Data Mining(中国人民大学) Yang qiang(香港科技大学) Han jia wei(UIUC) Classification Han jia wei(UIUC) Information Network Data Mining Pei jian（Simon Fraser University） Uncertainty Data and Model

DECISION TREE [Quinlan93] An internal node represents a test on an attribute. A branch represents an outcome of the test, e.g., Color=red. 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.

Training Set

Example Outlook sunny overcast rain humidity P windy high normal true false N P N P

Building Decision Tree [Q93] 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.

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 gini index

Which attribute to select?

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 “purest” nodes Popular impurity criterion: information gain Information gain increases with the average purity of the subsets that an attribute produces Underlying criterion: Entropy ,Gini, Misclassification error Strategy: choose attribute that results in greatest information gain The largest difference between Gini  impurity and entropy is that entropy peaks slower and as a result it tends to penalize mixed sets a little more heavily.  It is more commonly used than Gini impurity but easily swapped out.  

Computing information Information is measured in bits Given a probability distribution, the info required to predict an event is the distribution’s entropy Entropy gives the information required in bits (this can involve fractions of bits!) Formula for computing the entropy: Suppose a set S has n values: V1, V2, …Vn, where Vi has proportion pi, E.g., the weather data has 2 values: Play=P and Play=N. Thus, p1=9/14, p2=5/14. 在信息增益中，衡量标准是看特征能够为分类系统带来多少信息，带来的信息越多，该特征越重要。对一个特征而言，系统有它和没它时信息量将发生变化，而前后信息量的差值就是这个特征给系统带来的信息量。所谓信息量，其实就是熵。

Example: attribute “Outlook” “Outlook” = “Sunny”: “Outlook” = “Overcast”: “Outlook” = “Rainy”: Expected information for attribute: Note: this is normally not defined.

Computing the information gain Information gain: information before splitting – information after splitting Information gain for attributes from weather data:

Continuing to split

The final decision tree When all cases have the same class. The leaf node is labeled by this class. When there is no available attribute. The leaf node is labeled by the majority class. When the number of cases is less than a specified threshold. The leaf node is labeled by the majority class. Note: not all leaves need to be pure; sometimes identical instances have different classes  Splitting stops when data can’t be split any further

Highly-branching attributes Problematic: attributes with a large number of values (extreme case: ID code) 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) Another problem: fragmentation

The gain ratio Gain ratio: a modification of the information gain that reduces its bias on high-branch attributes 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 Also called split ratio Intrinsic information: entropy of distribution of instances into branches (i.e. how much info do we need to tell which branch an instance belongs to)

Gain Ratio IntrinsicInfo should be Large when data is evenly spread over all branches Small when all data belong to one branch Gain ratio (Quinlan’86) normalizes info gain by IntrinsicInfo: 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

Pruning Pruning simplifies a decision tree to prevent overfitting to noise in the data Two main pruning strategies: Postpruning: takes a fully-grown decision tree and discards unreliable parts Prepruning: stops growing a branch when information becomes unreliable Postpruning preferred in practice because of early stopping in prepruning