1.10. Decision Trees

1.10. Decision Trees

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.

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 visualised.
Requires little data preparation. Other techniques often require datanormalisation, dummy variables need to be created and blank values tobe removed. Note however that this module does not support missingvalues.
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. Other techniquesare usually specialised in analysing 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 notgeneralise the data well. This is called overfitting. Mechanismssuch as pruning (not currently supported), setting the minimumnumber of samples required at a leaf node or setting the maximumdepth of the tree are necessary 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.
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 size [n_samples, n_features] holding thetraining samples, and an array Y of integer values, size [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])

Alternatively, the probability of each class can be predicted, which is thefraction of training samples of the same class in a leaf:

>>>

>>> 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
>>> X, y = load_iris(return_X_y=True)
>>> clf = tree.DecisionTreeClassifier()
>>> clf = clf.fit(X, y)

Once trained, you can plot the tree with the plot_tree function:

>>>

>>> tree.plot_tree(clf.fit(iris.data, iris.target))

We can also export the tree in Graphviz format using the export_graphvizexporter. If you use the conda package manager, the graphviz binaries

and the python package can be installed with

conda install python-graphviz

Alternatively binaries for graphviz can be downloaded from the graphviz project homepage,and the Python wrapper installed from pypi with pip install graphviz.

Below is an example graphviz export of the above tree trained on the entireiris dataset; the results are saved in an output file iris.pdf:

>>>

>>> import graphviz 
>>> dot_data = tree.export_graphviz(clf, out_file=None) 
>>> graph = graphviz.Source(dot_data) 
>>> graph.render("iris")

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:

>>>

>>> 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

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:

>>>

>>> from sklearn.datasets import load_iris
>>> from sklearn.tree import DecisionTreeClassifier
>>> from sklearn.tree.export 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
<BLANKLINE>

Examples:

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])

Examples:

Decision Tree Regression

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 size [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 size [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 uponpredict_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.

Examples:

References:

M. Dumont et al, Fast multi-class image annotation with random subwindowsand multiple output randomized trees, International Conference onComputer Vision Theory and Applications 2009

1.10.4. Complexity

In general, the run time cost to construct a balanced binary tree is

and query time. 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 to find the feature that offers thelargest reduction in entropy. This has a cost of at each node, leading to atotal cost over the entire trees (by summing the cost at each node) of.

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.
Visualise your tree as you are training by using the exportfunction. Use max_depth=3 as 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_depth to controlthe size of the tree to prevent overfitting.
Use min_samples_split or min_samples_leaf to 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=5 as an initial value. If the sample size variesgreatly, a float number can be used as percentage in these two parameters.While min_samples_split can create arbitrarily small leaves,min_samples_leaf guarantees that each leaf has a minimum size, avoidinglow-variance, over-fit leaf nodes in regression problems. Forclassification with few classes, min_samples_leaf=1 is often the bestchoice.
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 as min_weight_fraction_leaf, will then be less biased towarddominant classes than criteria that are not aware of the sample weights,like min_samples_leaf.
If the samples are weighted, it will be easier to optimize the treestructure using weight-based pre-pruning criterion such asmin_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.float32 arrays 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 sparsecsc_matrix before calling fit and sparse csr_matrix before 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 generalise 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.These 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.

scikit-learn uses an optimised version of the CART algorithm; however, scikit-learnimplementation does not support categorical variables for now.

1.10.7. Mathematical formulation

Given training vectors

, i=1,…, l and a label vector, a decision tree recursively partitions the space suchthat the samples with the same labels are grouped together.

Let the data at node

be represented by. Foreach candidate split consisting of afeature and threshold, partition the data into and subsets

The impurity at

is computed using an impurity function, the choice of which depends on the task being solved(classification or regression)

Select the parameters that minimises the impurity

Recurse for subsets

and until the maximum allowable depth is reached, or.

1.10.7.1. Classification criteria

If a target is a classification outcome taking on values 0,1,…,K-1,for node

, representing a region withobservations, let

be the proportion of class k observations in node

Common measures of impurity are Gini

Entropy

and Misclassification

where

is the training data in node

1.10.7.2. Regression criteria

If the target is a continuous value, then for node

,representing a region with observations, commoncriteria to minimise as for determining locations for futuresplits are Mean Squared Error, which minimizes the L2 errorusing mean values at terminal nodes, and Mean Absolute Error, whichminimizes the L1 error using median values at terminal nodes.

Mean Squared Error:

Mean Absolute Error:

where

is the training data in node

1.10.8. 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

known as the complexity parameter. The complexityparameter is used to define the cost-complexity measure, ofa given tree:

where

is the number of terminal nodes in andis traditionally defined as the total misclassification rate of the terminalnodes. Alternatively, scikit-learn uses the total sample weighted impurity ofthe terminal nodes for. As shown above, the impurity of a nodedepends on the criterion. Minimal cost-complexity pruning finds the subtree of that minimizes.

The cost complexity measure of a single node is

. The branch,, is defined to be atree where node is its root. In general, the impurity of a nodeis greater than the sum of impurities of its terminal nodes,. However, the cost complexity measure of a node,, and its branch,, can be equal depending on. We define the effective of a node to be thevalue where they are equal, or. A non-terminal nodewith the smallest value of is the weakest link and willbe pruned. This process stops when the pruned tree’s minimal is greater than the ccp_alpha parameter.

Examples:

Post pruning decision trees with cost complexity pruning

References:

BRE
L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classificationand Regression Trees. Wadsworth, Belmont, CA, 1984.

https://en.wikipedia.org/wiki/Decision_tree_learning
https://en.wikipedia.org/wiki/Predictive_analytics
J.R. Quinlan. C4. 5: programs for machine learning. MorganKaufmann, 1993.
T. Hastie, R. Tibshirani and J. Friedman. Elements of StatisticalLearning, Springer, 2009.