Decision Trees (DTs) are a non-parametric supervised learning method usedfor classification and regression. The goal is to create a model that predicts the value of atarget variable by learning simple decision rules inferred from the datafeatures. A tree can be seen as a piecewise constant approximation.
For instance, in the example below, decision trees learn from data toapproximate a sine curve with a set of if-then-else decision rules. The deeperthe tree, the more complex the decision rules and the fitter the model.
Some advantages of decision trees are:
Simple to understand and to interpret. Trees can be visualized.
Requires little data preparation. Other techniques often require datanormalization, dummy variables need to be created and blank values tobe removed. Some tree and algorithm combinations supportmissing values.
The cost of using the tree (i.e., predicting data) is logarithmic in thenumber of data points used to train the tree.
Able to handle both numerical and categorical data. However, the scikit-learnimplementation does not support categorical variables for now. Othertechniques are usually specialized in analyzing datasets that have only one typeof variable. See algorithms for moreinformation.
Able to handle multi-output problems.
Uses a white box model. If a given situation is observable in a model,the explanation for the condition is easily explained by boolean logic.By contrast, in a black box model (e.g., in an artificial neuralnetwork), results may be more difficult to interpret.
Possible to validate a model using statistical tests. That makes itpossible to account for the reliability of the model.
Performs well even if its assumptions are somewhat violated bythe true model from which the data were generated.
The disadvantages of decision trees include:
Decision-tree learners can create over-complex trees that do notgeneralize the data well. This is called overfitting. Mechanismssuch as pruning, setting the minimum number of samples requiredat a leaf node or setting the maximum depth of the tree arenecessary to avoid this problem.
Decision trees can be unstable because small variations in thedata might result in a completely different tree being generated.This problem is mitigated by using decision trees within anensemble.
Predictions of decision trees are neither smooth nor continuous, butpiecewise constant approximations as seen in the above figure. Therefore,they are not good at extrapolation.
The problem of learning an optimal decision tree is known to beNP-complete under several aspects of optimality and even for simpleconcepts. Consequently, practical decision-tree learning algorithmsare based on heuristic algorithms such as the greedy algorithm wherelocally optimal decisions are made at each node. Such algorithmscannot guarantee to return the globally optimal decision tree. Thiscan be mitigated by training multiple trees in an ensemble learner,where the features and samples are randomly sampled with replacement.
There are concepts that are hard to learn because decision treesdo not express them easily, such as XOR, parity or multiplexer problems.
Decision tree learners create biased trees if some classes dominate.It is therefore recommended to balance the dataset prior to fittingwith the decision tree.
1.10.1. Classification¶
DecisionTreeClassifier is a class capable of performing multi-classclassification on a dataset.
As with other classifiers, DecisionTreeClassifier takes as input two arrays:an array X, sparse or dense, of shape (n_samples, n_features) holding thetraining samples, and an array Y of integer values, shape (n_samples,),holding the class labels for the training samples:
>>> from sklearn import tree>>> X = [[0, 0], [1, 1]]>>> Y = [0, 1]>>> clf = tree.DecisionTreeClassifier()>>> clf = clf.fit(X, Y)
After being fitted, the model can then be used to predict the class of samples:
>>> clf.predict([[2., 2.]])array([1])
In case that there are multiple classes with the same and highestprobability, the classifier will predict the class with the lowest indexamongst those classes.
As an alternative to outputting a specific class, the probability of each classcan be predicted, which is the fraction of training samples of the class in aleaf:
>>> clf.predict_proba([[2., 2.]])array([[0., 1.]])
DecisionTreeClassifier is capable of both binary (where thelabels are [-1, 1]) classification and multiclass (where the labels are[0, …, K-1]) classification.
Using the Iris dataset, we can construct a tree as follows:
>>> from sklearn.datasets import load_iris>>> from sklearn import tree>>> iris = load_iris()>>> X, y = iris.data, iris.target>>> clf = tree.DecisionTreeClassifier()>>> clf = clf.fit(X, y)
Once trained, you can plot the tree with the plot_tree function:
We can also export the tree in Graphviz format using the export_graphvizexporter. If you use the conda package manager, the graphviz binariesand the python package can be installed with Alternatively binaries for graphviz can be downloaded from the graphviz project homepage,and the Python wrapper installed from pypi with Below is an example graphviz export of the above tree trained on the entireiris dataset; the results are saved in an output file The export_graphviz exporter also supports a variety of aestheticoptions, including coloring nodes by their class (or value for regression) andusing explicit variable and class names if desired. Jupyter notebooks alsorender these plots inline automatically: Alternatively, the tree can also be exported in textual format with thefunction export_text. This method doesn’t require the installationof external libraries and is more compact:Alternative ways to export treesClick for more details¶
conda install python-graphviz.pip install graphviz.iris.pdf:>>> import graphviz >>> dot_data = tree.export_graphviz(clf, out_file=None) >>> graph = graphviz.Source(dot_data) >>> graph.render("iris")
>>> dot_data = tree.export_graphviz(clf, out_file=None, ... feature_names=iris.feature_names, ... class_names=iris.target_names, ... filled=True, rounded=True, ... special_characters=True) >>> graph = graphviz.Source(dot_data) >>> graph
>>> from sklearn.datasets import load_iris>>> from sklearn.tree import DecisionTreeClassifier>>> from sklearn.tree import export_text>>> iris = load_iris()>>> decision_tree = DecisionTreeClassifier(random_state=0, max_depth=2)>>> decision_tree = decision_tree.fit(iris.data, iris.target)>>> r = export_text(decision_tree, feature_names=iris['feature_names'])>>> print(r)|--- petal width (cm) <= 0.80| |--- class: 0|--- petal width (cm) > 0.80| |--- petal width (cm) <= 1.75| | |--- class: 1| |--- petal width (cm) > 1.75| | |--- class: 2
1.10.2. Regression¶
Decision trees can also be applied to regression problems, using theDecisionTreeRegressor class.
As in the classification setting, the fit method will take as argument arrays Xand y, only that in this case y is expected to have floating point valuesinstead of integer values:
>>> from sklearn import tree>>> X = [[0, 0], [2, 2]]>>> y = [0.5, 2.5]>>> clf = tree.DecisionTreeRegressor()>>> clf = clf.fit(X, y)>>> clf.predict([[1, 1]])array([0.5])
1.10.3. Multi-output problems¶
A multi-output problem is a supervised learning problem with several outputsto predict, that is when Y is a 2d array of shape (n_samples, n_outputs).
When there is no correlation between the outputs, a very simple way to solvethis kind of problem is to build n independent models, i.e. one for eachoutput, and then to use those models to independently predict each one of the noutputs. However, because it is likely that the output values related to thesame input are themselves correlated, an often better way is to build a singlemodel capable of predicting simultaneously all n outputs. First, it requireslower training time since only a single estimator is built. Second, thegeneralization accuracy of the resulting estimator may often be increased.
With regard to decision trees, this strategy can readily be used to supportmulti-output problems. This requires the following changes:
Store n output values in leaves, instead of 1;
Use splitting criteria that compute the average reduction across alln outputs.
This module offers support for multi-output problems by implementing thisstrategy in both DecisionTreeClassifier andDecisionTreeRegressor. If a decision tree is fit on an output array Yof shape (n_samples, n_outputs) then the resulting estimator will:
Output n_output values upon
predict;Output a list of n_output arrays of class probabilities upon
predict_proba.
The use of multi-output trees for regression is demonstrated inMulti-output Decision Tree Regression. In this example, the inputX is a single real value and the outputs Y are the sine and cosine of X.
The use of multi-output trees for classification is demonstrated inFace completion with a multi-output estimators. In this example, the inputsX are the pixels of the upper half of faces and the outputs Y are the pixels ofthe lower half of those faces.
M. Dumont et al, Fast multi-class image annotation with random subwindowsand multiple output randomized trees, International Conference onComputer Vision Theory and Applications 2009ReferencesClick for more details¶
1.10.4. Complexity¶
In general, the run time cost to construct a balanced binary tree is\(O(n_{samples}n_{features}\log(n_{samples}))\) and query time\(O(\log(n_{samples}))\). Although the tree construction algorithm attemptsto generate balanced trees, they will not always be balanced. Assuming that thesubtrees remain approximately balanced, the cost at each node consists ofsearching through \(O(n_{features})\) to find the feature that offers thelargest reduction in the impurity criterion, e.g. log loss (which is equivalent to aninformation gain). This has a cost of\(O(n_{features}n_{samples}\log(n_{samples}))\) at each node, leading to atotal cost over the entire trees (by summing the cost at each node) of\(O(n_{features}n_{samples}^{2}\log(n_{samples}))\).
1.10.5. Tips on practical use¶
Decision trees tend to overfit on data with a large number of features.Getting the right ratio of samples to number of features is important, sincea tree with few samples in high dimensional space is very likely to overfit.
Consider performing dimensionality reduction (PCA,ICA, or Feature selection) beforehand togive your tree a better chance of finding features that are discriminative.
Understanding the decision tree structure will helpin gaining more insights about how the decision tree makes predictions, which isimportant for understanding the important features in the data.
Visualize your tree as you are training by using the
exportfunction. Usemax_depth=3as an initial tree depth to get a feel forhow the tree is fitting to your data, and then increase the depth.Remember that the number of samples required to populate the tree doublesfor each additional level the tree grows to. Use
max_depthto controlthe size of the tree to prevent overfitting.Use
min_samples_splitormin_samples_leafto ensure that multiplesamples inform every decision in the tree, by controlling which splits willbe considered. A very small number will usually mean the tree will overfit,whereas a large number will prevent the tree from learning the data. Trymin_samples_leaf=5as an initial value. If the sample size variesgreatly, a float number can be used as percentage in these two parameters.Whilemin_samples_splitcan create arbitrarily small leaves,min_samples_leafguarantees that each leaf has a minimum size, avoidinglow-variance, over-fit leaf nodes in regression problems. Forclassification with few classes,min_samples_leaf=1is often the bestchoice.Note that
min_samples_splitconsiders samples directly and independent ofsample_weight, if provided (e.g. a node with m weighted samples is stilltreated as having exactly m samples). Considermin_weight_fraction_leaformin_impurity_decreaseif accounting for sample weights is required at splits.Balance your dataset before training to prevent the tree from being biasedtoward the classes that are dominant. Class balancing can be done bysampling an equal number of samples from each class, or preferably bynormalizing the sum of the sample weights (
sample_weight) for eachclass to the same value. Also note that weight-based pre-pruning criteria,such asmin_weight_fraction_leaf, will then be less biased towarddominant classes than criteria that are not aware of the sample weights,likemin_samples_leaf.If the samples are weighted, it will be easier to optimize the treestructure using weight-based pre-pruning criterion such as
min_weight_fraction_leaf, which ensure that leaf nodes contain at leasta fraction of the overall sum of the sample weights.All decision trees use
np.float32arrays internally.If training data is not in this format, a copy of the dataset will be made.If the input matrix X is very sparse, it is recommended to convert to sparse
csc_matrixbefore calling fit and sparsecsr_matrixbefore callingpredict. Training time can be orders of magnitude faster for a sparsematrix input compared to a dense matrix when features have zero values inmost of the samples.
1.10.6. Tree algorithms: ID3, C4.5, C5.0 and CART¶
What are all the various decision tree algorithms and how do they differfrom each other? Which one is implemented in scikit-learn?
ID3 (Iterative Dichotomiser 3) was developed in 1986 by Ross Quinlan.The algorithm creates a multiway tree, finding for each node (i.e. ina greedy manner) the categorical feature that will yield the largestinformation gain for categorical targets. Trees are grown to theirmaximum size and then a pruning step is usually applied to improve theability of the tree to generalize to unseen data. C4.5 is the successor to ID3 and removed the restriction that featuresmust be categorical by dynamically defining a discrete attribute (basedon numerical variables) that partitions the continuous attribute valueinto a discrete set of intervals. C4.5 converts the trained trees(i.e. the output of the ID3 algorithm) into sets of if-then rules.The accuracy of each rule is then evaluated to determine the orderin which they should be applied. Pruning is done by removing a rule’sprecondition if the accuracy of the rule improves without it. C5.0 is Quinlan’s latest version release under a proprietary license.It uses less memory and builds smaller rulesets than C4.5 while beingmore accurate. CART (Classification and Regression Trees) is very similar to C4.5, butit differs in that it supports numerical target variables (regression) anddoes not compute rule sets. CART constructs binary trees using the featureand threshold that yield the largest information gain at each node.Various decision tree algorithmsClick for more details¶
scikit-learn uses an optimized version of the CART algorithm; however, thescikit-learn implementation does not support categorical variables for now.
1.10.7. Mathematical formulation¶
Given training vectors \(x_i \in R^n\), i=1,…, l and a label vector\(y \in R^l\), a decision tree recursively partitions the feature spacesuch that the samples with the same labels or similar target values are groupedtogether.
Let the data at node \(m\) be represented by \(Q_m\) with \(n_m\)samples. For each candidate split \(\theta = (j, t_m)\) consisting of afeature \(j\) and threshold \(t_m\), partition the data into\(Q_m^{left}(\theta)\) and \(Q_m^{right}(\theta)\) subsets
\[ \begin{align}\begin{aligned}Q_m^{left}(\theta) = \{(x, y) | x_j \leq t_m\}\\Q_m^{right}(\theta) = Q_m \setminus Q_m^{left}(\theta)\end{aligned}\end{align} \]
The quality of a candidate split of node \(m\) is then computed using animpurity function or loss function \(H()\), the choice of which depends onthe task being solved (classification or regression)
\[G(Q_m, \theta) = \frac{n_m^{left}}{n_m} H(Q_m^{left}(\theta))+ \frac{n_m^{right}}{n_m} H(Q_m^{right}(\theta))\]
Select the parameters that minimises the impurity
\[\theta^* = \operatorname{argmin}_\theta G(Q_m, \theta)\]
Recurse for subsets \(Q_m^{left}(\theta^*)\) and\(Q_m^{right}(\theta^*)\) until the maximum allowable depth is reached,\(n_m < \min_{samples}\) or \(n_m = 1\).
1.10.7.1. Classification criteria¶
If a target is a classification outcome taking on values 0,1,…,K-1,for node \(m\), let
\[p_{mk} = \frac{1}{n_m} \sum_{y \in Q_m} I(y = k)\]
be the proportion of class k observations in node \(m\). If \(m\) is aterminal node, predict_proba for this region is set to \(p_{mk}\).Common measures of impurity are the following.
Gini:
\[H(Q_m) = \sum_k p_{mk} (1 - p_{mk})\]
Log Loss or Entropy:
\[H(Q_m) = - \sum_k p_{mk} \log(p_{mk})\]
The entropy criterion computes the Shannon entropy of the possible classes. Ittakes the class frequencies of the training data points that reached a givenleaf \(m\) as their probability. Using the Shannon entropy as tree nodesplitting criterion is equivalent to minimizing the log loss (also known ascross-entropy and multinomial deviance) between the true labels \(y_i\)and the probabilistic predictions \(T_k(x_i)\) of the tree model \(T\) for class \(k\). To see this, first recall that the log loss of a tree model \(T\)computed on a dataset \(D\) is defined as follows: \[\mathrm{LL}(D, T) = -\frac{1}{n} \sum_{(x_i, y_i) \in D} \sum_k I(y_i = k) \log(T_k(x_i))\] where \(D\) is a training dataset of \(n\) pairs \((x_i, y_i)\). In a classification tree, the predicted class probabilities within leaf nodesare constant, that is: for all \((x_i, y_i) \in Q_m\), one has:\(T_k(x_i) = p_{mk}\) for each class \(k\). This property makes it possible to rewrite \(\mathrm{LL}(D, T)\) as thesum of the Shannon entropies computed for each leaf of \(T\) weighted bythe number of training data points that reached each leaf: \[\mathrm{LL}(D, T) = \sum_{m \in T} \frac{n_m}{n} H(Q_m)\]Shannon entropyClick for more details¶
1.10.7.2. Regression criteria¶
If the target is a continuous value, then for node \(m\), commoncriteria to minimize as for determining locations for future splits are MeanSquared Error (MSE or L2 error), Poisson deviance as well as Mean AbsoluteError (MAE or L1 error). MSE and Poisson deviance both set the predicted valueof terminal nodes to the learned mean value \(\bar{y}_m\) of the nodewhereas the MAE sets the predicted value of terminal nodes to the median\(median(y)_m\).
Mean Squared Error:
\[ \begin{align}\begin{aligned}\bar{y}_m = \frac{1}{n_m} \sum_{y \in Q_m} y\\H(Q_m) = \frac{1}{n_m} \sum_{y \in Q_m} (y - \bar{y}_m)^2\end{aligned}\end{align} \]
Half Poisson deviance:
\[H(Q_m) = \frac{1}{n_m} \sum_{y \in Q_m} (y \log\frac{y}{\bar{y}_m}- y + \bar{y}_m)\]
Setting criterion="poisson" might be a good choice if your target is a countor a frequency (count per some unit). In any case, \(y >= 0\) is anecessary condition to use this criterion. Note that it fits much slower thanthe MSE criterion.
Mean Absolute Error:
\[ \begin{align}\begin{aligned}median(y)_m = \underset{y \in Q_m}{\mathrm{median}}(y)\\H(Q_m) = \frac{1}{n_m} \sum_{y \in Q_m} |y - median(y)_m|\end{aligned}\end{align} \]
Note that it fits much slower than the MSE criterion.
1.10.8. Missing Values Support¶
DecisionTreeClassifier and DecisionTreeRegressorhave built-in support for missing values when splitter='best' and criterion is'gini', 'entropy’, or 'log_loss', for classification or'squared_error', 'friedman_mse', or 'poisson' for regression.
For each potential threshold on the non-missing data, the splitter will evaluatethe split with all the missing values going to the left node or the right node.
Decisions are made as follows:
By default when predicting, the samples with missing values are classifiedwith the class used in the split found during training:
>>> from sklearn.tree import DecisionTreeClassifier>>> import numpy as np>>> X = np.array([0, 1, 6, np.nan]).reshape(-1, 1)>>> y = [0, 0, 1, 1]>>> tree = DecisionTreeClassifier(random_state=0).fit(X, y)>>> tree.predict(X)array([0, 0, 1, 1])
If the criterion evaluation is the same for both nodes,then the tie for missing value at predict time is broken by going to theright node. The splitter also checks the split where all the missingvalues go to one child and non-missing values go to the other:
>>> from sklearn.tree import DecisionTreeClassifier>>> import numpy as np>>> X = np.array([np.nan, -1, np.nan, 1]).reshape(-1, 1)>>> y = [0, 0, 1, 1]>>> tree = DecisionTreeClassifier(random_state=0).fit(X, y)>>> X_test = np.array([np.nan]).reshape(-1, 1)>>> tree.predict(X_test)array([1])
If no missing values are seen during training for a given feature, then duringprediction missing values are mapped to the child with the most samples:
>>> from sklearn.tree import DecisionTreeClassifier>>> import numpy as np>>> X = np.array([0, 1, 2, 3]).reshape(-1, 1)>>> y = [0, 1, 1, 1]>>> tree = DecisionTreeClassifier(random_state=0).fit(X, y)>>> X_test = np.array([np.nan]).reshape(-1, 1)>>> tree.predict(X_test)array([1])
1.10.9. Minimal Cost-Complexity Pruning¶
Minimal cost-complexity pruning is an algorithm used to prune a tree to avoidover-fitting, described in Chapter 3 of [BRE]. This algorithm is parameterizedby \(\alpha\ge0\) known as the complexity parameter. The complexityparameter is used to define the cost-complexity measure, \(R_\alpha(T)\) ofa given tree \(T\):
\[R_\alpha(T) = R(T) + \alpha|\widetilde{T}|\]
where \(|\widetilde{T}|\) is the number of terminal nodes in \(T\) and \(R(T)\)is traditionally defined as the total misclassification rate of the terminalnodes. Alternatively, scikit-learn uses the total sample weighted impurity ofthe terminal nodes for \(R(T)\). As shown above, the impurity of a nodedepends on the criterion. Minimal cost-complexity pruning finds the subtree of\(T\) that minimizes \(R_\alpha(T)\).
The cost complexity measure of a single node is\(R_\alpha(t)=R(t)+\alpha\). The branch, \(T_t\), is defined to be atree where node \(t\) is its root. In general, the impurity of a nodeis greater than the sum of impurities of its terminal nodes,\(R(T_t)<R(t)\). However, the cost complexity measure of a node,\(t\), and its branch, \(T_t\), can be equal depending on\(\alpha\). We define the effective \(\alpha\) of a node to be thevalue where they are equal, \(R_\alpha(T_t)=R_\alpha(t)\) or\(\alpha_{eff}(t)=\frac{R(t)-R(T_t)}{|T|-1}\). A non-terminal nodewith the smallest value of \(\alpha_{eff}\) is the weakest link and willbe pruned. This process stops when the pruned tree’s minimal\(\alpha_{eff}\) is greater than the ccp_alpha parameter.
[BRE] L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classificationand Regression Trees. Wadsworth, Belmont, CA, 1984. J.R. Quinlan. C4. 5: programs for machine learning. MorganKaufmann, 1993. T. Hastie, R. Tibshirani and J. Friedman. Elements of StatisticalLearning, Springer, 2009.ReferencesClick for more details¶
