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Decision Trees

Decision Trees. a n introduction. Entropy over the class attribute. A class attribute contains uncertainty over the values uncertainty captured by entropy H( p ) of the target Increase certainty about the class by considering other attributes

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Decision Trees

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  1. Decision Trees an introduction

  2. Entropy over the class attribute • A class attribute contains uncertainty over the values • uncertainty captured by entropy H(p) of the target • Increase certainty about the class by considering other attributes • Conditioning (splitting) on an informative attribute produces splits with lower entropy • information gain: entropy before split compared to entropy after split

  3. Yes Age < 35 Rent No Age ≥ 35 Yes Price < 200K Buy No Price ≥ 200K No Other Decision tree • An internal node is a test on an attribute • A branch represents an outcome of the test, e.g., house = Rent • A leaf node represents a class label or class label distribution • At each node, one attribute is chosen to split training examples into distinct classes as much as possible • A new case is classified by following a matching path to a leaf node

  4. Weather data: play tennis or not

  5. Example tree for “play tennis”

  6. Building decision tree, Quinlan 1993 • Top-down tree construction • At start, all training examples are at the root • Partition the examples recursively by choosing one attribute each time. • Bottom-up tree pruning • Remove subtrees or branches, in a bottom-up manner, to improve the estimated accuracy on new cases • Discussed next week

  7. Choosing the splitting attribute • At each node, available attributes are evaluated on the basis of separating the classes of the training examples • A goodness function is used for this purpose • Typical goodness functions: • information gain (ID3/C4.5) • information gain ratio • giniindex (not discussed)

  8. A criterion for attribute selection • Which is the best attribute? • The one which will result in the smallest tree • Heuristic: choose the attribute that produces the purestnodes • Pure and high entropy are opposites • Popular impurity criterion: information gain • Information gain uses entropy H(p) of the class attribute • Information gain increases with the average purity of the subsets that an attribute produces • Strategy: choose attribute that results in greatest information gain

  9. Which attribute to select?

  10. pure, 100% yes not pure at all, 40% yes pure, 100% yes not pure at all, 40% yes Consider entropy H(p) done allmost 1 bit of information required to distinguish yes and no

  11. Entropy Entropy: H(p) = – plg(p) – (1–p)lg(1–p) H(0) = 0 pure node, distribution is skewed H(1) = 0 pure node, distribution is skewed H(0.5) = 1 mixed node, equal distribution

  12. Information gain • Information before split minus information after split • gain(A) = H(p) – ΣH(pi)ni/n • pprobability of positive in current set • nnumber of examples in current set • piprobability of positive in branch i • ninumber of examples in branch i • i before split after split

  13. 0lg(0) is not defined, but we evaluate 0lg(0)as zero Example: attribute “Outlook” • Outlook = “Sunny”: H([2,3]) = H(0.4) = −0.4lg(0.4)−0.6lg(0.6) = 0.971 bit • Outlook = “Overcast”: H([4,0]) = H(1) = −1lg(1)−0lg(0) = 0 bit • Outlook = “Rainy”: H([3,2]) = H(0.6) = −0.6lg(0.6)−0.4lg(0.4)= 0.971 bit • Average entropy for Outlook: Weighted sum: (5/14)0.971 + (4/14)0 + (5/14)0.971 = 0.693

  14. Computing the information gain • Information gain for Outlook • gain(Outlook) = H([9,5]) – 0.693 = 0.94 – 0.693 = 0.247 bit • Information gain for attributes from weather data: • gain(Outlook) = 0.247 bit • gain(Temperature) = 0.029 bit • gain(Humidity) = 0.152 bit • gain(Windy) = 0.048 bit

  15. Continuing to split

  16. The final decision tree • Note: not all leaves need to be pure; sometimes identical examples have different classes  Splitting stops when data can’t be split any further

  17. Highly-branching attributes • Problematic: attributes with a large number of values (extreme case: customer ID) • Subsets are more likely to be pure if there is a large number of values • Information gain is biased towards choosing attributes with a large number of values • This may result in overfitting (selection of an attribute that is non-optimal for prediction)

  18. Weather data with ID

  19. Split for ID attribute Entropy of each branch = 0 (since each leaf node is pure, having only one case) Information gain is maximal for ID

  20. Gain ratio • Gain ratio: a modification of the information gain that reduces its bias on high-branch attributes • Gain ratio should be • Large when data is divided in few, even groups • Small when each example belongs to a separate branch • Gain ratio takes number and size of branches into account when choosing an attribute • It corrects the information gain by taking the intrinsic information of a split into account (i.e. how much info do we need to tell which branch an instance belongs to)

  21. Gain ratio and intrinsic information • Intrinsic information: entropy of distribution of instances into branches • Gain rationormalizes info gain by:

  22. Computing the gain ratio • Example: intrinsic information for ID • Importance of attribute decreases as intrinsic information gets larger • Example:

  23. Gain ratios for weather data

  24. More on the gain ratio • Outlook still comes out top • However: ID has greater gain ratio • Standard fix: ad hoc test to prevent splitting on that type of attribute • Problem with gain ratio: it may overcompensate • May choose an attribute just because its intrinsic information is very low • Standard fix: • First, only consider attributes with greater than average information gain • Then, compare them on gain ratio

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