Decision Trees. Common applications: Health diagnosis systems Bank credit analysis
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1 Decision Trees Rodrigo Fernandes de Mello Invited Professor at Télécom ParisTech Associate Professor at Universidade de São Paulo, ICMC, Brazil
2 Decision Trees They build up trees to organize the attributes as internal nodes and answers as leaves The importance of each attribute is related to its tree height Intended to be used on discrete data If not, one needs to discretize it Common applications: Health diagnosis systems Bank credit analysis
3 Decision Trees For example, consider the problem of Play some Sport Let s classify if a day is good to play Outlook Sunny Cloudy Rainy Moisture Yes Wind High Normal Strong Weak No Yes No Yes For example: Having the query instance: Output: No <Outlook=Sunny, Temperature=Hot, Moisture=High>
4 Decision Trees After building up the tree, one can organize the rules employed in a next classification stage: Sunny Outlook Cloudy Rainy Moisture Yes Wind High Normal Strong Weak No Yes No Yes <Outlook=Sunny AND Moisture=Normal> OR <Outlook=Cloudy> OR <Outlook=Rainy AND Wind=Weak>
5 Decision Trees Most well-known algorithms: ID3 (Quinlan, 1986) C4.5 (Quinlan, 1993) J48 The ID3 algorithm It builds up a tree based on a top-down approach It attempts to position the most discriminative attribute as close as possible to the tree root Every attribute must be tested in order to define their relevances For each attribute, branches are created according to its possible values
6 Decision Trees ID3 employs the Information Gain approach to decide on the most important attributes That depends on the Shannon s Entropy
7 Shannon s Entropy Entropy: A Thermodynamic property to determine the amount of useful energy in a given system Gibbs stated that the best interpretation for the Entropy in the Statistical Mechanics is as an Uncertainty Measure A little about the History: Entropy starts with Lazare Carnot (1803) Rudolf Clausius (1850s-1860s) brings new interpretations in physics Claude Shannon (1948) designs the concept of Entropy in the context of Information Theory
8 Shannon s Entropy Let the system start as follows: 1 And change to:
9 Shannon s Entropy This is the Equation proposed by Shannon: It measures the total energy of a system: It considers the system is at a given state i and transitions to j The function log 2 supports to find the number of bits of Information
10 Shannon s Entropy Thus: After modifying the behavior, the system agregated a greater level of uncertainty or energy (Ex: Win the Lottery) 1
11 Let a collection of instances S, including positive and negative examples i.e., two distinct and discrete classes/labels ID3 using the Entropy Let the probability of pertaining to each class of S The Entropy is given by:
12 ID3 using the Entropy For illustration purposes, consider S has 14 examples: 9 positives 5 negatives Then, the Entropy of such a set is: Observe a binary-class scenario has maximum Entropy equals to 1
13 ID3 using the Entropy Other scenarios: Having [7+, 7-] Having [0+, 14-] or [14+, 0-] Entropy measures the level of uncertainty about some event/ source
14 We may generalize Entropy to more classes: ID3 using the Entropy Several other systems can be studied using Entropy: For instance: Time Series, after some quantization process Why the Log function is used? It allows to measure information in bits A great example is about codifying some message Codify: TORONTO
15 After defining Entropy, we can define Information Gain It measures how effective an attribute is to classify some dataset A different point of view: ID3 uses Information Gain It measures the Entropy reduction when partitioning the dataset using a given attribute The second term measures the Entropy after partitioning the dataset with attribute A Therefore: IG measures the Entropy reduction, i.e. the uncertainty decrease, after selecting attribute A to compose the tree
16 To illustrate, take S and the attribute Wind (Weak or Strong) S contains 14 instances [9+, 5-] Now consider that: ID3 uses Information Gain 6 of the positive examples and 2 of the negative ones have Wind=Weak (8 in total) There are 3 instances with Wind=Strong for both the positive and the negative classes (6 in total) Then, the Information Gain while selecting the attribute Wind to take the root of our decision tree is given by: weak strong
17 So that: ID3 uses Information Gain weak strong weak weak strong weak
18 ID3 uses Information Gain Therefore: weak strong weak weak strong weak This is the Information Gain measure used by ID3 along the iterations to build up a decision tree Observe this scenario has only reduced the uncertainty a little Thus, is this attribute good enough to take the root? You will see it is not!
19 Running ID3 Consider the concept of play some sport Outlook Temperature Moisture Wind Play Tennis Sunny Warm High Weak No Sunny Warm High Strong No Cloudy Warm High Weak Yes Rainy Pleasant High Weak Yes Rainy Cold Normal Weak Yes Rainy Cold Normal Strong No Cloudy Cold Normal Strong Yes Sunny Sunny Pleasant Cold High Normal Weak Weak No Yes Rainy Pleasant Normal Weak Yes Sunny Pleasant Normal Strong Yes Cloudy Pleasant High Strong Yes Cloudy Warm Normal Weak Yes Rainy Pleasant High Strong No
20 Running ID3 First step: Compute the Information Gain for each of its attributes: Outlook weak Moisture Wind Temperature The attribute with greater IG is selected to be the tree root This is the one that reduces the most the uncertainty! The children nodes are created according to the possible values for the root attribute
21 Running ID3 Having the tree root: We must proceed in the same way to each of its branches We only consider the examples filtered by each branch If there is divergence S = {D1,..., D14} Outlook Sunny S = {D1, D2, D8, D9, D11} Cloudy S = {D3, D7, D12, D13} Rainy S = {D4, D5, D6, D10, D14}???
22 Running ID3 Observe there is no divergence for one of the branches, so its Entropy is zero and a leaf label is defined: S = {D1,..., D14} Outlook Sunny S = {D1, D2, D8, D9, D11} Cloudy S = {D3, D7, D12, D13} Rainy S = {D4, D5, D6, D10, D14}? Yes? The attributes already incorporated throughout the path to reach a node are not considered in the Information Gain Observe the Outlook will never be assessed again for both branches
23 Running ID3 Carrying on... S = {D1,..., D14} [9+, 5-] Outlook Sunny S = {D1, D2, D8, D9, D11} [2+, 3-] Cloudy S = {D3, D7, D12, D13} [4+, 0-] Rainy? Yes? S = {D4, D5, D6, D10, D14} [3+, 2-] Computing the Information Gain to the Sunny branch: Entropy is E(S = Sunny) Sunny weak
24 Computing the Information Gain to the Sunny branch: Running ID3 Moisture Temperature Wind Observe the Moisture is the best choice!
25 Carrying on... Running ID3 S = {D1,..., D14} [9+, 5-] Outlook Sunny S = {D1, D2, D8, D9, D11} [2+, 3-] Cloudy S = {D3, D7, D12, D13} [4+, 0-] Rainy Moisture Yes? S = {D4, D5, D6, D10, D14} [3+, 2-] High S = {D1, D2, D8} [0+, 3-] No Normal S = {D9, D11} [2+, 0-] Yes
26 And then... Running ID3 S = {D1,..., D14} [9+, 5-] Outlook Sunny S = {D1, D2, D8, D9, D11} [2+, 3-] Umidade Cloudy S = {D3, D7, D12, D13} [4+, 0-] Yes Rainy S = {D4, D5, D6, D10, D14} [3+, 2-] Wind High S = {D1, D2, D8} [0+, 3-] Normal S = {D9, D11} [2+, 0-] Strong S = {D6, D14} [0+, 2-] Weak S = {D4, D5, D10} [3+, 0-] No Yes No Yes
27 Running ID3 ID3 carries on until one of the following conditions is satisfied: (1) Either all attributes were included in the path from the root to the leaves (2) Or the training examples at a given branch have a single class/label Observe ID3 starts with a null tree Then it builds the tree up from the scratch And It builds up a single tree
28 Discussion C4.5 is an extension of ID3 It proceeds in the same way but it has a backtracking process afterwards in attempt to reduce the number of tree nodes It is a copyrighted solution J48 is based on the C4.5 documentation and is open The Information Gain is a way of reducing the tree height But why? William of Occam, 1320
29 Suggestion Implement ID3 Discrete version Continuous version Test it with the UCI datasets
30 Complementary references Tom Mitchell, Machine Learning, 1994 Hudson (2006) Signal Processing Using Mutual Information, IEEE Signal Processing Magazine Leo Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and Regression Trees. Statistics/Probability Series. Wadsworth Publishing Company, Belmont, California, U.S.A., 1984.
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