Random forests provide an improvement over bagged trees by way of a
-small tweak that decorrelates the trees.
-
+Random forests provide an improvement over bagged trees by way of a
+small tweak that decorrelates the trees.
+
+
+As in bagging, we build a
number of decision trees on bootstrapped training samples. But when
building these decision trees, each time a split in a tree is
considered, a random sample of \( m \) predictors is chosen as split
candidates from the full set of \( p \) predictors. The split is allowed to
-use only one of those \( m \) predictors.
-
+A fresh sample of \( m \) predictors is
taken at each split, and typically we choose
-
+In building a random forest, at
each split in the tree, the algorithm is not even allowed to consider
-a majority of the available predictors.
-
The reason for this is rather clever. Suppose that there is one very
+
+The reason for this is rather clever. Suppose that there is one very
strong predictor in the data set, along with a number of other
moderately strong predictors. Then in the collection of bagged
variable importance random forest trees, most or all of the trees will
@@ -2676,36 +2082,41 @@ lead to as large of a reduction in variance as averaging many
uncorrelated quantities. In particular, this means that bagging will
not lead to a substantial reduction in variance over a single tree in
this setting.
-
The algorithm described here can be applied to both classification and regression problems.
We will grow of forest of say \( B \) trees.
+The algorithm described here can be applied to both classification and regression problems.
+
+
+Recall that the cumulative gains curve shows the percentage of the
overall number of cases in a given category gained by targeting a
percentage of the total number of cases.
-
+Similarly, the receiver operating characteristic curve, or ROC curve,
displays the diagnostic ability of a binary classifier system as its
discrimination threshold is varied. It plots the true positive rate against the false positive rate.
-
-
+
+
-
+
-
Boosting, a Bird's Eye View
-
The basic idea is to combine weak classifiers in order to create a good
+
Boosting, a Bird's Eye View
+
+
+The basic idea is to combine weak classifiers in order to create a good
classifier. With a weak classifier we often intend a classifier which
produces results which are only slightly better than we would get by
random guesses.
-
-
This is done by applying in an iterative way a weak (or a standard
+
+This is done by applying in an iterative way a weak (or a standard
classifier like decision trees) to modify the data. In each iteration
we emphasize those observations which are misclassified by weighting
them with a factor.
-
+
-
What is boosting? Additive Modelling/Iterative Fitting
-
Boosting is a way of fitting an additive expansion in a set of
+
What is boosting? Additive Modelling/Iterative Fitting
+
+
+Boosting is a way of fitting an additive expansion in a set of
elementary basis functions like for example some simple polynomials.
Assume for example that we have a function
-
$$
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
$$
-
where \( \beta_m \) are the expansion parameters to be determined in a
+
+where \( \beta_m \) are the expansion parameters to be determined in a
minimization process and \( b(x;\gamma_m) \) are some simple functions of
the multivariable parameter \( x \) which is characterized by the
parameters \( \gamma_m \).
-
-
As an example, consider the Sigmoid function we used in logistic
+
+As an example, consider the Sigmoid function we used in logistic
regression. In that case, we can translate the function
\( b(x;\gamma_m) \) into the Sigmoid function
-
$$
\sigma(t) = \frac{1}{1+\exp{(-t)}},
$$
-
where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and
+
+where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and
\( \gamma_1 \) were determined by the Logistic Regression fitting
algorithm.
-
-
As another example, consider the cost function we defined for linear regression
+
+As another example, consider the cost function we defined for linear regression
$$
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
-
In this case the function \( f(x) \) was replaced by the design matrix
+
+In this case the function \( f(x) \) was replaced by the design matrix
\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \),
that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can
simply invert a matrix and obtain the parameters \( \beta \) by
-
$$
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
$$
-
In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
+
+In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
+
-
Iterative Fitting, Regression and Squared-error Cost Function
-
The way we proceed is as follows (here we specialize to the squared-error cost function)
+
Iterative Fitting, Regression and Squared-error Cost Function
+
+
+The way we proceed is as follows (here we specialize to the squared-error cost function)
- Establish a cost function, here \( {\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2 \) with \( f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m) \).
- Initialize with a guess \( f_0(x) \). It could be one or even zero or some random numbers.
- For \( m=1:M \)
+
- minimize \( \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2 \) wrt \( \gamma \) and \( \beta \)
- This gives the optimal values \( \beta_m \) and \( \gamma_m \)
- Determine then the new values \( f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m) \)
+
-
We could use any of the algorithms we have discussed till now. If we
+
+We could use any of the algorithms we have discussed till now. If we
use trees, \( \gamma \) parameterizes the split variables and split points
at the internal nodes, and the predictions at the terminal nodes.
-
+
-
Squared-Error Example and Iterative Fitting
-
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
+
Squared-Error Example and Iterative Fitting
-
For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \).
+
+To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
-
This means that for every iteration \( m \), we need to optimize
+
+For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \).
+
+
+This means that for every iteration \( m \), we need to optimize
$$
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
$$
-
We start our iteration by simply setting \( f_0(x)=0 \).
+
+We start our iteration by simply setting \( f_0(x)=0 \).
Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain
-
$$
\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
$$
-
and
+and
$$
\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
$$
-
We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) 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)
$$
\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
$$
-
which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have
+which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have
$$
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
$$
-
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 \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.
-
+
+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 \( \gamma \). 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
-\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
-
+
+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
+\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
+
-
Iterative Fitting, Classification and AdaBoost
-
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
+
Iterative Fitting, Classification and AdaBoost
+
+
+Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
observations. We define a classification function \( G(x) \) which produces a prediction taking one or the other of the two values
\( \{-1,1\} \).
-
-
The error rate of the training sample is then
+
+The error rate of the training sample is then
$$
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
$$
-
The iterative procedure starts with defining a weak classifier whose
+
+The iterative procedure starts with defining a weak classifier whose
error rate is barely better than random guessing. The iterative
procedure in boosting is to sequentially apply a weak
classification algorithm to repeatedly modified versions of the data
producing a sequence of weak classifiers \( G_m(x) \).
-
-
Here we will express our function \( f(x) \) in terms of \( G(x) \). That is
+
+Here we will express our function \( f(x) \) in terms of \( G(x) \). That is
$$
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
$$
-
will be a function of
+will be a function of
$$
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
$$
-
+
-
Adaptive Boosting, AdaBoost
-
In our iterative procedure we define thus
+
Adaptive Boosting, AdaBoost
+
+
+In our iterative procedure we define thus
$$
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
$$
-
The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
+
+The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
exponential cost/loss function defined as
-
$$
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
$$
-
We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case.
+
+We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case.
This is normally done in two steps. Let us however first rewrite the cost function as
-
$$
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
$$
-
where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
+where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
+
-
Building up AdaBoost
-
First, for any \( \beta > 0 \), we optimize \( G \) by setting
+
Building up AdaBoost
+
+
+First, for any \( \beta > 0 \), we optimize \( G \) by setting
$$
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
$$
-
which is the classifier that minimizes the weighted error rate in predicting \( y \).
+which is the classifier that minimizes the weighted error rate in predicting \( y \).
-
We can do this by rewriting
+
+We can do this by rewriting
$$
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
$$
-
which can be rewritten as
+which can be rewritten as
$$
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
$$
-
which leads to
+which leads to
$$
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
$$
-
where we have redefined the error as
+where we have redefined the error as
$$
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
$$
-
which leads to an update of
+which leads to an update of
$$
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
$$
-
This leads to the new weights
+This leads to the new weights
$$
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
$$
-
+
-
Adaptive boosting: AdaBoost, Basic Algorithm
-
The algorithm here is rather straightforward. Assume that our weak
+
Adaptive boosting: AdaBoost, Basic Algorithm
+
+
+The algorithm here is rather straightforward. Assume that our weak
classifier is a decision tree and we consider a binary set of outputs
with \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
observations. Our design matrix is given in terms of the
feature/predictor vectors
\( \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] \). Finally, we define also a
-classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \).
-
+classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \).
-
We have already defined the misclassification error \( \mathrm{err} \) as
+
+We have already defined the misclassification error \( \mathrm{err} \) as
$$
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
$$
-
where the function \( I() \) is one if we misclassify and zero if we classify correctly.
+where the function \( I() \) is one if we misclassify and zero if we classify correctly.
+
-
Basic Steps of AdaBoost
-
With the above definitions we are now ready to set up the algorithm for AdaBoost.
+
Basic Steps of AdaBoost
+
+
+With the above definitions we are now ready to set up the algorithm for AdaBoost.
The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
-
+
- We start by initializing all weights to \( w_i = 1/n \), with \( i=0,1,2,\dots n-1 \). It is easy to see that we must have \( \sum_{i=0}^{n-1}w_i = 1 \).
- We rewrite the misclassification error as
+
$$
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
$$
+
- Then we start looping over all attempts at classifying, namely we start an iterative process for \( m=1:M \), where \( M \) is the final number of classifications. Our given classifier could for example be a plain decision tree.
+
- Fit then a given classifier to the training set using the weights \( w_i \).
- Compute then \( \mathrm{err} \) and figure out which events are classified properly and which are classified wrongly.
- Define a quantity \( \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m} \)
- Set the new weights to \( w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)} \).
+
Compute the new classifier \( G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i) \).
-
For the iterations with \( m \le 2 \) the weights are modified
+
+For the iterations with \( m \le 2 \) the weights are modified
individually at each steps. The observations which were misclassified
at iteration \( m-1 \) have a weight which is larger than those which were
classified properly. As this proceeds, the observations which were
difficult to classifiy correctly are given a larger influence. Each
new classification step \( m \) is then forced to concentrate on those
observations that are missed in the previous iterations.
-
+
-
AdaBoost Examples
-
Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
+
AdaBoost Examples
+
+Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
+
+
-
+
-
Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent
-
Gradient boosting is again a similar technique to Adaptive boosting,
+
Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent
+
+
+Gradient boosting is again a similar technique to Adaptive boosting,
it combines so-called weak classifiers or regressors into a strong
method via a series of iterations.
-
-
In order to understand the method, let us illustrate its basics by
+
+In order to understand the method, let us illustrate its basics by
bringing back the essential steps in linear regression, where our cost
function was the least squares function.
-
+
-
The Squared-Error again! Steepest Descent
-
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
+
The Squared-Error again! Steepest Descent
+
+
+We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
This means that for every iteration, we need to optimize
-
$$
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
-
We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as
+
+We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as
$$
f_M(x) = \sum_{m=0}^M h_m(x).
$$
-
In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as
+
+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as
$$
g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
$$
-
With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that
+
+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that
the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \).
-
-
Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have
+
+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have
$$
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
$$
-
+
-
Steepest Descent Example
-
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
+
Steepest Descent Example
+
+
+Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
$$
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
$$
-
We can then proceed and compute
+We can then proceed and compute
$$
g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
$$
-
and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called gradient boosting.
+and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called
gradient boosting.
+
-
Gradient Boosting, algorithm
-
Steepest descent is however not much used, since it only optimizes \( f \) at a fixed set of \( n \) points,
+
Gradient Boosting, algorithm
+
+
+Steepest descent is however not much used, since it only optimizes \( f \) at a fixed set of \( n \) points,
so we do not learn a function that can generalize. However, we can modify the algorithm by
-fitting a weak learner to approximate the negative gradient signal.
-
+fitting a weak learner to approximate the negative gradient signal.
-
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
+
+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
$$
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
-
The way we proceed in an iterative fashion is to
+
+The way we proceed in an iterative fashion is to
+
- Initialize our estimate \( f_0(x) \).
- For \( m=1:M \), we
+
- compute the negative gradient vector \( \boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x) \) at \( f(x) = f_{m-1}(x) \);
- fit the so-called base-learner to the negative gradient \( h_m(u_m,x) \);
- update the estimate \( f_m(x) = f_{m-1}(x)+h_m(u_m,x) \);
+
The final estimate is then \( f_M(x) = \sum_{m=1}^M h_m(u_m,x) \).
+
-
Gradient Boosting, Examples of Regression
+
+
Gradient Boosting, Examples of Regression
+
-
+
-
Gradient Boosting, Classification Example
+
+
Gradient Boosting, Classification Example
+
-
+
-
XGBoost: Extreme Gradient Boosting
-
XGBoost or Extreme Gradient
+
XGBoost: Extreme Gradient Boosting
+
+
+XGBoost or Extreme Gradient
Boosting, is an optimized distributed gradient boosting library
designed to be highly efficient, flexible and portable. It implements
machine learning algorithms under the Gradient Boosting
framework. XGBoost provides a parallel tree boosting that solve many
data science problems in a fast and accurate way. See the article by Chen and Guestrin.
-
-
The authors design and build a highly scalable end-to-end tree
+
+The authors design and build a highly scalable end-to-end tree
boosting system. It has a theoretically justified weighted quantile
sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.
-
-
It is now the algorithm which wins essentially all ML competitions!!!
+
+It is now the algorithm which wins essentially all ML competitions!!!
+
-
Regression Case
+
Regression Case
+
+
-
+
-
Xgboost on the Cancer Data
-
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
+
Xgboost on the Cancer Data
+
+
+As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
+
-
+
+
+
© 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+
+
+
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diff --git a/doc/pub/week45/ipynb/week45.ipynb b/doc/pub/week45/ipynb/week45.ipynb
index 6a20f05e4..887022d0d 100644
--- a/doc/pub/week45/ipynb/week45.ipynb
+++ b/doc/pub/week45/ipynb/week45.ipynb
@@ -2,38 +2,20 @@
"cells": [
{
"cell_type": "markdown",
- "id": "999587fc",
- "metadata": {
- "editable": true
- },
- "source": [
- "\n",
- ""
- ]
- },
- {
- "cell_type": "markdown",
- "id": "f3ea736a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
+ "\n",
"# Week 45: Decisions Trees, Random Forests, Bagging and Boosting\n",
+ "\n",
+ " \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 9, 2021**\n",
+ "Date: **Nov 10, 2021**\n",
+ "\n",
+ "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
+ "\n",
+ "\n",
"\n",
- "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "f4a62cf4",
- "metadata": {
- "editable": true
- },
- "source": [
"## Overview of week 45\n",
"\n",
"* Thursday: Basics of Decision Trees, Bagging and Voting\n",
@@ -46,20 +28,18 @@
"\n",
"[Video on boosting methods by Hastie](https://www.youtube.com/watch?v=wPqtzj5VZus&ab_channel=H2O.ai).\n",
"\n",
+ "\n",
+ "\n",
"**Reading.**\n",
"\n",
- "1. Decision Trees: Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from [STK-IN4300, lecture 7](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf). Chapter 9.2 of Hastie et al contains also a good discussion."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "7d7e0a30",
- "metadata": {
- "editable": true
- },
- "source": [
+ "1. Decision Trees: Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from [STK-IN4300, lecture 7](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf). Chapter 9.2 of Hastie et al contains also a good discussion.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Decision trees, overarching aims\n",
"\n",
+ "\n",
"We start here with the most basic algorithm, the so-called decision\n",
"tree. With this basic algorithm we can in turn build more complex\n",
"networks, spanning from homogeneous and heterogenous forests (bagging,\n",
@@ -70,6 +50,7 @@
"Decision trees are supervised learning algorithms used for both,\n",
"classification and regression tasks.\n",
"\n",
+ "\n",
"The main idea of decision trees\n",
"is to find those descriptive features which contain the most\n",
"**information** regarding the target feature and then split the dataset\n",
@@ -79,16 +60,8 @@
"The descriptive features which reproduce best the target/output features are normally said\n",
"to be the most informative ones. The process of finding the **most\n",
"informative** feature is done until we accomplish a stopping criteria\n",
- "where we then finally end up in so called **leaf nodes**."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "f448055c",
- "metadata": {
- "editable": true
- },
- "source": [
+ "where we then finally end up in so called **leaf nodes**. \n",
+ "\n",
"## Basics of a tree\n",
"\n",
"A decision tree is typically divided into a **root node**, the **interior nodes**,\n",
@@ -99,58 +72,35 @@
"to our trained model. This is possible since the model has \n",
"learned the underlying structure of the training data and hence can,\n",
"given some assumptions, make predictions about the target feature value\n",
- "(class) of unseen query instances."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0cacfbd6",
- "metadata": {
- "editable": true
- },
- "source": [
+ "(class) of unseen query instances.\n",
+ "\n",
"## A Sketch of a Tree, Regression problem\n",
"\n",
- ""
- ]
- },
- {
- "cell_type": "markdown",
- "id": "09cac93a",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## A Sketch of a Tree, Classification problem\n",
"\n",
- ""
- ]
- },
- {
- "cell_type": "markdown",
- "id": "9e7e3a96",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## A typical Decision Tree with its pertinent Jargon, Classification Problem\n",
"\n",
"\n",
"\n",
"\n",
- "
![](\"DataFiles/cancer.png\")
Figure 1:
\n",
+ "
\n",
+ "
\n",
+ "\n",
"\n",
"\n",
- "This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using **Scikit-Learn**'s decision tree classifier. Here we have used the so-called **gini** index (see below) to split the various branches."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "260a1d3f",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using **Scikit-Learn**'s decision tree classifier. Here we have used the so-called **gini** index (see below) to split the various branches.\n",
+ "\n",
+ "\n",
+ "\n",
"## General Features\n",
"\n",
"The overarching approach to decision trees is a top-down approach.\n",
@@ -164,18 +114,12 @@
"* An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.\n",
"\n",
"This process is then repeated for the subtree rooted at the new\n",
- "node."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "71e35330",
- "metadata": {
- "editable": true
- },
- "source": [
+ "node.\n",
+ "\n",
+ "\n",
"## How do we set it up?\n",
"\n",
+ "\n",
"In simplified terms, the process of training a decision tree and\n",
"predicting the target features of query instances is as follows:\n",
"\n",
@@ -187,26 +131,20 @@
"\n",
"4. Show query instances to the tree and run down the tree until we arrive at leaf nodes\n",
"\n",
- "Then we are essentially done!"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c999b0f7",
- "metadata": {
- "editable": true
- },
- "source": [
+ "Then we are essentially done!\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Decision trees and Regression"
]
},
{
"cell_type": "code",
"execution_count": 1,
- "id": "d3154072",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -304,10 +242,7 @@
},
{
"cell_type": "markdown",
- "id": "f38fb41a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Building a tree, regression\n",
"\n",
@@ -326,10 +261,7 @@
},
{
"cell_type": "markdown",
- "id": "3f153956",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\sum_{j=1}^J\\sum_{i\\in R_j}(y_i-\\overline{y}_{R_j})^2,\n",
@@ -338,22 +270,11 @@
},
{
"cell_type": "markdown",
- "id": "be1e684e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $\\overline{y}_{R_j}$ is the mean response for the training observations \n",
- "within box $j$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "69f23ebf",
- "metadata": {
- "editable": true
- },
- "source": [
+ "within box $j$. \n",
+ "\n",
"## A top-down approach, recursive binary splitting\n",
"\n",
"Unfortunately, it is computationally infeasible to consider every\n",
@@ -366,16 +287,8 @@
"further down on the tree. It is greedy because at each step of the\n",
"tree-building process, the best split is made at that particular step,\n",
"rather than looking ahead and picking a split that will lead to a\n",
- "better tree in some future step."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0c7546ee",
- "metadata": {
- "editable": true
- },
- "source": [
+ "better tree in some future step.\n",
+ "\n",
"## Making a tree\n",
"\n",
"In order to implement the recursive binary splitting we start by selecting\n",
@@ -384,10 +297,7 @@
},
{
"cell_type": "markdown",
- "id": "901c465d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\left\\{X\\vert x_j < s\\right\\},\n",
@@ -396,20 +306,14 @@
},
{
"cell_type": "markdown",
- "id": "f193d5dc",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and"
]
},
{
"cell_type": "markdown",
- "id": "5b6fee06",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\left\\{X\\vert x_j \\geq s\\right\\},\n",
@@ -418,20 +322,14 @@
},
{
"cell_type": "markdown",
- "id": "fb7f787a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"so that we obtain the lowest MSE, that is"
]
},
{
"cell_type": "markdown",
- "id": "76179adf",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\sum_{i:x_i\\in R_j}(y_i-\\overline{y}_{R_1})^2+\\sum_{i:x_i\\in R_2}(y_i-\\overline{y}_{R_2})^2,\n",
@@ -440,10 +338,7 @@
},
{
"cell_type": "markdown",
- "id": "6c8ec2b3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which we want to minimize by considering all predictors\n",
"$x_1,x_2,\\dots,x_p$. We consider also all possible values of $s$ for\n",
@@ -468,16 +363,9 @@
"have three regions. Again, we look to split one of these three regions\n",
"further, so as to minimize the MSE. The process continues until a\n",
"stopping criterion is reached; for instance, we may continue until no\n",
- "region contains more than five observations."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "ee74a1ea",
- "metadata": {
- "editable": true
- },
- "source": [
+ "region contains more than five observations.\n",
+ "\n",
+ "\n",
"## Pruning the tree\n",
"\n",
"The above procedure is rather straightforward, but leads often to\n",
@@ -492,16 +380,8 @@
"we consider a sequence of trees indexed by a nonnegative tuning\n",
"parameter $\\alpha$.\n",
"\n",
- "Read more at the following [Scikit-Learn link on pruning](https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#sphx-glr-auto-examples-tree-plot-cost-complexity-pruning-py)."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "6ab80129",
- "metadata": {
- "editable": true
- },
- "source": [
+ "Read more at the following [Scikit-Learn link on pruning](https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#sphx-glr-auto-examples-tree-plot-cost-complexity-pruning-py).\n",
+ "\n",
"## Cost complexity pruning\n",
"\n",
"For each value of $\\alpha$ there corresponds a subtree $T \\in T_0$ such that"
@@ -509,10 +389,7 @@
},
{
"cell_type": "markdown",
- "id": "4fb3b414",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\sum_{m=1}^{\\overline{T}}\\sum_{i:x_i\\in R_m}(y_i-\\overline{y}_{R_m})^2+\\alpha\\overline{T},\n",
@@ -521,10 +398,7 @@
},
{
"cell_type": "markdown",
- "id": "4a53cd13",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"is as small as possible. Here $\\overline{T}$ is \n",
"the number of terminal nodes of the tree $T$ , $R_m$ is the\n",
@@ -539,25 +413,20 @@
"having a tree with many terminal nodes. The above equation will\n",
"tend to be minimized for a smaller subtree. \n",
"\n",
+ "\n",
"It turns out that as we increase $\\alpha$ from zero\n",
"branches get pruned from the tree in a nested and predictable fashion,\n",
"so obtaining the whole sequence of subtrees as a function of $\\alpha$ is\n",
"easy. We can select a value of $\\alpha$ using a validation set or using\n",
"cross-validation. We then return to the full data set and obtain the\n",
- "subtree corresponding to $\\alpha$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "d4972eb3",
- "metadata": {
- "editable": true
- },
- "source": [
+ "subtree corresponding to $\\alpha$. \n",
+ "\n",
+ "\n",
"## Schematic Regression Procedure\n",
"\n",
"**Building a Regression Tree.**\n",
"\n",
+ "\n",
"1. Use recursive binary splitting to grow a large tree on the training data, stopping only when each terminal node has fewer than some minimum number of observations.\n",
"\n",
"2. Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of $\\alpha$.\n",
@@ -570,16 +439,12 @@
"\n",
" * Finally we average the results for each value of $\\alpha$, and pick $\\alpha$ to minimize the average error.\n",
"\n",
- "4. Return the subtree from Step 2 that corresponds to the chosen value of $\\alpha$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "fa21cd89",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "4. Return the subtree from Step 2 that corresponds to the chosen value of $\\alpha$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## A Classification Tree\n",
"\n",
"A classification tree is very similar to a regression tree, except\n",
@@ -593,16 +458,8 @@
"classification tree, we are often interested not only in the class\n",
"prediction corresponding to a particular terminal node region, but\n",
"also in the class proportions among the training observations that\n",
- "fall into that region."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "ed9e140f",
- "metadata": {
- "editable": true
- },
- "source": [
+ "fall into that region. \n",
+ "\n",
"## Growing a classification tree\n",
"\n",
"The task of growing a\n",
@@ -620,16 +477,9 @@
"When building a classification tree, either the Gini index or the\n",
"entropy are typically used to evaluate the quality of a particular\n",
"split, since these two approaches are more sensitive to node purity\n",
- "than is the classification error rate."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "8339299a",
- "metadata": {
- "editable": true
- },
- "source": [
+ "than is the classification error rate. \n",
+ "\n",
+ "\n",
"## Classification tree, how to split nodes\n",
"\n",
"If our targets are the outcome of a classification process that takes\n",
@@ -644,22 +494,16 @@
},
{
"cell_type": "markdown",
- "id": "5a01e8b0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
- "p_{mk} = \\frac{1}{N_m}\\sum_{x_i\\in R_m}I(y_i=k).\n",
+ "p_{mk} = \\frac{1}{N_m}\\sum_{i\\in R_m}I(y_i=k).\n",
"$$"
]
},
{
"cell_type": "markdown",
- "id": "3b46b1a0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We let $p_{mk}$ represent the majority class of observations in region\n",
"$m$. The three most common ways of splitting a node are given by\n",
@@ -669,32 +513,23 @@
},
{
"cell_type": "markdown",
- "id": "bf8cf60e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
- "p_{mk} = \\frac{1}{N_m}\\sum_{x_i\\in R_m}I(y_i\\ne k) = 1-p_{mk}.\n",
+ "\\frac{1}{N_m}\\sum_{i\\in R_m}I(y_i\\ne k) = 1-p_{mk}.\n",
"$$"
]
},
{
"cell_type": "markdown",
- "id": "91bb9314",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"* Gini index $g$"
]
},
{
"cell_type": "markdown",
- "id": "21041ba3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"g = \\sum_{k=1}^K p_{mk}(1-p_{mk}).\n",
@@ -703,20 +538,14 @@
},
{
"cell_type": "markdown",
- "id": "27f57a10",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"* Information entropy or just entropy $s$"
]
},
{
"cell_type": "markdown",
- "id": "28610811",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"s = -\\sum_{k=1}^K p_{mk}\\log{p_{mk}}.\n",
@@ -725,10 +554,7 @@
},
{
"cell_type": "markdown",
- "id": "448db121",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Visualizing the Tree, Classification"
]
@@ -736,10 +562,8 @@
{
"cell_type": "code",
"execution_count": 2,
- "id": "daa33ec4",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -780,10 +604,7 @@
},
{
"cell_type": "markdown",
- "id": "c35660d4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Visualizing the Tree, The Moons"
]
@@ -791,10 +612,8 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "20d3782d",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -826,10 +645,7 @@
},
{
"cell_type": "markdown",
- "id": "4b50c098",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Other ways of visualizing the trees\n",
"\n",
@@ -839,10 +655,8 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "df24658e",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -857,10 +671,7 @@
},
{
"cell_type": "markdown",
- "id": "88327e54",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Printing out as text\n",
"\n",
@@ -871,10 +682,8 @@
{
"cell_type": "code",
"execution_count": 5,
- "id": "cabfc0eb",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -890,10 +699,7 @@
},
{
"cell_type": "markdown",
- "id": "f9b97781",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Algorithms for Setting up Decision Trees\n",
"\n",
@@ -905,16 +711,8 @@
"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."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "6d158b95",
- "metadata": {
- "editable": true
- },
- "source": [
+ "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",
@@ -927,10 +725,7 @@
},
{
"cell_type": "markdown",
- "id": "3916a0a9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}G_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}G_{\\mathrm{right}},\n",
@@ -939,10 +734,7 @@
},
{
"cell_type": "markdown",
- "id": "35e64400",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $G_{\\mathrm{left/right}}$ 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",
@@ -951,16 +743,8 @@
"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 $max\\_leaf\\_nodes$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "6b6f2758",
- "metadata": {
- "editable": true
- },
- "source": [
+ "$min\\_samples\\_leaf$, $min\\_weight\\_fraction\\_leaf$, 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",
@@ -969,10 +753,7 @@
},
{
"cell_type": "markdown",
- "id": "56a085e6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}\\mathrm{MSE}_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}\\mathrm{MSE}_{\\mathrm{right}}.\n",
@@ -981,20 +762,14 @@
},
{
"cell_type": "markdown",
- "id": "fa6f435c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"Here the MSE for a specific node is defined as"
]
},
{
"cell_type": "markdown",
- "id": "8afa690b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\mathrm{MSE}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}(\\overline{y}_{\\mathrm{node}}-y_i)^2,\n",
@@ -1003,20 +778,14 @@
},
{
"cell_type": "markdown",
- "id": "1b9c7d84",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"with"
]
},
{
"cell_type": "markdown",
- "id": "1af13ff9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\overline{y}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}y_i,\n",
@@ -1025,24 +794,14 @@
},
{
"cell_type": "markdown",
- "id": "aa4dc553",
- "metadata": {
- "editable": true
- },
+ "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."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "aa1afc4e",
- "metadata": {
- "editable": true
- },
- "source": [
+ "just like for classification tasks, is prone to overfitting.\n",
+ "\n",
+ "\n",
"## Computing the Gini index\n",
"\n",
"The example we will look at is a classical one in many Machine\n",
@@ -1057,7 +816,7 @@
"humidity and weak and strong for wind.\n",
"\n",
"The table here summarizes the various attributes and\n",
- "
\n",
+ "\n",
"\n",
"| Day | Outlook | Temperature | Humidity | Wind | Ride |
|---|
\n",
"\n",
@@ -1077,26 +836,16 @@
"| 13 | Overcast | Hot | Normal | Weak | 1 |
\n",
"| 14 | Rain | Mild | High | Strong | 0 |
\n",
"\n",
- "
"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "391b026f",
- "metadata": {
- "editable": true
- },
- "source": [
+ "
\n",
+ "\n",
"## Simple Python Code to read in Data and perform Classification"
]
},
{
"cell_type": "code",
"execution_count": 6,
- "id": "5d786a97",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1171,10 +920,7 @@
},
{
"cell_type": "markdown",
- "id": "2ed48016",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Computing the Gini Factor\n",
"\n",
@@ -1189,10 +935,8 @@
{
"cell_type": "code",
"execution_count": 7,
- "id": "a3b26fd1",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1260,10 +1004,7 @@
},
{
"cell_type": "markdown",
- "id": "2896b4f4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Entropy and the ID3 algorithm\n",
"\n",
@@ -1294,26 +1035,17 @@
"training examples according to their target classification.\n",
"\n",
"The ID3 algorithm uses this information gain measure to select among the candidate\n",
- "attributes at each step while growing the tree."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "fea0e5e6",
- "metadata": {
- "editable": true
- },
- "source": [
+ "attributes at each step while growing the tree.\n",
+ "\n",
+ "\n",
"## Cancer Data again now with Decision Trees and other Methods"
]
},
{
"cell_type": "code",
"execution_count": 8,
- "id": "1dbc948a",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1362,10 +1094,7 @@
},
{
"cell_type": "markdown",
- "id": "7b0d35e1",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Another example, the moons again"
]
@@ -1373,10 +1102,8 @@
{
"cell_type": "code",
"execution_count": 9,
- "id": "ef709f87",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1448,10 +1175,7 @@
},
{
"cell_type": "markdown",
- "id": "e7a8b237",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Playing around with regions"
]
@@ -1459,10 +1183,8 @@
{
"cell_type": "code",
"execution_count": 10,
- "id": "c5be70d1",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1490,10 +1212,7 @@
},
{
"cell_type": "markdown",
- "id": "0c4d5dab",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Regression trees"
]
@@ -1501,10 +1220,8 @@
{
"cell_type": "code",
"execution_count": 11,
- "id": "0963d7fc",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1519,10 +1236,8 @@
{
"cell_type": "code",
"execution_count": 12,
- "id": "689f738f",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1534,10 +1249,7 @@
},
{
"cell_type": "markdown",
- "id": "f476fc1b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Final regressor code"
]
@@ -1545,10 +1257,8 @@
{
"cell_type": "code",
"execution_count": 13,
- "id": "ec98869a",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1595,10 +1305,8 @@
{
"cell_type": "code",
"execution_count": 14,
- "id": "ae89b603",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1634,10 +1342,7 @@
},
{
"cell_type": "markdown",
- "id": "f4c99932",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Pros and cons of trees, pros\n",
"\n",
@@ -1653,16 +1358,8 @@
"\n",
"* Can model interactions between the different descriptive features\n",
"\n",
- "* Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "bcc7fb3d",
- "metadata": {
- "editable": true
- },
- "source": [
+ "* Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)\n",
+ "\n",
"## Disadvantages\n",
"\n",
"* Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches\n",
@@ -1681,16 +1378,9 @@
"\n",
"However, by aggregating many decision trees, using methods like\n",
"bagging, random forests, and boosting, the predictive performance of\n",
- "trees can be substantially improved."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "029c3b45",
- "metadata": {
- "editable": true
- },
- "source": [
+ "trees can be substantially improved.\n",
+ "\n",
+ "\n",
"## Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods\n",
"\n",
"As stated above and seen in many of the examples discussed here about\n",
@@ -1712,32 +1402,22 @@
"\n",
"4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)\n",
"\n",
- "We discuss these methods here."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "597ecf99",
- "metadata": {
- "editable": true
- },
- "source": [
+ "We discuss these methods here.\n",
+ "\n",
+ "\n",
"## An Overview of Ensemble Methods\n",
"\n",
"\n",
"\n",
"\n",
- "
![](\"DataFiles/ensembleoverview.png\")
Figure 1:
\n",
- ""
- ]
- },
- {
- "cell_type": "markdown",
- "id": "34ccd8a6",
- "metadata": {
- "editable": true
- },
- "source": [
+ "
\n",
+ "
\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Bagging\n",
"\n",
"The **plain** decision trees suffer from high\n",
@@ -1750,16 +1430,9 @@
"\n",
"**Bootstrap aggregation**, or just **bagging**, is a\n",
"general-purpose procedure for reducing the variance of a statistical\n",
- "learning method."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "60285855",
- "metadata": {
- "editable": true
- },
- "source": [
+ "learning method. \n",
+ "\n",
+ "\n",
"## More bagging\n",
"\n",
"Bagging typically results in improved accuracy\n",
@@ -1782,26 +1455,16 @@
"trees. A large value indicates an important predictor. Similarly, in\n",
"the context of bagging classification trees, we can add up the total\n",
"amount that the Gini index is decreased by splits over a given\n",
- "predictor, averaged over all $B$ trees."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "da86704c",
- "metadata": {
- "editable": true
- },
- "source": [
+ "predictor, averaged over all $B$ trees.\n",
+ "\n",
"## Simple Voting Example, head or tail"
]
},
{
"cell_type": "code",
"execution_count": 15,
- "id": "86f01d7b",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1822,10 +1485,7 @@
},
{
"cell_type": "markdown",
- "id": "66aa1558",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Using the Voting Classifier"
]
@@ -1833,10 +1493,8 @@
{
"cell_type": "code",
"execution_count": 16,
- "id": "1782fb88",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1887,10 +1545,7 @@
},
{
"cell_type": "markdown",
- "id": "f5aa4b7b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Please, not the moons again! Voting and Bagging"
]
@@ -1898,10 +1553,8 @@
{
"cell_type": "code",
"execution_count": 17,
- "id": "3c1a494b",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1928,10 +1581,8 @@
{
"cell_type": "code",
"execution_count": 18,
- "id": "15cc6ef1",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1946,10 +1597,8 @@
{
"cell_type": "code",
"execution_count": 19,
- "id": "d5380589",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1966,10 +1615,8 @@
{
"cell_type": "code",
"execution_count": 20,
- "id": "2630974e",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -1983,10 +1630,7 @@
},
{
"cell_type": "markdown",
- "id": "4bb8f29b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Bagging Examples"
]
@@ -1994,10 +1638,8 @@
{
"cell_type": "code",
"execution_count": 21,
- "id": "47d6424f",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2014,10 +1656,8 @@
{
"cell_type": "code",
"execution_count": 22,
- "id": "554b7c3e",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2028,10 +1668,8 @@
{
"cell_type": "code",
"execution_count": 23,
- "id": "5cdabae8",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2044,10 +1682,8 @@
{
"cell_type": "code",
"execution_count": 24,
- "id": "511564a7",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2082,10 +1718,7 @@
},
{
"cell_type": "markdown",
- "id": "2179d349",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Making your own Bootstrap: Changing the Level of the Decision Tree\n",
"\n",
@@ -2096,10 +1729,8 @@
{
"cell_type": "code",
"execution_count": 25,
- "id": "42afc36e",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2165,10 +1796,7 @@
},
{
"cell_type": "markdown",
- "id": "5b790359",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Why Voting?\n",
"\n",
@@ -2184,16 +1812,8 @@
"In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in\n",
"each iteration. \n",
"\n",
- "Decision trees play an important role as our weak classifier. They serve as the basic method."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c3c7a9fd",
- "metadata": {
- "editable": true
- },
- "source": [
+ "Decision trees play an important role as our weak classifier. They serve as the basic method. \n",
+ "\n",
"## Tossing coins\n",
"\n",
"The simplest case is a so-called voting ensemble. To illustrate this,\n",
@@ -2215,26 +1835,16 @@
"\n",
"The example here shows how we can implement the coin tossing case,\n",
"clealry demostrating that after some tosses we see the [law of large](https://en.wikipedia.org/wiki/Law_of_large_numbers)\n",
- "numbers kicking in."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "50fc15a0",
- "metadata": {
- "editable": true
- },
- "source": [
+ "numbers kicking in.\n",
+ "\n",
"## Standard imports first"
]
},
{
"cell_type": "code",
"execution_count": 26,
- "id": "c0b99f8c",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2279,10 +1889,7 @@
},
{
"cell_type": "markdown",
- "id": "69f8628e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Simple Voting Example, head or tail"
]
@@ -2290,10 +1897,8 @@
{
"cell_type": "code",
"execution_count": 27,
- "id": "805d4227",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2324,10 +1929,7 @@
},
{
"cell_type": "markdown",
- "id": "49717d4a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Using the Voting Classifier\n",
"\n",
@@ -2337,10 +1939,8 @@
{
"cell_type": "code",
"execution_count": 28,
- "id": "34a0cc41",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2390,10 +1990,7 @@
},
{
"cell_type": "markdown",
- "id": "452d0656",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Voting and Bagging"
]
@@ -2401,10 +1998,8 @@
{
"cell_type": "code",
"execution_count": 29,
- "id": "de6fd29b",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2431,10 +2026,8 @@
{
"cell_type": "code",
"execution_count": 30,
- "id": "37edc304",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2449,10 +2042,8 @@
{
"cell_type": "code",
"execution_count": 31,
- "id": "e1f00841",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2469,10 +2060,8 @@
{
"cell_type": "code",
"execution_count": 32,
- "id": "08dced6b",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2486,10 +2075,7 @@
},
{
"cell_type": "markdown",
- "id": "0ed0228e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Random forests\n",
"\n",
@@ -2509,10 +2095,7 @@
},
{
"cell_type": "markdown",
- "id": "bc05ef74",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"m\\approx \\sqrt{p}.\n",
@@ -2521,10 +2104,7 @@
},
{
"cell_type": "markdown",
- "id": "ffb4b144",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"In building a random forest, at\n",
"each split in the tree, the algorithm is not even allowed to consider\n",
@@ -2541,16 +2121,9 @@
"lead to as large of a reduction in variance as averaging many\n",
"uncorrelated quantities. In particular, this means that bagging will\n",
"not lead to a substantial reduction in variance over a single tree in\n",
- "this setting."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c75cd923",
- "metadata": {
- "editable": true
- },
- "source": [
+ "this setting.\n",
+ "\n",
+ "\n",
"## Random Forest Algorithm\n",
"The algorithm described here can be applied to both classification and regression problems.\n",
"\n",
@@ -2567,26 +2140,18 @@
"\n",
"3. split the node into daughter nodes\n",
"\n",
- "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "2e732ead",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "\n",
+ "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem. \n",
+ "\n",
"## Random Forests Compared with other Methods on the Cancer Data"
]
},
{
"cell_type": "code",
"execution_count": 33,
- "id": "befbdbd7",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2660,10 +2225,7 @@
},
{
"cell_type": "markdown",
- "id": "9173c1f7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"Recall that the cumulative gains curve shows the percentage of the\n",
"overall number of cases in a given category *gained* by targeting a\n",
@@ -2671,26 +2233,17 @@
"\n",
"Similarly, the receiver operating characteristic curve, or ROC curve,\n",
"displays the diagnostic ability of a binary classifier system as its\n",
- "discrimination threshold is varied. It plots the true positive rate against the false positive rate."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "9cb41a55",
- "metadata": {
- "editable": true
- },
- "source": [
+ "discrimination threshold is varied. It plots the true positive rate against the false positive rate.\n",
+ "\n",
+ "\n",
"## Compare Bagging on Trees with Random Forests"
]
},
{
"cell_type": "code",
"execution_count": 34,
- "id": "d83311e2",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2702,10 +2255,8 @@
{
"cell_type": "code",
"execution_count": 35,
- "id": "8cd7ddc3",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -2720,10 +2271,7 @@
},
{
"cell_type": "markdown",
- "id": "27c06676",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Boosting, a Bird's Eye View\n",
"\n",
@@ -2735,16 +2283,9 @@
"This is done by applying in an iterative way a weak (or a standard\n",
"classifier like decision trees) to modify the data. In each iteration\n",
"we emphasize those observations which are misclassified by weighting\n",
- "them with a factor."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "42ba5198",
- "metadata": {
- "editable": true
- },
- "source": [
+ "them with a factor.\n",
+ "\n",
+ "\n",
"## What is boosting? Additive Modelling/Iterative Fitting\n",
"\n",
"Boosting is a way of fitting an additive expansion in a set of\n",
@@ -2754,10 +2295,7 @@
},
{
"cell_type": "markdown",
- "id": "7d10fac1",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
@@ -2766,10 +2304,7 @@
},
{
"cell_type": "markdown",
- "id": "9e3039f9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $\\beta_m$ are the expansion parameters to be determined in a\n",
"minimization process and $b(x;\\gamma_m)$ are some simple functions of\n",
@@ -2783,10 +2318,7 @@
},
{
"cell_type": "markdown",
- "id": "afc02aa3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n",
@@ -2795,10 +2327,7 @@
},
{
"cell_type": "markdown",
- "id": "a614ac0d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $t=\\gamma_0+\\gamma_1 x$ and the parameters $\\gamma_0$ and\n",
"$\\gamma_1$ were determined by the Logistic Regression fitting\n",
@@ -2809,10 +2338,7 @@
},
{
"cell_type": "markdown",
- "id": "a8df1b75",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n",
@@ -2821,10 +2347,7 @@
},
{
"cell_type": "markdown",
- "id": "f5fc30cf",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"In this case the function $f(x)$ was replaced by the design matrix\n",
"$\\boldsymbol{X}$ and the unknown linear regression parameters $\\boldsymbol{\\beta}$,\n",
@@ -2834,10 +2357,7 @@
},
{
"cell_type": "markdown",
- "id": "c7fe95ae",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -2846,21 +2366,11 @@
},
{
"cell_type": "markdown",
- "id": "2c6f0083",
- "metadata": {
- "editable": true
- },
- "source": [
- "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "b73d8c2d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
+ "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$.\n",
+ "\n",
+ "\n",
"## Iterative Fitting, Regression and Squared-error Cost Function\n",
"\n",
"The way we proceed is as follows (here we specialize to the squared-error cost function)\n",
@@ -2877,18 +2387,12 @@
"\n",
"c. Determine then the new values $f_m(x)=f_{m-1}(x) +\\beta_m b(x;\\gamma_m)$\n",
"\n",
+ "\n",
"We could use any of the algorithms we have discussed till now. If we\n",
"use trees, $\\gamma$ parameterizes the split variables and split points\n",
- "at the internal nodes, and the predictions at the terminal nodes."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "5c6f971b",
- "metadata": {
- "editable": true
- },
- "source": [
+ "at the internal nodes, and the predictions at the terminal nodes.\n",
+ "\n",
+ "\n",
"## Squared-Error Example and Iterative Fitting\n",
"\n",
"To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.\n",
@@ -2900,10 +2404,7 @@
},
{
"cell_type": "markdown",
- "id": "e0ed27b3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"(\\beta_m,\\gamma_m) = \\mathrm{argmin}_{\\beta,\\lambda}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2=\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta(1+\\gamma x_i))^2.\n",
@@ -2912,10 +2413,7 @@
},
{
"cell_type": "markdown",
- "id": "fc9f5f4f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We start our iteration by simply setting $f_0(x)=0$. \n",
"Taking the derivatives with respect to $\\beta$ and $\\gamma$ we obtain"
@@ -2923,10 +2421,7 @@
},
{
"cell_type": "markdown",
- "id": "02c2ec22",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial {\\cal C}}{\\partial \\beta} = -2\\sum_{i}(1+\\gamma x_i)(y_i-\\beta(1+\\gamma x_i))=0,\n",
@@ -2935,20 +2430,14 @@
},
{
"cell_type": "markdown",
- "id": "e2288bd2",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and"
]
},
{
"cell_type": "markdown",
- "id": "98153a85",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial {\\cal C}}{\\partial \\gamma} =-2\\sum_{i}\\beta x_i(y_i-\\beta(1+\\gamma x_i))=0.\n",
@@ -2957,20 +2446,14 @@
},
{
"cell_type": "markdown",
- "id": "c39e24a0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma \\boldsymbol{x})$ with $\\boldsymbol{e}$ being the unit vector)"
]
},
{
"cell_type": "markdown",
- "id": "afdfb641",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n",
@@ -2979,20 +2462,14 @@
},
{
"cell_type": "markdown",
- "id": "3039d39b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have"
]
},
{
"cell_type": "markdown",
- "id": "2ce0aa48",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n",
@@ -3001,25 +2478,16 @@
},
{
"cell_type": "markdown",
- "id": "119b9640",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which leads to $\\gamma =(\\boldsymbol{x}^T\\boldsymbol{y}-\\beta\\boldsymbol{x}^T\\boldsymbol{e})/(\\beta\\boldsymbol{x}^T\\boldsymbol{x})$. Inserting\n",
"for $\\beta$ gives us an equation for $\\gamma$. This is a non-linear equation in the unknown $\\gamma$ and has to be solved numerically. \n",
"\n",
"The solution to these two equations gives us in turn $\\beta_1$ and $\\gamma_1$ leading to the new expression for $f_1(x)$ as\n",
- "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "419822da",
- "metadata": {
- "editable": true
- },
- "source": [
+ "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. \n",
+ "\n",
+ "\n",
+ "\n",
"## Iterative Fitting, Classification and AdaBoost\n",
"\n",
"Let us consider a binary classification problem with two outcomes $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n",
@@ -3031,10 +2499,7 @@
},
{
"cell_type": "markdown",
- "id": "1430356a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n",
@@ -3043,10 +2508,7 @@
},
{
"cell_type": "markdown",
- "id": "c3a81493",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"The iterative procedure starts with defining a weak classifier whose\n",
"error rate is barely better than random guessing. The iterative\n",
@@ -3059,10 +2521,7 @@
},
{
"cell_type": "markdown",
- "id": "190028df",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
@@ -3071,20 +2530,14 @@
},
{
"cell_type": "markdown",
- "id": "bbbca2ac",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"will be a function of"
]
},
{
"cell_type": "markdown",
- "id": "7aee5d30",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n",
@@ -3093,10 +2546,7 @@
},
{
"cell_type": "markdown",
- "id": "16cc7ab6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Adaptive Boosting, AdaBoost\n",
"\n",
@@ -3105,10 +2555,7 @@
},
{
"cell_type": "markdown",
- "id": "0e525c09",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n",
@@ -3117,10 +2564,7 @@
},
{
"cell_type": "markdown",
- "id": "df0326b4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the\n",
"exponential cost/loss function defined as"
@@ -3128,10 +2572,7 @@
},
{
"cell_type": "markdown",
- "id": "07d19584",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}\\exp{(-y_i(f_{m-1}(x_i)+\\beta G(x_i))}.\n",
@@ -3140,10 +2581,7 @@
},
{
"cell_type": "markdown",
- "id": "76774593",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We optimize $\\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.\n",
"This is normally done in two steps. Let us however first rewrite the cost function as"
@@ -3151,10 +2589,7 @@
},
{
"cell_type": "markdown",
- "id": "bb66a839",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}w_i^{m}\\exp{(-y_i\\beta G(x_i))},\n",
@@ -3163,21 +2598,10 @@
},
{
"cell_type": "markdown",
- "id": "bcd0b507",
- "metadata": {
- "editable": true
- },
- "source": [
- "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0c203949",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
+ "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$.\n",
+ "\n",
"## Building up AdaBoost\n",
"\n",
"First, for any $\\beta > 0$, we optimize $G$ by setting"
@@ -3185,10 +2609,7 @@
},
{
"cell_type": "markdown",
- "id": "415377f0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"G_m(x) = \\mathrm{sign} \\sum_{i=0}^{n-1} w_i^m I(y_i \\ne G_(x_i)),\n",
@@ -3197,10 +2618,7 @@
},
{
"cell_type": "markdown",
- "id": "daaca279",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which is the classifier that minimizes the weighted error rate in predicting $y$.\n",
"\n",
@@ -3209,10 +2627,7 @@
},
{
"cell_type": "markdown",
- "id": "f7bfba93",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\exp{-(\\beta)}\\sum_{y_i=G(x_i)}w_i^m+\\exp{(\\beta)}\\sum_{y_i\\ne G(x_i)}w_i^m,\n",
@@ -3221,20 +2636,14 @@
},
{
"cell_type": "markdown",
- "id": "4f6a2f84",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which can be rewritten as"
]
},
{
"cell_type": "markdown",
- "id": "6858be89",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"(\\exp{(\\beta)}-\\exp{-(\\beta)})\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i))+\\exp{(-\\beta)}\\sum_{i=0}^{n-1}w_i^m=0,\n",
@@ -3243,20 +2652,14 @@
},
{
"cell_type": "markdown",
- "id": "eecfcfb5",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which leads to"
]
},
{
"cell_type": "markdown",
- "id": "cd73a579",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n",
@@ -3265,20 +2668,14 @@
},
{
"cell_type": "markdown",
- "id": "33302d95",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where we have redefined the error as"
]
},
{
"cell_type": "markdown",
- "id": "ce21db0f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\mathrm{\\overline{err}}_m=\\frac{1}{n}\\frac{\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i)}{\\sum_{i=0}^{n-1}w_i^m},\n",
@@ -3287,20 +2684,14 @@
},
{
"cell_type": "markdown",
- "id": "1fd9965f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which leads to an update of"
]
},
{
"cell_type": "markdown",
- "id": "2a092b92",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n",
@@ -3309,20 +2700,14 @@
},
{
"cell_type": "markdown",
- "id": "1553ca38",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"This leads to the new weights"
]
},
{
"cell_type": "markdown",
- "id": "60bd5373",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n",
@@ -3331,10 +2716,7 @@
},
{
"cell_type": "markdown",
- "id": "eace9c52",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Adaptive boosting: AdaBoost, Basic Algorithm\n",
"\n",
@@ -3351,10 +2733,7 @@
},
{
"cell_type": "markdown",
- "id": "8cb0caec",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n",
@@ -3363,21 +2742,10 @@
},
{
"cell_type": "markdown",
- "id": "4545fb70",
- "metadata": {
- "editable": true
- },
- "source": [
- "where the function $I()$ is one if we misclassify and zero if we classify correctly."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c8c0b69c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
+ "where the function $I()$ is one if we misclassify and zero if we classify correctly. \n",
+ "\n",
"## Basic Steps of AdaBoost\n",
"\n",
"With the above definitions we are now ready to set up the algorithm for AdaBoost.\n",
@@ -3389,10 +2757,7 @@
},
{
"cell_type": "markdown",
- "id": "9a715091",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\mathrm{\\overline{err}}_m=\\frac{\\sum_{i=0}^{n-1}w_i^m I(y_i\\ne G(x_i))}{\\sum_{i=0}^{n-1}w_i},\n",
@@ -3401,10 +2766,7 @@
},
{
"cell_type": "markdown",
- "id": "ebec62af",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.\n",
"\n",
@@ -3416,6 +2778,7 @@
"\n",
"d. Set the new weights to $w_i = w_i\\times \\exp{(\\alpha_m I(y_i\\ne G(x_i)}$.\n",
"\n",
+ "\n",
"5. Compute the new classifier $G(x)= \\sum_{i=0}^{n-1}\\alpha_m I(y_i\\ne G(x_i)$.\n",
"\n",
"For the iterations with $m \\le 2$ the weights are modified\n",
@@ -3424,16 +2787,10 @@
"classified properly. As this proceeds, the observations which were\n",
"difficult to classifiy correctly are given a larger influence. Each\n",
"new classification step $m$ is then forced to concentrate on those\n",
- "observations that are missed in the previous iterations."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "6df3a47f",
- "metadata": {
- "editable": true
- },
- "source": [
+ "observations that are missed in the previous iterations.\n",
+ "\n",
+ "\n",
+ "\n",
"## AdaBoost Examples\n",
"\n",
"Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here."
@@ -3442,10 +2799,8 @@
{
"cell_type": "code",
"execution_count": 36,
- "id": "a762adfc",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -3474,10 +2829,7 @@
},
{
"cell_type": "markdown",
- "id": "38d12d89",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent\n",
"\n",
@@ -3487,16 +2839,8 @@
"\n",
"In order to understand the method, let us illustrate its basics by\n",
"bringing back the essential steps in linear regression, where our cost\n",
- "function was the least squares function."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "1b792064",
- "metadata": {
- "editable": true
- },
- "source": [
+ "function was the least squares function.\n",
+ "\n",
"## The Squared-Error again! Steepest Descent\n",
"\n",
"We start again with our cost function ${\\cal C}(\\boldsymbol{y}m\\boldsymbol{f})=\\sum_{i=0}^{n-1}{\\cal L}(y_i, f(x_i))$ where we want to minimize\n",
@@ -3505,10 +2849,7 @@
},
{
"cell_type": "markdown",
- "id": "beaf036c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"(\\hat{\\boldsymbol{f}}) = \\mathrm{argmin}_{\\boldsymbol{f}}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n",
@@ -3517,20 +2858,14 @@
},
{
"cell_type": "markdown",
- "id": "702b65a0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as"
]
},
{
"cell_type": "markdown",
- "id": "2652a1da",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_M(x) = \\sum_{m=0}^M h_m(x).\n",
@@ -3539,20 +2874,14 @@
},
{
"cell_type": "markdown",
- "id": "be287ca2",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"In the steepest decent approach we approximate $h_m(x) = -\\rho_m g_m(x)$, where $\\rho_m$ is a scalar and $g_m(x)$ the gradient defined as"
]
},
{
"cell_type": "markdown",
- "id": "ff3901b6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"g_m(x_i) = \\left[ \\frac{\\partial {\\cal L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{m-1}(x_i)}.\n",
@@ -3561,10 +2890,7 @@
},
{
"cell_type": "markdown",
- "id": "5228f5fd",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"With the new gradient we can update $f_m(x) = f_{m-1}(x) -\\rho_m g_m(x)$. Using the above squared-error function we see that\n",
"the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.\n",
@@ -3574,10 +2900,7 @@
},
{
"cell_type": "markdown",
- "id": "1e30448f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"(\\rho_1) = \\mathrm{argmin}_{\\rho}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i+2\\rho y_i)^2.\n",
@@ -3586,10 +2909,7 @@
},
{
"cell_type": "markdown",
- "id": "5bd43cc1",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Steepest Descent Example\n",
"\n",
@@ -3598,10 +2918,7 @@
},
{
"cell_type": "markdown",
- "id": "acaae452",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"f_1(x) = f_{0}(x) -\\rho_1 g_1(x)=-y_i.\n",
@@ -3610,20 +2927,14 @@
},
{
"cell_type": "markdown",
- "id": "022645ac",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We can then proceed and compute"
]
},
{
"cell_type": "markdown",
- "id": "8e6bbfb4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"g_2(x_i) = \\left[ \\frac{\\partial {\\cal L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,\n",
@@ -3632,21 +2943,10 @@
},
{
"cell_type": "markdown",
- "id": "66ce0709",
- "metadata": {
- "editable": true
- },
- "source": [
- "and find a new value for $\\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "8ac79309",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
+ "and find a new value for $\\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**. \n",
+ "\n",
"## Gradient Boosting, algorithm\n",
"\n",
"Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,\n",
@@ -3658,10 +2958,7 @@
},
{
"cell_type": "markdown",
- "id": "d562f162",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{y},\\boldsymbol{f})=\\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n",
@@ -3670,10 +2967,7 @@
},
{
"cell_type": "markdown",
- "id": "9e609544",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"The way we proceed in an iterative fashion is to\n",
"1. Initialize our estimate $f_0(x)$.\n",
@@ -3686,26 +2980,17 @@
"\n",
"c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;\n",
"\n",
- "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M h_m(u_m,x)$."
- ]
- },
- {
- "cell_type": "markdown",
- "id": "eced6bdc",
- "metadata": {
- "editable": true
- },
- "source": [
+ "\n",
+ "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M h_m(u_m,x)$.\n",
+ "\n",
"## Gradient Boosting, Examples of Regression"
]
},
{
"cell_type": "code",
"execution_count": 37,
- "id": "613e05cd",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -3759,10 +3044,7 @@
},
{
"cell_type": "markdown",
- "id": "77c76d85",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Gradient Boosting, Classification Example"
]
@@ -3770,10 +3052,8 @@
{
"cell_type": "code",
"execution_count": 38,
- "id": "af5a3843",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -3821,13 +3101,11 @@
},
{
"cell_type": "markdown",
- "id": "0ca37c1d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## XGBoost: Extreme Gradient Boosting\n",
"\n",
+ "\n",
"[XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient\n",
"Boosting, is an optimized distributed gradient boosting library\n",
"designed to be highly efficient, flexible and portable. It implements\n",
@@ -3839,26 +3117,16 @@
"boosting system. It has a theoretically justified weighted quantile\n",
"sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.\n",
"\n",
- "It is now the algorithm which wins essentially all ML competitions!!!"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "b2dd3360",
- "metadata": {
- "editable": true
- },
- "source": [
+ "It is now the algorithm which wins essentially all ML competitions!!!\n",
+ "\n",
"## Regression Case"
]
},
{
"cell_type": "code",
"execution_count": 39,
- "id": "4bf9f894",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -3912,10 +3180,7 @@
},
{
"cell_type": "markdown",
- "id": "00b1f1af",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Xgboost on the Cancer Data\n",
"\n",
@@ -3925,10 +3190,8 @@
{
"cell_type": "code",
"execution_count": 40,
- "id": "4454900b",
"metadata": {
- "collapsed": false,
- "editable": true
+ "collapsed": false
},
"outputs": [],
"source": [
@@ -3989,5 +3252,5 @@
],
"metadata": {},
"nbformat": 4,
- "nbformat_minor": 5
+ "nbformat_minor": 4
}
diff --git a/doc/src/week45/week45.do.txt b/doc/src/week45/week45.do.txt
index f7dc247d1..3bc775cc3 100644
--- a/doc/src/week45/week45.do.txt
+++ b/doc/src/week45/week45.do.txt
@@ -412,7 +412,7 @@ observations of this class in the region $R_m$ as
!bt
\[
-p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k).
+p_{mk} = \frac{1}{N_m}\sum_{i\in R_m}I(y_i=k).
\]
!et
@@ -422,7 +422,7 @@ $m$. The three most common ways of splitting a node are given by
* Misclassification error
!bt
\[
-p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}.
+\frac{1}{N_m}\sum_{i\in R_m}I(y_i\ne k) = 1-p_{mk}.
\]
!et
* Gini index $g$
