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CS B551: D ECISION T REES

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A GENDA Decision trees Complexity Learning curves Combatting overfitting Boosting

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R ECAP Still in supervised setting with logical attributes Find a representation of CONCEPT in the form: CONCEPT(x) S(A,B, …) where S(A,B,…) is a sentence built with the observable attributes, e.g.: CONCEPT(x) A(x) ( B(x) v C(x))

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P REDICATE AS A D ECISION T REE The predicate CONCEPT(x) A(x) ( B(x) v C(x)) can be represented by the following decision tree: A? B? C? True FalseTrue False Example: A mushroom is poisonous iff it is yellow and small, or yellow, big and spotted x is a mushroom CONCEPT = POISONOUS A = YELLOW B = BIG C = SPOTTED

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P REDICATE AS A D ECISION T REE The predicate CONCEPT(x) A(x) ( B(x) v C(x)) can be represented by the following decision tree: A? B? C? True FalseTrue False Example: A mushroom is poisonous iff it is yellow and small, or yellow, big and spotted x is a mushroom CONCEPT = POISONOUS A = YELLOW B = BIG C = SPOTTED D = FUNNEL-CAP E = BULKY

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T RAINING S ET Ex. #ABCDECONCEPT 1False TrueFalseTrueFalse 2 TrueFalse 3 True False 4 TrueFalse 5 True False 6TrueFalseTrueFalse True 7 False TrueFalseTrue 8 FalseTrueFalseTrue 9 FalseTrue 10True 11True False 12True False TrueFalse 13TrueFalseTrue

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P OSSIBLE D ECISION T REE D CE B E AA A T F F FF F T T T TT

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D CE B E AA A T F F FF F T T T TT CONCEPT (D ( E v A)) v ( D (C (B v ( B ((E A) v ( E A)))))) A? B? C? True FalseTrue False CONCEPT A ( B v C)

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P OSSIBLE D ECISION T REE D CE B E AA A T F F FF F T T T TT A? B? C? True FalseTrue False CONCEPT A ( B v C) KIS bias Build smallest decision tree Computationally intractable problem greedy algorithm CONCEPT (D ( E v A)) v ( D (C (B v ( B ((E A) v ( E A))))))

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T OP -D OWN I NDUCTION OF A DT DTL( , Predicates) 1. If all examples in are positive then return True 2. If all examples in are negative then return False 3. If Predicates is empty then return majority rule 4. A error-minimizing predicate in Predicates 5. Return the tree whose: - root is A, - left branch is DTL( +A,Predicates-A), - right branch is DTL( -A,Predicates-A) A C True B False

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L EARNABLE C ONCEPTS Some simple concepts cannot be represented compactly in DTs Parity(x) = X 1 xor X 2 xor … xor X n Majority(x) = 1 if most of X i ’s are 1, 0 otherwise Exponential size in # of attributes Need exponential # of examples to learn exactly The ease of learning is dependent on shrewdly (or luckily) chosen attributes that correlate with CONCEPT

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve size of training set % correct on test set 100 Typical learning curve

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve size of training set % correct on test set 100 Typical learning curve Some concepts are unrealizable within a machine’s capacity

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve Overfitting Risk of using irrelevant observable predicates to generate an hypothesis that agrees with all examples in the training set size of training set % correct on test set 100 Typical learning curve

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve Overfitting Tree pruning Risk of using irrelevant observable predicates to generate an hypothesis that agrees with all examples in the training set Terminate recursion when # errors / information gain is small

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve Overfitting Tree pruning Terminate recursion when # errors / information gain is small Risk of using irrelevant observable predicates to generate an hypothesis that agrees with all examples in the training set The resulting decision tree + majority rule may not classify correctly all examples in the training set

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P ERFORMANCE I SSUES Assessing performance: Training set and test set Learning curve Overfitting Tree pruning Incorrect examples Missing data Multi-valued and continuous attributes

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U SING I NFORMATION T HEORY Rather than minimizing the probability of error, minimize the expected number of questions needed to decide if an object x satisfies CONCEPT Use the information-theoretic quantity known as information gain Split on variable with highest information gain

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E NTROPY / I NFORMATION GAIN Entropy: encodes the quantity of uncertainty in a random variable H(X) = - x Val(X) P(x) log P(x) Properties H(X) = 0 if X is known, i.e. P(x)=1 for some value x H(X) > 0 if X is not known with certainty H(X) is maximal if P(X) is uniform distribution Information gain: measures the reduction in uncertainty in X given knowledge of Y I(X,Y) = E y [H(X) – H(X|Y)] = y P(y) x [P(x|y) log P(x|y) – P(x)log P(x)] Properties Always nonnegative = 0 if X and Y are independent If Y is a choice, maximizing IG = > minimizing E y [H(X|Y)]

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M AXIMIZING IG / MINIMIZING CONDITIONAL ENTROPY IN DECISION TREES E y [H(X|Y)] = y P(y) x P(x|y) log P(x|y) Let n be # of examples Let n +,n - be # of examples on T/F branches of Y Let p +,p - be accuracy on true/false branches of Y P(Y) = (p + n + +p - n - )/n P(correct|Y) = p +, P(correct|-Y) = p - E y [H(X|Y)] n + [p + log p + + (1-p + )log (1-p + )] + n - [p - log p - + (1-p - ) log (1-p - )]

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C ONTINUOUS A TTRIBUTES Continuous attributes can be converted into logical ones via thresholds X => X<a When considering splitting on X, pick the threshold a to minimize # of errors / entropy 7765654543454567

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M ULTI -V ALUED A TTRIBUTES Simple change: consider splits on all values A can take on Caveat: the more values A can take on, the more important it may appear to be, even if it is irrelevant More values => dataset split into smaller example sets when picking attributes Smaller example sets => more likely to fit well to spurious noise

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S TATISTICAL M ETHODS FOR A DDRESSING O VERFITTING / N OISE There may be few training examples that match the path leading to a deep node in the decision tree More susceptible to choosing irrelevant/incorrect attributes when sample is small Idea: Make a statistical estimate of predictive power (which increases with larger samples) Prune branches with low predictive power Chi-squared pruning

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T OP - DOWN DT PRUNING Consider an inner node X that by itself (majority rule) predicts p examples correctly and n examples incorrectly At k leaf nodes, number of correct/incorrect examples are p 1 /n 1,…,p k /n k Chi-squared statistical significance test: Null hypothesis: example labels randomly chosen with distribution p/(p+n) (X is irrelevant) Alternate hypothesis: examples not randomly chosen (X is relevant) Prune X if testing X is not statistically significant

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C HI -S QUARED TEST Let Z = i (p i – p i ’) 2 /p i ’ + (n i – n i ’) 2 /n i ’ Where p i ’ = p i (p i +n i )/(p+n), n i ’ = n i (p i +n i )/(p+n) are the expected number of true/false examples at leaf node i if the null hypothesis holds Z is a statistic that is approximately drawn from the chi-squared distribution with k degrees of freedom Look up p-Value of Z from a table, prune if p- Value > for some (usually ~.05)

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E NSEMBLE L EARNING (B OOSTING )

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I DEA It may be difficult to search for a single hypothesis that explains the data Construct multiple hypotheses (ensemble), and combine their predictions “Can a set of weak learners construct a single strong learner?” – Michael Kearns, 1988

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M OTIVATION 5 classifiers with 60% accuracy On a new example, run them all, and pick the prediction using majority voting If errors are independent, new classifier has 94% accuracy! (In reality errors will not be independent, but we hope they will be mostly uncorrelated)

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B OOSTING Main idea: If learner 1 fails to learn an example correctly, this example is more important for learner 2 If learner 1 and 2 fail to learn an example correctly, this example is more important for learner 3 … Weighted training set Weights encode importance

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B OOSTING Weighted training set Ex. #WeightABCDECONCEPT 1w1w1 False TrueFalseTrueFalse 2w2w2 TrueFalse 3w3w3 True False 4w4w4 TrueFalse 5w5w5 True False 6w6w6 TrueFalseTrueFalse True 7w7w7 False TrueFalseTrue 8w8w8 FalseTrueFalseTrue 9w9w9 FalseTrue 10w 10 True 11w 11 True False 12w 12 True False TrueFalse 13w 13 TrueFalseTrue

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B OOSTING Start with uniform weights w i =1/N Use learner 1 to generate hypothesis h 1 Adjust weights to give higher importance to misclassified examples Use learner 2 to generate hypothesis h 2 … Weight hypotheses according to performance, and return weighted majority

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M USHROOM E XAMPLE “Decision stumps” - single attribute DT Ex. #WeightABCDECONCEPT 11/13False TrueFalseTrueFalse 21/13FalseTrueFalse 31/13FalseTrue False 41/13False TrueFalse 51/13False True False 61/13TrueFalseTrueFalse True 71/13TrueFalse TrueFalseTrue 81/13TrueFalseTrueFalseTrue 91/13True FalseTrue 101/13True 111/13True False 121/13True False TrueFalse 131/13TrueFalseTrue

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M USHROOM E XAMPLE Pick C first, learns CONCEPT = C Ex. #WeightABCDECONCEPT 11/13False TrueFalseTrueFalse 21/13FalseTrueFalse 31/13FalseTrue False 41/13False TrueFalse 51/13False True False 61/13TrueFalseTrueFalse True 71/13TrueFalse TrueFalseTrue 81/13TrueFalseTrueFalseTrue 91/13True FalseTrue 101/13True 111/13True False 121/13True False TrueFalse 131/13TrueFalseTrue

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M USHROOM E XAMPLE Pick C first, learns CONCEPT = C Ex. #WeightABCDECONCEPT 11/13False TrueFalseTrueFalse 21/13FalseTrueFalse 31/13FalseTrue False 41/13False TrueFalse 51/13False True False 61/13TrueFalseTrueFalse True 71/13TrueFalse TrueFalseTrue 81/13TrueFalseTrueFalseTrue 91/13True FalseTrue 101/13True 111/13True False 121/13True False TrueFalse 131/13TrueFalseTrue

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M USHROOM E XAMPLE Update weights (precise formula given in R&N) Ex. #WeightABCDECONCEPT 1.125False TrueFalseTrueFalse 2.056FalseTrueFalse 3.125FalseTrue False 4.125False TrueFalse 5.056False True False 6.056TrueFalseTrueFalse True 7.125TrueFalse TrueFalseTrue 8.056TrueFalseTrueFalseTrue 9.056True FalseTrue 10.056True 11.056True False 12.056True False TrueFalse 13.056TrueFalseTrue

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M USHROOM E XAMPLE Next try A, learn CONCEPT=A Ex. #WeightABCDECONCEPT 1.125False TrueFalseTrueFalse 2.056FalseTrueFalse 3.125FalseTrue False 4.125False TrueFalse 5.056False True False 6.056TrueFalseTrueFalse True 7.125TrueFalse TrueFalseTrue 8.056TrueFalseTrueFalseTrue 9.056True FalseTrue 10.056True 11.056True False 12.056True False TrueFalse 13.056TrueFalseTrue

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M USHROOM E XAMPLE Next try A, learn CONCEPT=A Ex. #WeightABCDECONCEPT 1.125False TrueFalseTrueFalse 2.056FalseTrueFalse 3.125FalseTrue False 4.125False TrueFalse 5.056False True False 6.056TrueFalseTrueFalse True 7.125TrueFalse TrueFalseTrue 8.056TrueFalseTrueFalseTrue 9.056True FalseTrue 10.056True 11.056True False 12.056True False TrueFalse 13.056TrueFalseTrue

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M USHROOM E XAMPLE Update weights Ex. #WeightABCDECONCEPT 10.07False TrueFalseTrueFalse 20.03FalseTrueFalse 30.07FalseTrue False 40.07False TrueFalse 50.03False True False 60.03TrueFalseTrueFalse True 70.07TrueFalse TrueFalseTrue 80.03TrueFalseTrueFalseTrue 90.03True FalseTrue 100.03True 110.25True False 120.25True False TrueFalse 130.03TrueFalseTrue

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M USHROOM E XAMPLE Next try E, learn CONCEPT=E Ex. #WeightABCDECONCEPT 10.07False TrueFalseTrueFalse 20.03FalseTrueFalse 30.07FalseTrue False 40.07False TrueFalse 50.03False True False 60.03TrueFalseTrueFalse True 70.07TrueFalse TrueFalseTrue 80.03TrueFalseTrueFalseTrue 90.03True FalseTrue 100.03True 110.25True False 120.25True False TrueFalse 130.03TrueFalseTrue

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M USHROOM E XAMPLE Next try E, learn CONCEPT= E Ex. #WeightABCDECONCEPT 10.07False TrueFalseTrueFalse 20.03FalseTrueFalse 30.07FalseTrue False 40.07False TrueFalse 50.03False True False 60.03TrueFalseTrueFalse True 70.07TrueFalse TrueFalseTrue 80.03TrueFalseTrueFalseTrue 90.03True FalseTrue 100.03True 110.25True False 120.25True False TrueFalse 130.03TrueFalseTrue

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M USHROOM E XAMPLE Update Weights… Ex. #WeightABCDECONCEPT 10.07False TrueFalseTrueFalse 20.03FalseTrueFalse 30.07FalseTrue False 40.07False TrueFalse 50.03False True False 60.03TrueFalseTrueFalse True 70.07TrueFalse TrueFalseTrue 80.03TrueFalseTrueFalseTrue 90.03True FalseTrue 100.03True 110.25True False 120.25True False TrueFalse 130.03TrueFalseTrue

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M USHROOM E XAMPLE Final classifier, order C,A,E,D,B Weights on hypotheses determined by overall error Weighted majority weights A=2.1, B=0.9, C=0.8, D=1.4, E=0.09 100% accuracy on training set

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B OOSTING S TRATEGIES Prior weighting strategy was the popular AdaBoost algorithm see R&N pp. 667 Many other strategies Typically as the number of hypotheses increases, accuracy increases as well Does this conflict with Occam’s razor?

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A NNOUNCEMENTS Next class: Neural networks & function learning R&N 18.6-7

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