Machine Learning, Decision Tree Learning module_2_ppt.pptx

MODULE -2
DECISION TREE
LEARNING
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Decision tree learning is a method for approximating
discrete-valued target functions, in which the learned
function is represented by a decision tree.
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DECISION TREE REPRESENTATION
FIGURE: A
decision tree for the
concept PlayTennis.
An example is
classified by sorting
it through the tree to
the appropriate leaf
node, then returning
the classification
associated with this
leaf
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• Decision trees classify instances by sorting them down the tree from the root to
some leaf node, which provides the classification of the instance.
• Each node in the tree specifies a test of some attribute of the instance, and each
branch descending from that node corresponds to one of the possible values
for this attribute.
• 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. This process
is then repeated for the subtree rooted at the new node.

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APPROPRIATE PROBLEMS FOR
DECISION TREE LEARNING
Decision tree learning is generally best suited to
problems with the following characteristics:
1. Instances are represented by attribute-value pairs – Instances are described by
a fixed set of attributes and their values
2. The target function has discrete output values – The decision tree assigns a
Boolean classification (e.g., yes or no) to each example. Decision tree
methods easily extend to learning functions with more than two possible
output values.
3. Disjunctive descriptions may be required

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4. The training data may contain errors – Decision tree learning methods are
robust to errors, both errors in classifications of the training examples and errors
in the attribute values that describe these examples.
5. The training data may contain missing attribute values – Decision tree
methods can be used even when some training examples have unknown values
• Decision tree learning has been applied to problems such as learning to classify
medical patients by their disease, equipment malfunctions by their cause,
and loan applicants by their likelihood of defaulting on payments.
• Such problems, in which the task is to classify examples into one of a discrete set
of possible categories, are often referred to as classification problems.

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THE BASIC DECISION TREE LEARNING
ALGORITHM
• 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. This approach is exemplified by the ID3
algorithm and its successor C4.5

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What is the ID3 algorithm?
• ID3 stands for Iterative Dichotomiser 3
• ID3 is a precursor to the C4.5 Algorithm.
• The ID3 algorithm was invented by Ross Quinlan in 1975
• Used to generate a decision tree from a given data set by employing a top-down,
greedy search, to test each attribute at every node of the tree.
• The resulting tree is used to classify future samples.

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ID3 algorithm
ID3(Examples, Target_attribute, Attributes)
Examples are the training examples. Target_attribute is the attribute whose value is to be predicted
by the tree. Attributes is a list of other attributes that may be tested by the learned decision tree.
Returns a decision tree that correctly classifies the given Examples.
 Create a Root node for the tree
 If all Examples are positive, Return the single-node tree Root, with label = +
 If all Examples are negative, Return the single-node tree Root, with label = -
 If Attributes is empty, Return the single-node tree Root, with label = most common value of
Target_attribute in Examples

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 Otherwise Begin
 A ← the attribute from Attributes that best* classifies Examples
 The decision attribute for Root ← A
 For each possible value, vi, of A,
 Add a new tree branch below Root, corresponding to the test A = vi
 Let Examples vi, be the subset of Examples that have value vi for A
 If Examples vi , is empty
 Then below this new branch add a leaf node with label = most common value of
Target_attribute in Examples
 Else below this new branch add the subtree
ID3(Examples vi, Targe_tattribute, Attributes – {A}))
 End
 Return Root
* The best attribute is the one with highest information gain

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Which Attribute Is the Best
Classifier?
• The central choice in the ID3 algorithm is selecting which attribute to test at each
node in the tree.
• A statistical property called information gain that measures how well a
given attribute separates the training examples according to their target
classification.
• ID3 uses information gain measure to select among the candidate attributes at
each step while growing the tree.

ENTROPY MEASURES HOMOGENEITY OF EXAMPLES
• To define information gain, we begin by defining a measure called entropy.
Entropy measures the impurity of a collection of examples.
• Given a collection S, containing positive and negative examples of some
target
concept, the entropy of S relative to this Boolean classification is
Where,
p+ is the proportion of positive examples in S
p- is the proportion of negative examples in S.
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Example: Entropy
• Suppose S is a collection of 14 examples of some boolean concept, including 9
positive and 5 negative examples. Then the entropy of S relative to this
boolean classification is
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• The entropy is 0 if all members of S belong to the same class
• The entropy is 1 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

INFORMATION GAIN MEASURES THE EXPECTED
REDUCTION IN ENTROPY
• Information gain, is the expected reduction in entropy caused by partitioning the
examples according to this attribute.
• The information gain, Gain(S, A) of an attribute A, relative to a collection
of examples S, is defined as
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Example: Information gain
Let, Values(Wind) = {Weak, Strong}
S
SWeak
SStrong
= [9+, 5−]
= [6+, 2−]
= [3+, 3−]
Information gain of attribute Wind:
Gain(S, Wind) = Entropy(S) − 8/14 Entropy (SWeak) − 6/14 Entropy (SStrong)
= 0.94 – (8/14)* 0.811 – (6/14) *1.00
= 0.048

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An Illustrative Example
• To illustrate the operation of ID3, consider the learning task represented by the
training examples of below table.
• Here the target attribute PlayTennis, which can have values yes or no
for
different days.
• Consider the first step through the algorithm, in which the topmost node of the
decision tree is created.

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Day Outlook Temperature Humidity Wind PlayTennis
D1 Sunny Hot High Weak No
D2 Sunny Hot High Strong No
D3 Overcast Hot High Weak Yes
D4 Rain Mild High Weak Yes
D5 Rain Cool Normal Weak Yes
D6 Rain Cool Normal Strong No
D7 Overcast Cool Normal Strong Yes
D8 Sunny Mild High Weak No
D9 Sunny Cool Normal Weak Yes
D10 Rain Mild Normal Weak Yes
D11 Sunny Mild Normal Strong Yes
D12 Overcast Mild High Strong Yes
D13 Overcast Hot Normal Weak Yes
D14 Rain Mild High Strong No

ID3 determines the information gain for each candidate attribute (i.e., Outlook,
Temperature, Humidity, and Wind), then selects the one with highest information
gain
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= 0.246
= 0.151
= 0.048
The information gain values for all four attributes
are
• Gain(S, Outlook)
• Gain(S, Humidity)
• Gain(S, Wind)
• Gain(S, Temperature)
= 0.029
• According to the information gain measure, the Outlook attribute provides the
best prediction of the target attribute, PlayTennis, over the training
examples. Therefore, Outlook is selected as the decision attribute for the
root node, and branches are created below the root for each of its
possible values i.e., Sunny, Overcast, and Rain.

SRain = { D4, D5, D6, D10, D14}
Gain (SRain , Humidity) = 0.970 – (2/5)1.0 – (3/5)0.917 = 0.019
Gain (SRain , Temperature) =0.970 – (0/5)0.0 – (3/5)0.918 – (2/5)1.0 = 0.019
Gain (SRain , Wind) =0.970 – (3/5)0.0 – (2/5)0.0 = 0.970
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HYPOTHESIS SPACE SEARCH IN DECISION TREE
LEARNING
• ID3 can be characterized as searching a space of hypotheses for one that fits the
training examples.
• The hypothesis space searched by ID3 is the set of possible decision trees.
• ID3 performs a simple-to complex, hill-climbing search through this hypothesis
space, beginning with the empty tree, then considering progressively
more elaborate hypotheses in search of a decision tree that correctly
classifies the training data

Figure:
• Hypothesis space search by ID3.
• ID3 searches through the space of
possible decision trees from
simplest to increasingly complex,
guided by the information gain
heuristic
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By viewing ID3 in terms of its search space and search strategy, we can get some
insight into its capabilities and limitations
1. ID3's hypothesis space of all decision trees is a complete space of finite discrete-
valued functions, relative to the available attributes. Because every finite discrete-
valued function can be represented by some decision tree
• ID3 avoids one of the major risks of methods that search incomplete hypothesis
spaces : that the hypothesis space might not contain the target function.

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2. ID3 maintains only a single current hypothesis as it searches through the
space of decision trees.
For example, with the earlier version space candidate elimination method,
which maintains the set of all hypotheses consistent with the available training
examples.
By determining only a single hypothesis, ID3 loses the capabilities that follow
from explicitly representing all consistent hypotheses.
For example, it does not have the ability to determine how many
alternative decision trees are consistent with the available training data, or
to pose new instance queries that optimally resolve among these competing
hypotheses

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3. ID3 in its pure form performs no backtracking in its search. Once it selects an
attribute to test at a particular level in the tree, it never backtracks to reconsider this
choice.
• In the case of ID3, a locally optimal solution corresponds to the decision tree it
selects along the single search path it explores. However, this locally
optimal solution may be less desirable than trees that would have been
encountered along a different branch of the search.
4. ID3 uses all training examples at each step in the search to make statistically
based decisions regarding how to refine its current hypothesis.
• One advantage of using statistical properties of all the examples is that
the resulting search is much less sensitive to errors in individual training
examples.
• ID3 can be easily extended to handle noisy training data by modifying
its termination criterion to accept hypotheses that imperfectly fit the training
data.

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INDUCTIVE BIAS IN DECISION TREE LEARNING
Inductive bias is the set of assumptions that, together with the training data,
deductively justify the classifications assigned by the learner to future instances
Given a collection of training examples, there are typically many decision trees
consistent with these examples. Which of these decision trees does ID3 choose?
ID3 search strategy
(a) selects in favour of shorter trees over longer ones
(b) selects trees that place the attributes with highest information gain closest to
the root.

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Approximate inductive bias of ID3: Shorter trees are preferred over larger trees
• Consider an algorithm that begins with the empty tree and searches breadth first
through progressively more complex trees.
• First considering all trees of depth 1, then all trees of depth 2, etc.
• Once it finds a decision tree consistent with the training data, it returns
the smallest consistent tree at that search depth (e.g., the tree with the fewest
nodes).
• Let us call this breadth-first search algorithm BFS-ID3.
• BFS-ID3 finds a shortest decision tree and thus exhibits the bias "shorter trees are
preferred over longer trees.

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A closer approximation to the inductive bias of ID3: Shorter trees are preferred
over longer trees. Trees that place high information gain attributes close to the root
are preferred over those that do not.
• ID3 can be viewed as an efficient approximation to BFS-ID3, using a greedy
heuristic search to attempt to find the shortest tree without conducting the
entire breadth-first search through the hypothesis space.
• Because ID3 uses the information gain heuristic and a hill climbing strategy, it
exhibits a more complex bias than BFS-ID3.
• In particular, it does not always find the shortest consistent tree, and it is biased to
favour trees that place attributes with high information gain closest to the root.

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Restriction Biases and Preference Biases
Difference between the types of inductive bias exhibited by ID3 and by the CANDIDATE-
ELIMINATION Algorithm.
ID3
• ID3 searches a complete hypothesis space
• It searches incompletely through this space, from simple to complex hypotheses,
until its
termination condition is met
• Its inductive bias is solely a consequence of the ordering of hypotheses by its search strategy. Its
hypothesis space introduces no additional bias
CANDIDATE-ELIMINATION Algorithm
• The version space CANDIDATE-ELIMINATION Algorithm searches an incomplete hypothesis
space
• It searches this space completely, finding every hypothesis consistent with the training data.
• Its inductive bias is solely a consequence of the expressive power of its
hypothesis representation. Its search strategy introduces no additional bias

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Restriction Biases and Preference Biases
• The inductive bias of ID3 is a preference for certain hypotheses over others (e.g.,
preference for shorter hypotheses over larger hypotheses), with no hard
restriction on the hypotheses that can be eventually enumerated. This form of
bias is called a preference bias or a search bias.
• The bias of the CANDIDATE ELIMINATION algorithm is in the form of a
categorical restriction on the set of hypotheses considered. This form of bias
is typically called a restriction bias or a language bias.

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Which type of inductive bias is preferred in order to generalize beyond the
training data, a preference bias or restriction bias?
• A preference bias is more desirable than a restriction bias, because it allows the
learner to work within a complete hypothesis space that is assured to contain
the unknown target function.
• In contrast, a restriction bias that strictly limits the set of potential hypotheses is
generally less desirable, because it introduces the possibility of excluding
the unknown target function altogether.

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Occam's razor
Occam's razor: is the problem-solving principle that the simplest solution tends to be
the right one. When presented with competing hypotheses to solve a problem, one
should select the solution with the fewest assumptions.
Occam's razor: “Prefer the simplest hypothesis that fits the data”.

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Why Prefer Short Hypotheses ?
Argument in favour:
Fewer short hypotheses than long ones:
• Short hypotheses fits the training data which are less likely to be
coincident
• Longer hypotheses fits the training data might be coincident.
Many complex hypotheses that fit the current training data but fail to
generalize correctly to subsequent data.

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Argument opposed:
• There are few small trees, and our priori chance of finding one consistent with an
arbitrary set of data is therefore small. The difficulty here is that there are
very many small sets of hypotheses that one can define but understood
by fewer learner.
• The size of a hypothesis is determined by the representation used internally by the
learner. Occam's razor will produce two different hypotheses from the
same training examples when it is applied by two learners, both
justifying their contradictory conclusions by Occam's razor. On this basis
we might be tempted to reject Occam's razor altogether.

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ISSUES IN DECISION TREE LEARNING
1. Avoiding Overfitting the Data
Reduced error
pruning Rule post-
pruning
2. Incorporating Continuous-
Valued Attributes
3. Alternative Measures for
Selecting Attributes
4. Handling Training Examples
with Missing Attribute
Values
5. Handling Attributes with
Differing Costs

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1. Avoiding Overfitting the Data
• The ID3 algorithm grows each branch of the tree just deeply enough to perfectly
classify the training examples but 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. This algorithm
can produce trees that overfit the training examples.
• Definition - Overfit: Given a hypothesis space H, a hypothesis h ∈ H is said
to overfit the training data if there exists some alternative hypothesis h' ∈
H, such that h has smaller error than h' over the training examples, but
h' has a smaller error than h over the entire distribution of instances.

• The below figure illustrates the impact of overfitting in a typical application of decision
tree learning.
• The horizontal axis of this plot indicates the total number of nodes in the decision tree, as the tree is being
constructed. The vertical axis indicates the accuracy of predictions made by the tree.
• The solid line shows the accuracy of the decision tree over the training examples. The broken line shows
accuracy measured over an independent set of test example
• The accuracy of the tree over the training examples increases monotonically as the tree is grown. The
accuracy measured over the independent test examples first increases, then decreases.
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How can it be possible for tree h to fit the training examples better than h', but for it to
perform more poorly over subsequent examples?
1. Overfitting can occur when the training examples contain random errors or noise
2. When small numbers of examples are associated with leaf nodes.
Noisy Training Example
Example 15: <Sunny, Hot, Normal, Strong, ->
• Example is noisy because the correct label is +
• Previously constructed tree misclassifies it
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Approaches to avoiding overfitting in decision tree learning
• Pre-pruning (avoidance): Stop growing the tree earlier, before it reaches the point where
it perfectly classifies the training data
• Post-pruning (recovery): Allow the tree to overfit the data, and then post-prune the tree
Criterion used to determine the correct final tree size
• Use a separate set of examples, distinct from the training examples, to evaluate the utility of
post-pruning nodes from the tree
• Use all the available data for training, but apply a statistical test to estimate whether
expanding (or pruning) a particular node is likely to produce an improvement beyond the
training set
• Use measure of the complexity for encoding the training examples and the decision tree,
halting growth of the tree when this encoding size is minimized. This approach is called the
Minimum Description Length
MDL – Minimize : size(tree) + size (misclassifications(tree))

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Reduced-Error Pruning
• Reduced-error pruning, is to consider each of the decision nodes in the tree to be
candidates for pruning
• Pruning a decision node consists of removing the subtree rooted at that node,
making it a leaf node, and assigning it the most common classification of
the training examples affiliated with that node
• Nodes are removed only if the resulting pruned tree performs no worse than-the
original over the validation set.
• Reduced error pruning has the effect that any leaf node added due to coincidental
regularities in the training set is likely to be pruned because these
same coincidences are unlikely to occur in the validation set

The impact of reduced-error pruning on the accuracy of the decision tree is illustrated in below
figure
• The additional line in figure shows accuracy over the test examples as the tree is pruned. When
pruning begins, the tree is at its maximum size and lowest accuracy over the test set. As
pruning proceeds, the number of nodes is reduced and accuracy over the test set increases.
• The available data has been split into three subsets: the training examples, the validation examples
used for pruning the tree, and a set of test examples used to provide an unbiased estimate
of accuracy over future unseen examples. The plot shows accuracy over the training and test
sets.
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Pros and Cons
Pro: Produces smallest version of most accurate T (subtree of T)
Con: Uses less data to construct T
Can afford to hold out Dvalidation?. If not (data is too limited), may make error worse
(insufficient Dtrain)

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Rule Post-Pruning
Rule post-pruning is successful method for finding high accuracy hypotheses
Rule post-pruning involves the following steps:
1. Infer the decision tree from the training set, growing the tree until the training
data is fit as well as possible and allowing overfitting to occur.
2. Convert the learned tree into an equivalent set of rules by creating one rule for
each path from the root node to a leaf node.
3. Prune (generalize) each rule by removing any preconditions that
result in improving its estimated accuracy.
4. Sort the pruned rules by their estimated accuracy, and consider them in
this sequence when classifying subsequent instances.

Converting a Decision Tree into Rules
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For example, consider the decision tree. The leftmost
path of the tree in below figure is translated into the rule.
IF (Outlook = Sunny) ^ (Humidity = High)
THEN PlayTennis = No
Given the above rule, rule post-pruning would consider removing the preconditions
(Outlook = Sunny) and (Humidity = High)
• It would select whichever of these pruning steps produced the
greatest
improvement in estimated rule accuracy, then consider pruning the second
precondition as a further pruning step.
• No pruning step is performed if it reduces the estimated rule accuracy.

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There are three main advantages by converting the decision tree to rules before
pruning
• Converting to rules allows distinguishing among the different contexts in which a
decision node is used. Because each distinct path through the decision tree
node produces a distinct rule, the pruning decision regarding that attribute test
can be made differently for each path.
• Converting to rules removes the distinction between attribute tests that occur near
the root of the tree and those that occur near the leaves. Thus, it avoid
messy bookkeeping issues such as how to reorganize the tree if the root node is
pruned while retaining part of the subtree below this test.
• Converting to rules improves readability. Rules are often easier for to understand.

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2. Incorporating Continuous-Valued Attributes
Continuous-valued decision attributes can be incorporated into the learned tree.
There are two methods for Handling Continuous Attributes
1. Define new discrete valued attributes that partition the continuous attribute
value into a discrete set of intervals.
E.g., {high ≡ Temp > 35º C, med ≡ 10º C < Temp ≤ 35º C, low ≡ Temp ≤ 10º C}
2. Using thresholds for splitting nodes
e.g., A ≤ a produces subsets A ≤ a and A > a

What threshold-based boolean attribute should be defined based on Temperature?
Pick a threshold, c, that produces the greatest information gain
• In the current example, there are two candidate thresholds, corresponding to the
values of Temperature at which the value of PlayTennis changes: (48 + 60)/2,
and (80 + 90)/2. The information gain can then be computed for each of the
candidate attributes, Temperature >54, and Temperature >85 and the best
can be selected (Temperature >54)
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3. Alternative Measures for Selecting Attributes
The problem is if attributes with many values, Gain will select it ?
Example: consider the attribute Date, which has a very large number of possible
values. (e.g., March 4, 1979).
• If this attribute is added to the PlayTennis data, it would have the highest
information gain of any of the attributes. This is because Date alone
perfectly predicts the target attribute over the training data. Thus, it would be
selected as the decision attribute for the root node of the tree and lead
to a tree of depth one, which perfectly classifies the training data.
• This decision tree with root node Date is not a useful predictor because it perfectly
separates the training data, but poorly predict on subsequent examples.

One Approach: Use GainRatio instead of Gain
• The gain ratio measure penalizes attributes by incorporating a split information,
that is sensitive to how broadly and uniformly the attribute splits the data
• where Si is subset of S, for which attribute A has value vi
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4. Handling Training Examples with Missing Attribute Values
The data which is available may contain missing values for some attributes
Example: Medical diagnosis
• <Fever = true, Blood-Pressure = normal, …, Blood-Test = ?, …>
• Sometimes values truly unknown, sometimes low priority (or cost too
high)

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Strategies for dealing with the missing attribute value
• If node n test A, assign most common value of A among other training examples
sorted to node n
• Assign most common value of A among other training examples with same target
value
• Assign a probability pi to each of the possible values vi of A rather than simply
assigning the most common value to A(x)

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5. Handling Attributes with Differing Costs
In some learning tasks the instance attributes may have associated costs.
For example:
• In learning to classify medical diseases, the patients described in terms of
attributes such as Temperature, BiopsyResult, Pulse, BloodTestResults, etc.
• These attributes vary significantly in their costs, both in terms of monetary
cost and cost to patient comfort
• Decision trees use low-cost attributes where possible, depends only on high-
cost
attributes only when needed to produce reliable classifications

How to Learn A Consistent Tree with Low Expected
Cost?
One approach is replace Gain by Cost-Normalized-Gain
Examples of normalization functions
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