Theses slides are based on the slides by CISC 4631 Data Mining Lecture 04: Decision Trees Theses slides are based on the slides by Tan, Steinbach and Kumar (textbook authors) Eamonn Koegh (UC Riverside) Raymond Mooney (UT Austin)

Classification: Definition Given a collection of records (training set ) Each record contains a set of attributes, one of the attributes is the class. Find a model for class attribute as a function of the values of other attributes. Goal: previously unseen records should be assigned a class as accurately as possible. A test set is used to determine the accuracy of the model. Usually, the given data set is divided into training and test sets, with training set used to build the model and test set used to validate it.

Illustrating Classification Task

Classification Techniques Decision Tree based Methods Rule-based Methods Memory based reasoning Neural Networks Naïve Bayes and Bayesian Belief Networks Support Vector Machines

Example of a Decision Tree categorical continuous class Splitting Attributes Refund Yes No NO MarSt Single, Divorced Married TaxInc NO < 80K > 80K NO YES Training Data Model: Decision Tree

Another Example of Decision Tree categorical categorical continuous class MarSt Single, Divorced Married NO Refund No Yes NO TaxInc < 80K > 80K NO YES There could be more than one tree that fits the same data!

Decision Tree Classification Task

Apply Model to Test Data Start from the root of tree. Refund MarSt TaxInc YES NO Yes No Married Single, Divorced < 80K > 80K

Apply Model to Test Data Refund MarSt TaxInc YES NO Yes No Married Single, Divorced < 80K > 80K

Apply Model to Test Data Refund Yes No NO MarSt Single, Divorced Married TaxInc NO < 80K > 80K NO YES

Apply Model to Test Data Refund Yes No NO MarSt Single, Divorced Married TaxInc NO < 80K > 80K NO YES

Apply Model to Test Data Refund Yes No NO MarSt Single, Divorced Married TaxInc NO < 80K > 80K NO YES

Apply Model to Test Data Refund Yes No NO MarSt Married Assign Cheat to “No” Single, Divorced TaxInc NO < 80K > 80K NO YES

Decision Tree Terminology

Decision Tree Classification Task

Decision Tree Induction Many Algorithms: Hunt’s Algorithm (one of the earliest) CART ID3, C4.5 SLIQ,SPRINT John Ross Quinlan is a computer science researcher in data mining and decision theory. He has contributed extensively to the development of decision tree algorithms, including inventing the canonical C4.5 and ID3 algorithms. 

Decision Tree Classifier 10 1 2 3 4 5 6 7 8 9 Ross Quinlan Abdomen Length > 7.1? Antenna Length no yes Antenna Length > 6.0? Katydid no yes Grasshopper Katydid Abdomen Length

Decision trees predate computers Antennae shorter than body? Yes No 3 Tarsi? Grasshopper Yes No Foretiba has ears? Yes No Cricket Decision trees predate computers Katydids Camel Cricket

Definition Decision tree is a classifier in the form of a tree structure Decision node: specifies a test on a single attribute Leaf node: indicates the value of the target attribute Arc/edge: split of one attribute Path: a disjunction of test to make the final decision Decision trees classify instances or examples by starting at the root of the tree and moving through it until a leaf node.

Decision Tree Classification Decision tree generation consists of two phases Tree construction At start, all the training examples are at the root Partition examples recursively based on selected attributes Tree pruning Identify and remove branches that reflect noise or outliers Use of decision tree: Classifying an unknown sample Test the attribute values of the sample against the decision tree

Decision Tree Representation Each internal node tests an attribute Each branch corresponds to attribute value Each leaf node assigns a classification outlook sunny overcast rain humidity yes wind high normal strong weak no yes no yes

How do we construct the decision tree? Basic algorithm (a greedy algorithm) Tree is constructed in a top-down recursive divide-and-conquer manner At start, all the training examples are at the root Attributes are categorical (if continuous-valued, they can be discretized in advance) Examples are partitioned recursively based on selected attributes. Test attributes are selected on the basis of a heuristic or statistical measure (e.g., information gain) Conditions for stopping partitioning All samples for a given node belong to the same class There are no remaining attributes for further partitioning – majority voting is employed for classifying the leaf There are no samples left

Top-Down Decision Tree Induction Main loop: A  the “best” decision attribute for next node Assign A as decision attribute for node For each value of A, create new descendant of node Sort training examples to leaf nodes If training examples perfectly classified, Then STOP, Else iterate over new leaf nodes

Tree Induction Greedy strategy. Issues Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting

How To Split Records Random Split Principled Criterion The tree can grow huge These trees are hard to understand. Larger trees are typically less accurate than smaller trees. Principled Criterion Selection of an attribute to test at each node - choosing the most useful attribute for classifying examples. How? Information gain 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

Tree Induction Greedy strategy: Split the records based on an attribute test that optimizes certain criterion: Hunt’s algorithm: recursively partition training records into successively purer subsets. How to measure purity/impurity Entropy and information gain (covered in the lectures slides) Gini (covered in the textbook) Classification error

How to determine the Best Split Before Splitting: 10 records of class 0, 10 records of class 1 Gender Which test condition is the best? Why is student id a bad feature to use?

How to determine the Best Split Greedy approach: Nodes with homogeneous class distribution are preferred Need a measure of node impurity: Non-homogeneous, High degree of impurity Homogeneous, Low degree of impurity

Picking a Good Split Feature Goal is to have the resulting tree be as small as possible, per Occam’s razor. Finding a minimal decision tree (nodes, leaves, or depth) is an NP-hard optimization problem. Top-down divide-and-conquer method does a greedy search for a simple tree but does not guarantee to find the smallest. General lesson in Machine Learning and Data Mining: “Greed is good.” Want to pick a feature that creates subsets of examples that are relatively “pure” in a single class so they are “closer” to being leaf nodes. There are a variety of heuristics for picking a good test, a popular one is based on information gain that originated with the ID3 system of Quinlan (1979). R. Mooney, UT Austin

Information Theory Think of playing "20 questions": I am thinking of an integer between 1 and 1,000 -- what is it? What is the first question you would ask? What question will you ask? Why? Entropy measures how much more information you need before you can identify the integer. Initially, there are 1000 possible values, which we assume are equally likely. What is the maximum number of question you need to ask?

Entropy Entropy (disorder, impurity) of a set of examples, S, relative to a binary classification is: where p1 is the fraction of positive examples in S and p0 is the fraction of negatives. If all examples are in one category, entropy is zero (we define 0log(0)=0) If examples are equally mixed (p1=p0=0.5), entropy is a maximum of 1. Entropy can be viewed as the number of bits required on average to encode the class of an example in S where data compression (e.g. Huffman coding) is used to give shorter codes to more likely cases. For multi-class problems with c categories, entropy generalizes to: R. Mooney, UT Austin

Entropy Plot for Binary Classification The entropy is 0 if the outcome is certain. The entropy is maximum if we have no knowledge of the system (or any outcome is equally possible). Entropy of a 2-class problem with regard to the portion of one of the two groups

Information Gain Is the expected reduction in entropy caused by partitioning the examples according to this attribute. is the number of bits saved when encoding the target value of an arbitrary member of S, by knowing the value of attribute A.

Information Gain in Decision Tree Induction Assume that using attribute A, a current set will be partitioned into some number of child sets The encoding information that would be gained by branching on A Note: entropy is at its minimum if the collection of objects is completely uniform

Examples for Computing Entropy NOTE: p( j | t) is computed as the relative frequency of class j at node t P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Entropy = – 0 log2 0 – 1 log2 1 = – 0 – 0 = 0 P(C1) = 1/6 P(C2) = 5/6 Entropy = – (1/6) log2 (1/6) – (5/6) log2 (5/6) = 0.65 P(C1) = 2/6 P(C2) = 4/6 Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92 P(C1) = 3/6=1/2 P(C2) = 3/6 = 1/2 Entropy = – (1/2) log2 (1/2) – (1/2) log2 (1/2) = -(1/2)(-1) – (1/2)(-1) = ½ + ½ = 1

How to Calculate log2x Many calculators only have a button for log10x and logex (note log typically means log10) You can calculate the log for any base b as follows: logb(x) = logk(x) / logk(b) Thus log2(x) = log10(x) / log10(2) Since log10(2) = .301, just calculate the log base 10 and divide by .301 to get log base 2. You can use this for HW if needed

Splitting Based on INFO... Information Gain: Parent Node, p is split into k partitions; ni is number of records in partition i Measures Reduction in Entropy achieved because of the split. Choose the split that achieves most reduction (maximizes GAIN) Used in ID3 and C4.5 Disadvantage: Tends to prefer splits that result in large number of partitions, each being small but pure.

Continuous Attribute? (more on it later) Each non-leaf node is a test, its edge partitioning the attribute into subsets (easy for discrete attribute). For continuous attribute Partition the continuous value of attribute A into a discrete set of intervals Create a new boolean attribute Ac , looking for a threshold c, How to choose c ?

Person Homer 0” 250 36 M Marge 10” 150 34 F Bart 2” 90 10 Lisa 6” 78 8 Hair Length Weight Age Class Homer 0” 250 36 M Marge 10” 150 34 F Bart 2” 90 10 Lisa 6” 78 8 Maggie 4” 20 1 Abe 1” 170 70 Selma 8” 160 41 Otto 180 38 Krusty 200 45 Comic 8” 290 38 ?

Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 yes no Hair Length <= 5? Let us try splitting on Hair length Entropy(1F,3M) = -(1/4)log2(1/4) - (3/4)log2(3/4) = 0.8113 Entropy(3F,2M) = -(3/5)log2(3/5) - (2/5)log2(2/5) = 0.9710 Gain(Hair Length <= 5) = 0.9911 – (4/9 * 0.8113 + 5/9 * 0.9710 ) = 0.0911

Gain(Weight <= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900 Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 yes no Weight <= 160? Let us try splitting on Weight Entropy(4F,1M) = -(4/5)log2(4/5) - (1/5)log2(1/5) = 0.7219 Entropy(0F,4M) = -(0/4)log2(0/4) - (4/4)log2(4/4) = 0 Gain(Weight <= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900

Gain(Age <= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183 Entropy(4F,5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 yes no age <= 40? Let us try splitting on Age Entropy(3F,3M) = -(3/6)log2(3/6) - (3/6)log2(3/6) = 1 Entropy(1F,2M) = -(1/3)log2(1/3) - (2/3)log2(2/3) = 0.9183 Gain(Age <= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183

This time we find that we can split on Hair length, and we are done! Of the 3 features we had, Weight was best. But while people who weigh over 160 are perfectly classified (as males), the under 160 people are not perfectly classified… So we simply recurse! yes no Weight <= 160? This time we find that we can split on Hair length, and we are done! yes no Hair Length <= 2?

Male Male Female Weight <= 160? We don’t need to keep the data around, just the test conditions. Weight <= 160? yes no How would these people be classified? Hair Length <= 2? Male yes no Male Female

Male Male Female Weight <= 160? Rules to Classify Males/Females It is trivial to convert Decision Trees to rules… Weight <= 160? yes no Hair Length <= 2? Male yes no Male Female Rules to Classify Males/Females If Weight greater than 160, classify as Male Elseif Hair Length less than or equal to 2, classify as Male Else classify as Female

Once we have learned the decision tree, we don’t even need a computer! This decision tree is attached to a medical machine, and is designed to help nurses make decisions about what type of doctor to call. Decision tree for a typical shared-care setting applying the system for the diagnosis of prostatic obstructions.

The worked examples we have seen were performed on small datasets The worked examples we have seen were performed on small datasets. However with small datasets there is a great danger of overfitting the data… When you have few datapoints, there are many possible splitting rules that perfectly classify the data, but will not generalize to future datasets. Yes No Wears green? Female Male For example, the rule “Wears green?” perfectly classifies the data, so does “Mothers name is Jacqueline?”, so does “Has blue shoes”…

How to Find the Best Split: GINI Before Splitting: M0 A? B? Yes No Yes No Node N1 Node N2 Node N3 Node N4 M1 M2 M3 M4 M12 M34 Gain = M0 – M12 vs M0 – M34

Measure of Impurity: GINI (at node t) Gini Index for a given node t with classes j NOTE: p( j | t) is computed as the relative frequency of class j at node t Example: Two classes C1 & C2 and node t has 5 C1 and 5 C2 examples. Compute Gini(t) 1 – [p(C1|t) + p(C2|t)] = 1 – [(5/10)2 + [(5/10)2 ] 1 – [¼ + ¼] = ½. Do you think this Gini value indicates a good split or bad split? Is it an extreme value?

More on Gini Worst Gini corresponds to probabilities of 1/nc, where nc is the number of classes. For 2-class problems the worst Gini will be ½ How do we get the best Gini? Come up with an example for node t with 10 examples for classes C1 and C2 10 C1 and 0 C2 Now what is the Gini? 1 – [(10/10)2 + (0/10)2 = 1 – [1 + 0] = 0 So 0 is the best Gini So for 2-class problems: Gini varies from 0 (best) to ½ (worst).

Some More Examples Below we see the Gini values for 4 nodes with different distributions. They are ordered from best to worst. See next slide for details Note that thus far we are only computing GINI for one node. We need to compute it for a split and then compute the change in Gini from the parent node.

Examples for computing GINI P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0 P(C1) = 1/6 P(C2) = 5/6 Gini = 1 – (1/6)2 – (5/6)2 = 0.278 P(C1) = 2/6 P(C2) = 4/6 Gini = 1 – (2/6)2 – (4/6)2 = 0.444

Splitting Criteria based on Classification Error Classification error at a node t : Measures misclassification error made by a node. Maximum (1 - 1/nc) when records are equally distributed among all classes, implying least interesting information Minimum (0.0) when all records belong to one class, implying most interesting information

Examples for Computing Error P(C1) = 0/6 = 0 P(C2) = 6/6 = 1 Error = 1 – max (0, 1) = 1 – 1 = 0 P(C1) = 1/6 P(C2) = 5/6 Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6 P(C1) = 2/6 P(C2) = 4/6 Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3

Comparison among Splitting Criteria For a 2-class problem:

Discussion Error rate is often the metric used to evaluate a classifier (but not always) So it seems reasonable to use error rate to determine the best split That is, why not just use a splitting metric that matches the ultimate evaluation metric? But this is wrong! The reason is related to the fact that decision trees use a greedy strategy, so we need to use a splitting metric that leads to globally better results The other metrics will empirically outperform error rate, although there is no proof for this.

DTs in practice... Growing to purity is bad (overfitting) x2: sepal width x1: petal length

DTs in practice... Growing to purity is bad (overfitting) x2: sepal width x1: petal length

DTs in practice... Growing to purity is bad (overfitting) Terminate growth early Grow to purity, then prune back

DTs in practice... Growing to purity is bad (overfitting) Not statistically supportable leaf Remove split & merge leaves x2: sepal width x1: petal length

(more on overfitting later) Avoid Overfitting in Classification (more on overfitting later) The generated tree may overfit the training data Too many branches, some may reflect anomalies due to noise or outliers Result is in poor accuracy for unseen samples Two approaches to avoid overfitting Prepruning: Halt tree construction early—do not split a node if this would result in the goodness measure falling below a threshold Difficult to choose an appropriate threshold Postpruning: Remove branches from a “fully grown” tree—get a sequence of progressively pruned trees Use a set of data different from the training data to decide which is the “best pruned tree”

Tree Induction Greedy strategy. Issues Split the records based on an attribute test that optimizes certain criterion. Issues Determine how to split the records How to specify the attribute test condition? How to determine the best split? Determine when to stop splitting

How to Specify Test Condition? Depends on attribute types Nominal Ordinal Continuous Depends on number of ways to split 2-way split Multi-way split

Splitting Based on Nominal Attributes Multi-way split: Use as many partitions as distinct values. Binary split: Divides values into two subsets. Need to find optimal partitioning. CarType Family Sports Luxury CarType {Sports, Luxury} {Family} CarType {Family, Luxury} {Sports} OR

Splitting Based on Ordinal Attributes Multi-way split: Use as many partitions as distinct values. Binary split: Divides values into two subsets. Need to find optimal partitioning. What about this split? Size Small Medium Large Size {Small, Medium} {Large} Size {Medium, Large} {Small} OR Size {Small, Large} {Medium}

Splitting Based on Continuous Attributes Different ways of handling Discretization to form an ordinal categorical attribute Static – discretize once at the beginning Dynamic – ranges can be found by equal interval bucketing, equal frequency bucketing (percentiles), or clustering. Binary Decision: (A < v) or (A  v) consider all possible splits and finds the best cut can be more compute intensive

Splitting Based on Continuous Attributes

Data Fragmentation Number of instances gets smaller as you traverse down the tree Number of instances at the leaf nodes could be too small to make any statistically significant decision

Search Strategy Finding an optimal decision tree is NP-hard The algorithm presented so far uses a greedy, top-down, recursive partitioning strategy to induce a reasonable solution

Expressiveness Decision tree provides expressive representation for learning discrete-valued function But they do not generalize well to certain types of Boolean functions Example: parity function: Class = 1 if there is an even number of Boolean attributes with truth value = True Class = 0 if there is an odd number of Boolean attributes with truth value = True For accurate modeling, must have a complete tree Not expressive enough for modeling continuous variables Particularly when test condition involves only a single attribute at-a-time

Decision Boundary Border line between two neighboring regions of different classes is known as decision boundary Decision boundary is parallel to axes because test condition involves a single attribute at-a-time

Oblique Decision Trees x + y < 1 Class = + Class = Test condition may involve multiple attributes More expressive representation Finding optimal test condition is computationally expensive

Vertical/Horizontal Boundaries 500 circular and 500 triangular data points. Circular points: 0.5  sqrt(x12+x22)  1 Triangular points: sqrt(x12+x22) > 0.5 or sqrt(x12+x22) < 1

Tree Replication Same subtree appears in multiple branches

Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates?

? Deep Bushy Tree Useless Which of the “Problems” can be solved by a Decision Tree? 10 1 2 3 4 5 6 7 8 9 Deep Bushy Tree Useless ? 10 1 2 3 4 5 6 7 8 9 100 10 20 30 40 50 60 70 80 90 The Decision Tree has a hard time with correlated attributes

Advantages/Disadvantages of Decision Trees Easy to understand (Doctors love them!) Easy to generate rules Disadvantages: May suffer from overfitting. Classifies by rectangular partitioning (so does not handle correlated features very well). Can be quite large – pruning is necessary. Does not handle streaming data easily