G_{\mathrm{left\left}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \) + is the number of instances in the left/right subset + +

    +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the \( min\_samples\_split \), +\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \). +











    The CART algorithm for Regression

    +

    +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +$$ + +Here the MSE for a specific node is defined as +$$ +\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, +$$ + +with +$$ +\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, +$$ + +the mean value of all observations in a specific node. + +

    +Without any regularization, the regression task for decision trees, +just like for classification tasks, is prone to overfitting. +











    diff --git a/doc/pub/DecisionTrees/html/DecisionTrees.html b/doc/pub/DecisionTrees/html/DecisionTrees.html index 61f9eb3dd..9d22847ca 100644 --- a/doc/pub/DecisionTrees/html/DecisionTrees.html +++ b/doc/pub/DecisionTrees/html/DecisionTrees.html @@ -228,7 +228,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 13, 2019

    +

    Nov 24, 2019












    @@ -760,18 +760,66 @@ Two algorithms stand out in the set up of decision trees:

  • The ID3 algorithm based on the computation of the information gain for classification
  • -We discuss both algorithms with applications here. The popular library Scikit-Learn uses the CART algorithm. For classification problems you can use either the gini index or the entropy to split a tree in two branches. +We discuss both algorithms with applications here. The popular library +Scikit-Learn uses the CART algorithm. For classification problems +you can use either the gini index or the entropy to split a tree +in two branches.











    The CART algorithm for Classification

    +

    +For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). +This could be for example a threshold set by a number below a certain circumference of a malign tumor. + +

    +How do we find these two quantities? +We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). +The cost function it tries to minimize is then +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +$$ + +where \( G_{\mathrm{left\left}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \) + is the number of instances in the left/right subset + +

    +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the \( min\_samples\_split \), +\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \). +











    The CART algorithm for Regression

    +

    +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +$$ + +Here the MSE for a specific node is defined as +$$ +\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, +$$ + +with +$$ +\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, +$$ + +the mean value of all observations in a specific node. + +

    +Without any regularization, the regression task for decision trees, +just like for classification tasks, is prone to overfitting. +











    diff --git a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb index 5f72bcd04..aaf2eaca4 100644 --- a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb +++ b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Nov 13, 2019**\n", + "Date: **Nov 24, 2019**\n", "\n", "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -609,13 +609,98 @@ "\n", "2. The ID3 algorithm based on the computation of the information gain for classification\n", "\n", - "We discuss both algorithms with applications here. The popular library **Scikit-Learn** uses the CART algorithm. For classification problems you can use either the **gini** index or the **entropy** to split a tree in two branches.\n", + "We discuss both algorithms with applications here. The popular library\n", + "**Scikit-Learn** uses the CART algorithm. For classification problems\n", + "you can use either the **gini** index or the **entropy** to split a tree\n", + "in two branches.\n", "\n", "## The CART algorithm for Classification\n", "\n", + "For classification, the CART algorithm splits the data set in two subsets using a single feature $k$ and a threshold $t_k$.\n", + "This could be for example a threshold set by a number below a certain circumference of a malign tumor.\n", + "\n", + "How do we find these two quantities?\n", + "We search for the pair $(k,t_k)$ that produces the purest subset using for example the **gini** factor $G$.\n", + "The cost function it tries to minimize is then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}G_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}G_{\\mathrm{right}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $G_{\\mathrm{left\\left}}$ measures the impurity of the left/right subset and $m_{\\mathrm{left/right}}$\n", + " is the number of instances in the left/right subset\n", + "\n", + "Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets\n", + "and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the\n", + "$max\\_depth$ hyperparameter), or if it cannot find a split that will reduce impurity. A few other\n", + "hyperparameters control additional stopping conditions such as the $min\\_samples\\_split$,\n", + "$min\\_samples\\_leaf$, $min\\_weight\\_fraction\\_leaf$, and $max\\_leaf\\_nodes$.\n", "\n", "## The CART algorithm for Regression\n", "\n", + "The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the\n", + "training set in a way that minimizes say the **gini** or **entropy** impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}\\mathrm{MSE}_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}\\mathrm{MSE}_{\\mathrm{right}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here the MSE for a specific node is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{MSE}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}(\\overline{y}_{\\mathrm{node}}-y_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\overline{y}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}y_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "the mean value of all observations in a specific node.\n", + "\n", + "Without any regularization, the regression task for decision trees, \n", + "just like for classification tasks, is prone to overfitting.\n", "\n", "\n", "## Computing the Gini index\n", diff --git a/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz b/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz index 43c2508b62c318899cf313e21e011c75135d9115..c08f488f1acc4501141e17944d6ca4bb321b73be 100644 GIT binary patch delta 30 mcmZ4cQ*iB1L3a6W4hFxUw;I`7*%@2enOfPIx3aS=s{sJDkqTV^ delta 30 mcmZ4cQ*iB1L3a6W4u-&rGmY%6?2N7KOs(w9TiIEb)c^pi$_dc` diff --git a/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf b/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf index 519b8268ce8f048306b412ff9afd42ace4ae4640..6d8201a6e2b2b946ba51e538a265c84bd0ae7db9 100644 GIT binary patch delta 97708 zcmZs>Q*b6w^X{9xv2AB!+qP}nw%?dDv29Q6WMbR4ZS#Ep^PRKz#ojloy82gZU9GC8 z`{~a6B3Wu6t>Xd!)TkPH0I?9@gy;8(A(K!MlTe_olTZK!UvL&qcILFZd;k@23Rgew 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zrsfuGxlqO{FP~^B=%WaW8kROI=3ou85|_Vr!cI1%cIw=jHVg`vmMVVNCbRX}qxV^! zhnD!AW2MOT9fJuSElx5>;f5g_52cd(x?V^$AyFFrI&LY^_CHw|QQ^ze@WA8RvABN{ zR_;ApkHMwd9XwT3qO?5qe}*Cpu8B!YQA;j{hf`DdU%Z3|WlO6R(`%$uDvYlUHbCGa zWh)+XwrLNJSt0_)_Dh&BUw)#f9tqr36jCj|Re?m#@L=rIT;n`M`tguF+!RE;H3ttk zsV2?0Jv8brEqxNU(g0EqMbe5pUl+B=$2NG?Wbni`ueIa%4v^K-7!S9-s%5=p^YNu0 z)}++jHC~0h4GGioDMY@@9b6fy#n`Q)``AU=Z8NLIX^_1DZA{fEfR`cZ_bdgq*#*SS zMtC1nhQBm_;Qp0dR$N4gG!h`0Qk+M??=*Y>NjZZsM>t4>N{Bh&#!c~_S=9y!W94pp j_}d74p^!%`VX6G)43;bq89Fa7A1^;D9i5DtEb9LQQ3}4? diff --git a/doc/src/DecisionTrees/DecisionTrees.do.txt b/doc/src/DecisionTrees/DecisionTrees.do.txt index 1b2dd2ae9..c6f31cbd8 100644 --- a/doc/src/DecisionTrees/DecisionTrees.do.txt +++ b/doc/src/DecisionTrees/DecisionTrees.do.txt @@ -471,15 +471,60 @@ Two algorithms stand out in the set up of decision trees: o The CART (Classification And Regression Tree) algorithm for both classification and regression o The ID3 algorithm based on the computation of the information gain for classification -We discuss both algorithms with applications here. The popular library _Scikit-Learn_ uses the CART algorithm. For classification problems you can use either the _gini_ index or the _entropy_ to split a tree in two branches. +We discuss both algorithms with applications here. The popular library +_Scikit-Learn_ uses the CART algorithm. For classification problems +you can use either the _gini_ index or the _entropy_ to split a tree +in two branches. !split ===== The CART algorithm for Classification ===== +For classification, the CART algorithm splits the data set in two subsets using a single feature $k$ and a threshold $t_k$. +This could be for example a threshold set by a number below a certain circumference of a malign tumor. + +How do we find these two quantities? +We search for the pair $(k,t_k)$ that produces the purest subset using for example the _gini_ factor $G$. +The cost function it tries to minimize is then +!bt +\[ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +\] +!et +where $G_{\mathrm{left\left}}$ measures the impurity of the left/right subset and $m_{\mathrm{left/right}}$ + is the number of instances in the left/right subset + +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +$max\_depth$ hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the $min\_samples\_split$, +$min\_samples\_leaf$, $min\_weight\_fraction\_leaf$, and $max\_leaf\_nodes$. !split ===== The CART algorithm for Regression ===== +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the _gini_ or _entropy_ impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +!bt +\[ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +\] +!et +Here the MSE for a specific node is defined as +!bt +\[ +\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, +\] +!et +with +!bt +\[ +\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, +\] +!et +the mean value of all observations in a specific node. + +Without any regularization, the regression task for decision trees, +just like for classification tasks, is prone to overfitting. !split