Artificial Intelligence Decision Trees
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1 Artificial Intelligence Decision Trees Andrea Torsello
2 Decision Trees Complex decisions can often be expressed in terms of a series of questions: What to do this Weekend? If my parents are visiting We ll go to the cinema If not Then, if it s sunny I ll play tennis But if it s windy and I m rich, I ll go shopping If it s windy and I m poor, I ll go to the cinema If it s rainy, I ll stay in
3 Decision Trees These decision can be expressed in terms of a tree Root of tree Leaves
4 Decision Trees Decision trees can be written as Horn clauses in first order logic Read from the root to every tip If this and this and this and this, then do this In our example: If no_parents and sunny_day, then play_tennis no_parents sunny_day play_tennis
5 Decision Tree Learning Decision tree can be seen as rules for performing a categorization E.g., what kind of weekend will this be? Remember that we re learning from examples Not turning thought processes into decision trees We need examples put into categories We also need attributes for the examples Attributes describe examples (background knowledge) Each attribute takes only a finite set of values
6 The ID3 Algorithm The major question in decision tree learning Which nodes to put in which positions Including the root node and the leaf nodes ID3 uses a measure called Information Gain Based on a notion of entropy Impurity in the data Used to choose which node to put in next Node with the highest information gain is chosen When there are no choices, a leaf node is put on
7 Entropy For examples in a binary categorization Where p+ is the proportion of positives Given a set of examples, S And p- is the proportion of negatives For examples in categorizations c 1 to c n Where pn is the proportion of examples in c n
8 Entropy Each category adds to the whole measure When pi is near to 1 (Nearly) all the examples are in this category So it should score low for its bit of the entropy log2(pi) gets closer and closer to 0 And this part dominates the overall calculation So the overall calculation comes to nearly 0 (which is good) When pi is near to 0 (Very) few examples are in this category So it should score low for its bit of the entropy log2(pi) gets larger (more negative), but does not dominate Hence overall calculation comes to nearly 0 (which is good)
9 Information Gain Given set of examples S and an attribute A A can take values v1 v m Let Sv = {examples which take value v for attribute A} Calculate Gain(S,A) Estimates the reduction in entropy we get if we know the value of attribute A for the examples in S
10 Example Calculation of Information Gain Suppose we have a set of examples S = {s1, s 2, s 3, s 4 } In a binary categorization With one positive example and three negative examples The positive example is s 1 And Attribute A Which takes values v1, v 2, v 3 S1 takes value v 2 for A S2 takes value v 2 for A S3 takes value v 3 for A S4 takes value v 1 for A
11 Recall that Entropy(S) = -p+ log 2 (p + ) p - log 2 (p - ) From binary categorisation, we know that p+ = ¼ and p - = ¾ Hence Entropy(S) = -(1/4)log2 (1/4) (3/4)log 2 (3/4) = Note that, by convention: 0*log 2 (0) is taken to be 0
12 Remember that And that S v = {set of example with value V for A} So, Sv1 = {s 4 }, S v2 = {s 1,s 2 }, S v3 ={s 3 } Now, ( S v1 / S ) * Entropy(S v1 ) = (1/4) * (-(0/1)*log 2 (0/1)-(1/1)log 2 (1/1)) = (1/4) * (0 - (1)log 2 (1)) = (1/4)(0-0) = 0 Similarly, ( S v2 / S ) = 0.5 and ( S v3 / S ) = 0 So, we have: Gain(S,A) = (0+1/2+0) = 0.311
13 The ID3 Algorithm Given a set of examples, S Described by a set of attributes Ai Categorised into categories cj 1. Choose the root node to be attribute A Such that A scores highest for information gain Relative to S, i.e., gain(s,a) is the highest over all attributes 2. For each value v that A can take Draw a branch and label each with corresponding v Then see the options in the next slide!
14 The ID3 Algorithm For each branch you ve just drawn (for value v) If Sv only contains examples in category c Then put that category as a leaf node in the tree If Sv is empty Then find the default category (which contains the most examples from S) Put this default category as a leaf node in the tree Otherwise Remove A from attributes which can be put into nodes Replace S with S v Find new attribute A scoring best for Gain(S, A) Start again at part 2 Make sure you replace S with S v
15
16 Example Weekend Weather Parents Money Decision (Category) W1 Sunny Yes Rich Cinema W2 Sunny No Rich Tennis W3 Windy Yes Rich Cinema W4 Rainy Yes Poor Cinema W5 Rainy No Rich Stay in W6 Rainy Yes Poor Cinema W7 Windy No Poor Cinema W8 Windy No Rich Shopping W9 Windy Yes Rich Cinema W10 Sunny No Rich Tennis
17 S = {W1,W2,,W10} Firstly, we need to calculate: Entropy(S) = = Next, we need to calculate information gain For all the attributes we currently have available (which is all of them at the moment) Gain(S, weather) = = 0.7 Gain(S, parents) = = 0.61 Gain(S, money) = = Hence, the weather is the first attribute to split on Because this gives us the biggest information gain
18 So, this is the top of our tree: Now, we look at each branch in turn In particular, we look at the examples with the attribute prescribed by the branch S sunny = {W1,W2,W10} Categorisations are cinema, tennis and tennis for W1,W2 and W10 What does the algorithm say? Set is neither empty, nor a single category So we have to replace S by S sunny and start again
19 Weekend Weather Parents Money Decision W1 Sunny Yes Rich Cinema W2 Sunny No Rich Tennis W10 Sunny No Rich Tennis Need to choose a new attribute to split on Cannot be weather, of course we ve already had that So, calculate information gain again: Gain(Ssunny, parents) = = Gain(Ssunny, money) = = 0 Hence we choose to split on parents
20 If it s sunny and the parents have turned up Then, looking at the table in previous slide There s only one answer: go to cinema If it s sunny and the parents haven t turned up Then, again, there s only one answer: play tennis Hence our decision tree looks like this
21 Avoiding Overfitting Decision trees can be learned to perfectly fit the data given This is probably overfitting The answer is a memorisation, rather than generalisation Avoidance method 1: Stop growing the tree before it reaches perfection Avoidance method 2: Grow to perfection, then prune it back aftwerwards Most useful of two methods in practice Decision tree algorithms are fairly robust to errors In the actual classifications In the attribute-value pairs In missing information
22 Decision Boundary All the attributes used for categorization form an n-dimensional space Decisions can be of the form x i >constant Divides the space into axis aligned rectangles
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