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Decision tree. LING 572 Fei Xia 1/10/06. Outline. Basic concepts Main issues Advanced topics. Basic concepts. A classification problem. Classification and estimation problems. Given a finite set of (input) attributes: features Ex: District, House type, Income, Previous customer

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Decision tree

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decision tree

Decision tree

LING 572

Fei Xia


  • Basic concepts
  • Main issues
  • Advanced topics
classification and estimation problems
Classification and estimation problems
  • Given
    • a finite set of (input) attributes: features
      • Ex: District, House type, Income, Previous customer
    • a target attribute: the goal
      • Ex: Outcome: {Nothing, Respond}
    • training data: a set of classified examples in attribute-value representation
      • Ex: the previous table
  • Predict the value of the goal given the values of input attributes
    • The goal is a discrete variable  classification problem
    • The goal is a continuous variable  estimation problem
decision tree6








House type

Previous customer





No (2/2)







Decision tree
decision tree representation
Decision tree representation
  • Each internal node is a test:
    • Theoretically, a node can test multiple attributes
    • In most systems, a node tests exactly one attribute
  • Each branch corresponds to test results
    • A branch corresponds to an attribute value or a range of attribute values
  • Each leaf node assigns
    • a class: decision tree
    • a real value: regression tree
what s a best decision tree
What’s a best decision tree?
  • “Best”: You need a bias (e.g., prefer the “smallest” tree): least depth? Fewest nodes? Which trees are the best predictors of unseen data?
  • Occam's Razor: we prefer the simplest hypothesis that fits the data.

 Find a decision tree that is as small as possible and fits the data

finding a smallest decision tree
Finding a smallest decision tree
  • A decision tree can represent any discrete function of the inputs: y=f(x1, x2, …, xn)
  • The space of decision trees is too big for systemic search for a smallest decision tree.
  • Solution: greedy algorithm
basic algorithm top down induction
Basic algorithm: top-down induction
  • Find the “best” decision attribute, A, and assign A as decision attribute for node
  • For each value of A, create new branch, and divide up training examples
  • Repeat the process until the gain is small enough
major issues12
Major issues

Q1: Choosing best attribute: what quality measure to use?

Q2: Determining when to stop splitting: avoid overfitting

Q3: Handling continuous attributes

Q4: Handling training data with missing attribute values

Q5: Handing attributes with different costs

Q6: Dealing with continuous goal attribute

q1 what quality measure
Q1: What quality measure
  • Information gain
  • Gain Ratio
entropy of a training set
Entropy of a training set
  • S is a sample of training examples
  • Entropy is one way of measuring the impurity of S
  • Pc is the proportion of examples in S whose target attribute has value c.
information gain
Information Gain
  • Gain(S,A)=expected reduction in entropy due to sorting on A.
  • Choose the A with the max information gain.

(a.k.a. choose the A with the min average entropy)

an example















An example





InfoGain(S, Humidity)



InfoGain(S, Wind)



other quality measures
Other quality measures
  • Problem of information gain:
    • Information Gain prefers attributes with many values.
  • An alternative: Gain Ratio

Where Si is subset of S for which A has value vi.

q2 avoiding overfitting
Q2: avoiding overfitting
  • Overfitting occurs when our decision tree characterizes too much detail, or noise in our training data.
  • Consider error of hypothesis h over
    • Training data: ErrorTrain(h)
    • Entire distribution D of data: ErrorD(h)
  • A hypothesis h overfits training data if there is an alternative hypothesis h’, such that
    • ErrorTrain(h) < ErrorTrain(h’), and
    • ErrorD(h) > errorD(h’)
how to avoiding overfitting
How to avoiding Overfitting
  • Stop growing the tree earlier
    • Ex: InfoGain < threshold
    • Ex: Size of examples in a node < threshold
  • Grow full tree, then post-prune

 In practice, the latter works better than the former.

post pruning
  • Split data into training and validation set
  • Do until further pruning is harmful:
    • Evaluate impact on validation set of pruning each possible node (plus those below it)
    • Greedily remove the ones that don’t improve the performance on validation set
  • Produces a smaller tree with best performance measure
performance measure
Performance measure
  • Accuracy:
    • on validation data
    • K-fold cross validation
  • Misclassification cost: Sometimes more accuracy is desired for some classes than others.
  • MDL: size(tree) + errors(tree)
rule post pruning
Rule post-pruning
  • Convert tree to equivalent set of rules
  • Prune each rule independently of others
  • Sort final rules into desired sequence for use
  • Perhaps most frequently used method (e.g., C4.5)
q3 handling numeric attributes
Q3: handling numeric attributes
  • Continuous attribute  discrete attribute
  • Example
    • Original attribute: Temperature = 82.5
    • New attribute: (temperature > 72.3) = t, f

 Question: how to choose thresholds?

choosing thresholds for a continuous attribute
Choosing thresholds for a continuous attribute
  • Sort the examples according to the continuous attribute.
  • Identify adjacent examples that differ in their target classification  a set of candidate thresholds
  • Choose the candidate with the highest information gain.
q4 unknown attribute values
Q4: Unknown attribute values
  • Assume an attribute can take the value “blank”.
  • Assign most common value of A among training data at node n.
  • Assign most common value of A among training data at node n which have the same target class.
  • Assign prob pi to each possible value vi of A
    • Assign a fraction (pi) of example to each descendant in tree
    • This method is used in C4.5.
q5 attributes with cost
Q5: Attributes with cost
  • Consider medical diagnosis (e.g., blood test) has a cost
  • Question: how to learn a consistent tree with low expected cost?
  • One approach: replace gain by
    • Tan and Schlimmer (1990)
q6 dealing with continuous goal attribute regression tree
Q6: Dealing with continuous goal attribute  Regression tree
  • A variant of decision trees
  • Estimation problem: approximate real-valued functions: e.g., the crime rate
  • A leaf node is marked with a real value or a linear function: e.g., the mean of the target values of the examples at the node.
  • Measure of impurity: e.g., variance, standard deviation, …
summary of major issues
Summary of Major issues

Q1: Choosing best attribute: different quality measure.

Q2: Determining when to stop splitting: stop earlier or post-pruning

Q3: Handling continuous attributes: find the breakpoints

summary of major issues cont
Summary of major issues (cont)

Q4: Handling training data with missing attribute values: blank value, most common value, or fractional count

Q5: Handing attributes with different costs: use a quality measure that includes the cost factors.

Q6: Dealing with continuous goal attribute: various ways of building regression trees.

  • Proposed by Quinlan (so is C4.5)
  • Can handle basic cases: discrete attributes, no missing information, etc.
  • Information gain as quality measure
  • An extension of ID3:
    • Several quality measures
    • Incomplete information (missing attribute values)
    • Numerical (continuous) attributes
    • Pruning of decision trees
    • Rule derivation
    • Random mood and batch mood
  • CART (classification and regression tree)
  • Proposed by Breiman et. al. (1984)
  • Constant numerical values in leaves
  • Variance as measure of impurity
strengths of decision tree methods
Strengths of decision tree methods
  • Ability to generate understandable rules
  • Ease of calculation at classification time
  • Ability to handle both continuous and categorical variables
  • Ability to clearly indicate best attributes
the weaknesses of decision tree methods
The weaknesses of decision tree methods
  • Greedy algorithm: no global optimization
  • Error-prone with too many classes: numbers of training examples become smaller quickly in a tree with many levels/branches.
  • Expensive to train: sorting, combination of attributes, calculating quality measures, etc.
  • Trouble with non-rectangular regions: the rectangular classification boxes that may not correspond well with the actual distribution of records in the decision space.
combining multiple models
Combining multiple models
  • The inherent instability of top-down decision tree induction: different training datasets from a given problem domain will produce quite different trees.
  • Techniques:
    • Bagging
    • Boosting
  • Introduced by Breiman
  • It first creates multiple decision trees based on different training sets.
  • Then, it compares each tree and incorporates the best features of each.
  • This addresses some of the problems inherent in regular ID3.
  • Introduced by Freund and Schapire
  • It examines the trees that incorrectly classify an instance and assign them a weight.
  • These weights are used to eliminate hypotheses or refocus the algorithm on the hypotheses that are performing well.
  • Basic case:
    • Discrete input attributes
    • Discrete goal attribute
    • No missing attribute values
    • Same cost for all tests and all kinds of misclassification.
  • Extended cases:
    • Continuous attributes
    • Real-valued goal attribute
    • Some examples miss some attribute values
    • Some tests are more expensive than others.
    • Some misclassifications are more serious than others.
summary cont
Summary (cont)
  • Basic algorithm:
    • greedy algorithm
    • top-down induction
    • Bias for small trees
  • Major issues:
uncovered issues
Uncovered issues
  • Incremental decision tree induction?
  • How can a decision relate to other decisions? what's the order of making the decisions? (e.g., POS tagging)
  • What's the difference between decision tree and decision list?