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Decision Trees

Decision trees are powerful and popular tools for classification and prediction. The attractiveness of decision trees is due to the fact that, in contrast to neural networks, decision trees represent rules. Rules can readily be expressed so that humans can understand them or even directly used in a database access language like SQL so that records falling into a particular category may be retrieved.

In some applications, the accuracy of a classification or prediction is the only thing that matters. In such situations we do not necessarily care how or why the model works. In other situations, the ability to explain the reason for a decision, is crucial. In marketing one has describe the customer segments to marketing professionals, so that they can utilize this knowledge in launching a successful marketing campaign. This domain experts must recognize and approve this discovered knowledge, and for this we need good descriptions. There are a variety of algorithms for building decision trees that share the desirable quality of interpretability. A well known and frequently used over the years is C4.5 (or improved, but commercial version See5/C5.0).

What is a decision tree ?

Decision tree is a classifier in the form of a tree structure (see Figure 1), where each node is either:

A decision tree can be used to classify an example by starting at the root of the tree and moving through it until a leaf node, which provides the classification of the instance.

Decision tree induction is a typical inductive approach to learn knowledge on classification. The key requirements to do mining with decision trees are:

Figure 1: An example of a simple decision tree

Constructing decision trees

Most algorithms that have been developed for learning decision trees are variations on a core algorithm that employs a top-down, greedy search through the space of possible decision trees. Decision tree programs construct a decision tree T from a set of training cases.

J. Ross Quinlan originally developed ID3 at the University of Sydney. He first presented ID3 in 1975 in a book, Machine Learning, vol. 1, no. 1. ID3 is based on the Concept Learning System (CLS) algorithm.

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function ID3
Input:   (R: a set of non-target attributes,
          C: the target attribute,
          S: a training set) returns a decision tree;
begin
   If S is empty, return a single node with 
      value Failure;
   If S consists of records all with the same 
      value for the target attribute, 
      return a single leaf node with that value;
   If R is empty, then return a single node 
      with the value of the most frequent of the
      values of the target attribute that are 
      found in records of S; [in that case 
      there may be be errors, examples 
      that will be improperly classified];
   Let A be the attribute with largest 
      Gain(A,S) among attributes in R;
   Let {aj| j=1,2, .., m} be the values of 
      attribute A;
   Let {Sj| j=1,2, .., m} be the subsets of 
      S consisting respectively of records 
      with value aj for A;
   Return a tree with root labeled A and arcs 
      labeled a1, a2, .., am going respectively 
      to the trees (ID3(R-{A}, C, S1), ID3(R-{A}, C, S2),
      .....,ID3(R-{A}, C, Sm);
   Recursively apply ID3 to subsets {Sj| j=1,2, .., m}
      until they are empty
end
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Figure 2: ID3 Decision Tree Algorithm

ID3 searches through the attributes of the training instances and extracts the attribute that best separates the given examples. If the attribute perfectly classifies the training sets then ID3 stops; otherwise it recursively operates on the m (where m = number of possible values of an attribute) partitioned subsets to get their "best" attribute. The algorithm uses a greedy search, that is, it picks the best attribute and never looks back to reconsider earlier choices. Note that ID3 may misclassify data.

The central focus of the decision tree growing algorithm is selecting which attribute to test at each node in the tree. For the selection of the attribute with the most inhomogeneous class distribution the algorithm uses the concept of entropy, which is explained next

Which attribute is the best classifier?

The estimation criterion in the decision tree algorithm is the selection of an attribute to test at each decision node in the tree. The goal is to select the attribute that is most useful for classifying examples. A good quantitative measure of the worth of an attribute is a statistical property called information gain that measures how well a given attribute separates the training examples according to their target classification. This measure is used to select among the candidate attributes at each step while growing the tree.

Entropy - a measure of homogeneity of the set of examples

In order to define information gain precisely, we need to define a measure commonly used in information theory, called entropy, that characterizes the (im)purity of an arbitrary collection of examples. Given a set S, containing only positive and negative examples of some target concept (a 2 class problem), the entropy of set S relative to this simple, binary classification is defined as:

Entropy(S) = - pplog2 pp pnlog2 pn

where ppis the proportion of positive examples in S and pn is the proportion of negative examples in S. In all calculations involving entropy we define 0log0 to be 0.

To illustrate, suppose S is a collection of 25 examples, including 15 positive and 10 negative examples [15+, 10-]. Then the entropy of S relative to this classification is

Entropy(S) = - (15/25) log2 (15/25) - (10/25) log2 (10/25) = 0.970

Notice that the entropy is 0 if all members of S belong to the same class. For example, if all members are positive (pp= 1 ), then pn is 0, and Entropy(S) = -1 log2(1) - 0 log20 = -1 0 - 0 log20 = 0. Note the entropy is 1 (at its maximum!) when the collection contains an equal number of positive and negative examples. If the collection contains unequal numbers of positive and negative examples, the entropy is between 0 and 1. Figure 3 shows the form of the entropy function relative to a binary classification, as p+ varies between 0 and 1.

Figure 3: The entropy function relative to a binary classification, as the proportion of positive

examples pp varies between 0 and 1.

One interpretation of entropy from information theory is that it specifies the minimum number of bits of information needed to encode the classification of an arbitrary member of S (i.e., a member of S drawn at random with uniform probability). For example, if pp is 1, the receiver knows the drawn example will be positive, so no message need be sent, and the entropy is 0. On the other hand, if pp is 0.5, one bit is required to indicate whether the drawn example is positive or negative. If pp is 0.8, then a collection of messages can be encoded using on average less than 1 bit per message by assigning shorter codes to collections of positive examples and longer codes to less likely negative examples.

Thus far we have discussed entropy in the special case where the target classification is binary. If the target attribute takes on c different values, then the entropy of S relative to this c-wise classification is defined as

where pi is the proportion of S belonging to class i. Note the logarithm is still base 2 because entropy is a measure of the expected encoding length measured in bits. Note also that if the target attribute can take on c possible values, the maximum possible entropy is log2c.

Information gain measures the expected reduction in entropy

Given entropy as a measure of the impurity in a collection of training examples, we can now define a measure of the effectiveness of an attribute in classifying the training data. The measure we will use, called information gain, is simply the expected reduction in entropy caused by partitioning the examples according to this attribute. More precisely, the information gain, Gain (S, A) of an attribute A, relative to a collection of examples S, is defined as

where Values(A) is the set of all possible values for attribute A, and Sv is the subset of S for which attribute A has value v (i.e., Sv = {s S | A(s) = v}). Note the first term in the equation for Gain is just the entropy of the original collection S and the second term is the expected value of the entropy after S is partitioned using attribute A. The expected entropy described by this second term is simply the sum of the entropies of each subset Sv, weighted by the fraction of examples |Sv|/|S| that belong to Sv. Gain (S,A) is therefore the expected reduction in entropy caused by knowing the value of attribute A. Put another way, Gain(S,A) is the information provided about the target attribute value, given the value of some other attribute A. The value of Gain(S,A) is the number of bits saved when encoding the target value of an arbitrary member of S, by knowing the value of attribute A.

The process of selecting a new attribute and partitioning the training examples is now repeated for each non-terminal descendant node, this time using only the training examples associated with that node. Attributes that have been incorporated higher in the tree are excluded, so that any given attribute can appear at most once along any path through the tree. This process continues for each new leaf node until either of two conditions is met:

    1. every attribute has already been included along this path through the tree, or
    2. the training examples associated with this leaf node all have the same target attribute value (i.e., their entropy is zero).

Issues in data mining with decision trees

Practical issues in learning decision trees include determining how deeply to grow the decision tree, handling continuous attributes, choosing an appropriate attribute selection measure, handling training data with missing attribute values, handing attributes with differing costs, and improving computational efficiency. Below we discuss each of these issues and extensions to the basic ID3 algorithm that address them.

Avoiding over-fitting the data

In principle decision tree algorithm described in Figure 2 can grow each branch of the tree just deeply enough to perfectly classify the training examples. While this is sometimes a reasonable strategy, in fact it can lead to difficulties when there is noise in the data, or when the number of training examples is too small to produce a representative sample of the true target function. In either of these cases, this simple algorithm can produce trees that over-fit the training examples.

Over-fitting is a significant practical difficulty for decision tree learning and many other learning methods. There are several approaches to avoiding over-fitting in decision tree learning. These can be grouped into two classes:

Although the first of these approaches might seem more direct, the second approach of post-pruning over-fit trees has been found to be more successful in practice. This is due to the difficulty in the first approach of estimating precisely when to stop growing the tree.

Regardless of whether the correct tree size is found by stopping early or by post-pruning, a key question is what criterion is to be used to determine the correct final tree size. Approaches include:

The first of the above approaches is the most common and is often referred to as a training and validation set approach. In this approach, the available data are separated into two sets of examples: a training set, which is used to form the learned hypothesis, and a separate validation set, which is used to evaluate the accuracy of this hypothesis over subsequent data and, in particular, to evaluate the impact of pruning this hypothesis.

Incorporating Continuous-Valued Attributes

The initial definition of ID3 is restricted to attributes that take on a discrete set of values. First, the target attribute whose value is predicted by the learned tree must be discrete valued. Second, the attributes tested in the decision nodes of the tree must also be discrete valued. This second restriction can easily be removed so that continuous-valued decision attributes can be incorporated into the learned tree. This can be accomplished by dynamically defining new discrete-valued attributes that partition the continuous attribute value into a discrete set of intervals. In particular, for an attribute A that is continuous-valued, the algorithm can dynamically create a new Boolean attribute Ac that is true if A < c and false otherwise. The only question is how to select the best value for the threshold c. Clearly, we would like to pick a threshold, c, that produces the greatest information gain. By sorting the examples according to the continuous attribute A, then identifying adjacent examples that differ in their target classification, we can generate a set of candidate thresholds midway between the corresponding values of A. It can be shown that the value of c that maximizes information gain must always lie at such a boundary. These candidate thresholds can then be evaluated by computing the information gain associated with each. The information gain can then be computed for each of the candidate attributes, and the best can be selected. This dynamically created Boolean attribute can then compete with the other discrete-valued candidate attributes available for growing the decision tree.

Handling Training Examples with Missing Attribute Values

In certain cases, the available data may be missing values for some attributes. For example, in a medical domain in which we wish to predict patient outcome based on various laboratory tests, it may be that the lab test Blood-Test-Result is available only for a subset of the patients. In such cases, it is common to estimate the missing attribute value based on other examples for which this attribute has a known value.

Consider the situation in which Gain(S, A) is to be calculated at node n in the decision tree to evaluate whether the attribute A is the best attribute to test at this decision node. Suppose that <x, c(x)> is one of the training examples in S and that the value A(x) is unknown, where c(x) is the class label of x.

One strategy for dealing with the missing attribute value is to assign it the value that is most common among training examples at node n. Alternatively, we might assign it the most common value among examples at node n that have the classification c(x). The elaborated training example using this estimated value for A(x) can then be used directly by the existing decision tree learning algorithm.

A second, more complex procedure is to assign a probability to each of the possible values of A rather than simply assigning the most common value to A(x). These probabilities can be estimated again based on the observed frequencies of the various values for A among the examples at node n. For example, given a Boolean attribute A, if node n contains six known examples with A = 1 and four with A = 0, then we would say the probability that A(x) = 1 is 0.6, and the probability that A(x) = 0 is 0.4. A fractional 0.6 of instance x is now distributed down the branch for A = 1 and a fractional 0.4 of x down the other tree branch. These fractional examples are used for the purpose of computing information Gain and can be further subdivided at subsequent branches of the tree if a second missing attribute value must be tested. This same fractioning of examples can also be applied after learning, to classify new instances whose attribute values are unknown. In this case, the classification of the new instance is simply the most probable classification, computed by summing the weights of the instance fragments classified in different ways at the leaf nodes of the tree. This method for handling missing attribute values is used in C4.5.

Strengths and Weakness of Decision Tree Methods

The strengths of decision tree methods are:

The weaknesses of decision tree methods

Links to online tutorials on decision trees

Overview of Decision Trees
by H.Hamilton. E. Gurak, L. Findlater W. Olive
http://www.cs.uregina.ca/~dbd/cs831/notes/ml/dtrees/4_dtrees1.html




© 2001 LIS - Rudjer Boskovic Institute
Last modified: April 16 2014 11:57:48.

----- Decision Trees