# Lecture 7: Decision Trees

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1 Lecture 7: Decision Trees Instructor:

2 Outline 1 Geometric Perspective of Classification 2 Decision Trees

3 Geometric Perspective of Classification

4 Perspective of Classification Algorithmic Geometric Probabilistic...

5 Geometric Perspective of Classification Gives some intuition for model selection Understand the distribution of data Understand the expressiveness and limitations of various classifiers

6 Feature Space 1 Feature Vector: d-dimensional vector of features describing the object Feature Space: The vector space associated with feature vectors 1 DMA Book

7 Feature Space in Classification

8 Geometric Perspective of Classification Decision Region: A partition of feature space such that all feature vectors in it are assigned to same class. Decision Boundary: Boundaries between neighboring decision regions

9 Geometric Perspective of Classification Objective of a classifier is to approximate the real decision boundary as much as possible Most classification algorithm has specific expressiveness and limitations If they align, then classifier does a good approximation

10 Linear Decision Boundary

11 Piecewise Linear Decision Boundary 2 2 ISLR Book

12 Quadratic Decision Boundary 3 3 Figshare.com

13 Non-linear Decision Boundary 4 4 ISLR Book

14 Complex Decision Boundary 5 5 ISLR Book

15 Classifier Selection Tips If decision boundary is linear, most linear classifiers will do well If decision boundary is non-linear, we sometimes have to use kernels If decision boundary is piece-wise, decision trees can do well If decision boundary is too complex, k-nn might be a good choice

16 k-nn Decision Boundary 6 Asymptotically Consistent: With infinite training data and large enough k, k-nn approaches the best possible classifier (Bayes Optimal) With infinite training data and large enough k, k-nn could approximate most possible decision boundaries 6 ISLR Book

17 Decision Trees

18 Strategies for Classifiers Parametric Models: Makes some assumption about data distribution such as density and often use explicit probability models Non-parametric Models: No prior assumption of data and determine decision boundaries directly. k-nn Decision tree

19 Tree 7 7 http:

20 Binary Decision Tree 8 8 http:

21 20 Question Intuition 9 9

22 Decision Tree for Selfie Stick The Oatmeal Comics

23 Decision Trees and Rules

25 Building Decision Trees Intuition 13 Horsepower Weight Mileage 95 low low 90 low low 70 low high 86 low high 76 high low 88 high low Table: Car Mileage Prediction from Scalable-Distributed-Decision-Trees-in-Spark-Made-Das-Sparks-Talwalkar. pdf

26 Building Decision Trees Intuition Horsepower Weight Mileage 95 low low 90 low low 70 low high 86 low high 76 high low 88 high low Table: Car Mileage Prediction from 1971

27 Building Decision Trees Intuition

28 Building Decision Trees Intuition Horsepower Weight Mileage 95 low low 90 low low 70 low high 86 low high Table: Car Mileage Prediction from 1971

29 Building Decision Trees Intuition

30 Building Decision Trees Intuition

31 Building Decision Trees Intuition Prediction:

32 Building Decision Trees Intuition Prediction:

33 Learning Decision Trees

34 Decision Trees Defined by a hierarchy of rules (in form of a tree) Rules form the internal nodes of the tree (topmost internal node = root) Each rule (internal node) tests the value of some property the data Leaf nodes make the prediction

35 Decision Tree Learning Objective: Use the training data to construct a good decision tree Use the constructed Decision tree to predict labels for test inputs

36 Decision Tree Learning Identifying the region (blue or green) a point lies in A classification problem (blue vs green) Each input has 2 features: co-ordinates {x 1, x 2 } in the 2D plane Once learned, the decision tree can be used to predict the region (blue/green) of a new test point

37 Decision Tree Learning

38 Expressiveness of Decision Trees

39 Expressiveness of Decision Trees Decision tree divides feature space into axis-parallel rectangles Each rectangle is labelled with one of the C classes Any partition of feature space by recursive binary splitting can be simulated by Decision Trees

40 Expressiveness of Decision Trees

41 Expressiveness of Decision Trees Feature space on left can be simulated by Decision tree but not the one on right.

42 Expressiveness of Decision Tree Can express any logical function on input attributes Can express any boolean function For boolean functions, path to leaf gives truth table row Could require exponentially many nodes cyl = 3 (cyl = 4 (maker = asia maker = europe))...

43 Hypothesis Space Exponential search space wrt set of attributes If there are d boolean attributes, then the search space has 2 2d trees If d = 6, then it is approximately 18, 446, 744, 073, 709, 551, 616 (or approximately ) If there are d boolean attributes, each truth table has 2 d rows Hence there must be 2 2d truth tables that can take all possible variations Alternate argument: the number of trees is same as number of bolean functions with d variables = number of distinct truth tables with 2 d rows = 2 2d NP-Complete to find optimal decision tree Idea: Use greedy approach to find a locally optimal tree

44 Decision Tree Learning Algorithms 1966: Hunt and colleagues from Psychology developed first known algorithm for human concept learning 1977: Breiman, Friedman and others from Statistics developed CART 1979: Quinlan developed proto-id3 1986: Quinlan published ID3 paper 1993: Quinlan s updated algorithm C s and 90 s: Improvements for handling noise, continuous attributes, missing data, non-axis parallel DTs, better heuristics for pruning, overfitting, combining DTs

45 Decision Tree Learning Algorithms Main Loop: 1 Let A be the best decision attribute for next node 2 Assign A as decision attribute for node 3 For each value of A, create a new descendent of node 4 Sort training examples to leaf nodes 5 If training examples are perfectly classified, then STOP else iterate over leaf nodes

46 Recursive Algorithm for Learning Decision Trees

47 Decision Tree Learning Greedy Approach: Build tree, top-down by choosing one attribute at a time Choices are locally optimal and may or may not be globally optimal Major issues Selecting the next attribute Given an attribute, how to specify the split condition Determining termination condition

48 Termination Condition Stop expanding a node further when:

49 Termination Condition Stop expanding a node further when: It consist of examples all having the same label Or we run out of features to test!

50 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

51 Splitting based on Nominal Attributes

52 Splitting based on Ordinal Attributes

53 Splitting based on Continuous Attributes How to split continuous attributes such as Age, Income etc

54 Splitting based on Continuous Attributes How to split continuous attributes such as Age, Income etc Discretization to form an ordinal categorical attribute Static: discretize once at the beginning Dynamic: find ranges by equal interval bucketing, equal frequency bucketing, percentiles, clustering etc Binary Decision: (A < v)or(a v) Consider all possible split and find the best cut Often, computationally intensive

55 Splitting based on Continuous Attributes

56 Choosing the next Attribute - I

57 Choosing the next Attribute - II InformationGain.pdf

58 Choosing the next Attribute - III

59 Choosing an Attribute Good Attribute

60 Choosing an Attribute Good Attribute for one value we get all instances as positive for other value we get all instances as negative Bad Attribute

61 Choosing an Attribute Good Attribute for one value we get all instances as positive for other value we get all instances as negative Bad Attribute it provides no discrimination attribute is immaterial to the decision for each value we have same number of positive and negative instances

62 How to Find the Best Split?

63 Measures of Node Impurity Gini Index Entropy Misclassification Error

64 Gini Index An important measure of statistical dispersion Used in Economics to measure income inequality in countries Proposed by Corrado Gini

65 Gini Index

66 Gini Index

67 Splitting Based on Gini Used in CART, SLIQ, SPRINT When a node p is split into k partitions (children), the quality of split is computed as, Gini split = k i=1 n i n Gini(i) n i = number of records at child i n = number of records at node p

68 Gini Index for Binary Attributes

69 Gini Index for Categorical Attributes

70 Entropy and Information Gain You are watching a set of independent random samples of a random variable X Suppose the probabilities are equal: P(X = A) = P(X = B) = P(X = C) = P(X = D) = 1 4 Suppose you see a text like BAAC You want to transmit this information in a binary communication channel How many bits will you need to transmit this information?

71 Entropy and Information Gain You are watching a set of independent random samples of a random variable X Suppose the probabilities are equal: P(X = A) = P(X = B) = P(X = C) = P(X = D) = 1 4 Suppose you see a text like BAAC You want to transmit this information in a binary communication channel How many bits will you need to transmit this information? Simple idea: Represent each character via 2 bits: A = 00, B = 01, C = 10, D = 11 So, BAAC becomes Communication Complexity: 2 on average bits per symbol

72 Entropy and Information Gain Suppose you knew probabilities are unequal: P(X = A) = 1 2, P(X = B) = 1 4, P(X = C) = P(X = D) = 1 8 It is now possible to send information 1.75 bits on average per symbol

73 Entropy and Information Gain Suppose you knew probabilities are unequal: P(X = A) = 1 2, P(X = B) = 1 4, P(X = C) = P(X = D) = 1 8 It is now possible to send information 1.75 bits on average per symbol Choose a frequency based code! A = 0, B = 10, C = 110, D = 111

74 Entropy and Information Gain Suppose you knew probabilities are unequal: P(X = A) = 1 2, P(X = B) = 1 4, P(X = C) = P(X = D) = 1 8 It is now possible to send information 1.75 bits on average per symbol Choose a frequency based code! A = 0, B = 10, C = 110, D = 111 BAAC becomes Requires only 7 bits for transmitting BAAC

75 Entropy

76 Entropy

77 Entropy

78 Entropy

79 Splitting based on Classification Error Classification error at node t is Error(t) = 1 max i P(i t) Measures misclassification error made by a node. Minimum (0.0) when all records belong to one class, implying most interesting information Maximum (1 1 n c ) when records are equally distributed among all classes, implying least interesting information

80 Classification Error

81 Comparison among Splitting Criteria

82 Splitting Criteria Gini Index (CART, SLIQ, SPRINT): select attribute that minimize impurity of a split Information Gain (ID3, C4.5) select attribute with largest information gain Normalized Gain ratio (C4.5) normalize different domains of attributes Distance normalized measures (Lopez de Mantaras) define a distance metric between partitions of the data chose the one closest to the perfect partition χ 2 contingency table statistics (CHAID) measures correlation between each attribute and the class label select attribute with maximal correlation

83 Overfitting in Decision Trees Decision trees will always overfit in the absence of label noise Simple strategies for fixing: Fixed depth Fixed number of leaves Grow the tree till the gain is above some threshold Post pruning

84 Trees vs Linear Models

85 Advantages and Disadvantages Very easy to explain to people. Some people believe that decision trees more closely mirror human decision-making Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small) Trees can easily handle qualitative predictors without the need to create dummy variables. Inexpensive to construct Extremely fast at classifying new data Unfortunately, trees generally do not have the same level of predictive accuracy as other classifiers

86 Summary Major Concepts: Geometric interpretation of Classification Decision trees

87 Slide Material References Slides from ISLR book Slides by Piyush Rai Slides for Chapter 4 from Introduction to Data Mining book by Tan, Steinbach, Kumar Slides from Andrew Moore, CMU See also the footnotes

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