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- 1. ID3 Algorithm & ROC Analysis Talha KABAKUŞtalha.kabakus@ibu.edu.tr
- 2. Agenda● Where are we now?● Decision Trees● What is ID3?● Entropy● Information Gain● Pros and Cons of ID3● An Example - The Simpsons● What is ROC Analysis?● ROC Space● ROC Space Example over predictions
- 3. Where are we now?
- 4. Decision Trees● One of the most used classification approach because of its clear model and presentation● Classification by using data attributes● Aim is to reaching estimating destination field value using source fields● Tree Induction ○ Create tree ○ Apply data into tree to classify● Each branch node represents a choice between a number of alternatives● Each leaf node represents a classification or decision● Leaf Count = Rule Count
- 5. Decision Trees (Cont.)● Leafs are inserted through top to bottom A B C D E F G
- 6. Sample Decision Tree
- 7. Creating Tree Model by Training Data
- 8. Decision Tree Classification Task
- 9. Apply Model to Test Data
- 10. Apply Model to Test Data (Cont.)
- 11. Apply Model to Test Data (Cont.)
- 12. Apply Model to Test Data (Cont.)
- 13. Apply Model to Test Data (Cont.)
- 14. Apply Model to Test Data (Cont.)
- 15. Decision Tree Algorithms● Classification and Regression Algorithms ○ Twoig ○ Gini● Entropy-based Algorithms ○ ID3 ○ C4.5● Memory-based (Sample-based) Classification Algorithms
- 16. Decision Trees by Variable Type● Single Variable Decision Trees ○ Classifications are done with asking questions over only one variable● Hybrid Decision Trees ○ Classifications are done with asking questions over both single and multiple variables● Multiple Variables Decision Trees ○ Classifications are done with asking questions over multiple variables
- 17. ID3 Algorithm● Iterative Dichotomizer 3● Developed by J. Ross Quinlan in 1979● Based on Entropy● Only works for discrete data● Can not work with defective data● Advantage over Hunts algorithm is choosing the right attribute while classification. (Hunts algorithm chooses randomly)
- 18. Entropy● A formula to calculate the homogeneity of a sample; gives idea about how much information gain provides each leaf● A complete homogeneous sample entropy value is 0● An equally divided sample entropy value is 1● Formula:
- 19. Information Gain (IG)● Information Gain calculates effective change in entropy after making a decision based on the value of an attribute.● Which attribute creates the most homogeneous branches?● First the entropy of the total dataset is calculated.● The dataset is then split on the different attributes.
- 20. Information Gain (Cont.)● The entropy for each branch is calculated. Then it is added proportionally, to get total entropy for the split.● The resulting entropy is subtracted from the entropy before the split.● The result is the Information Gain, or decrease in entropy.● The attribute that yields the largest IG is chosen for the decision node.
- 21. Information Gain (Cont.)● A branch set with entropy of 0 is a leaf node.● Otherwise, the branch needs further splitting to classify its dataset.● The ID3 algorithm is run recursively on the non-leaf branches, until all data is classified.
- 22. ID3 Algorithm Stepsfunction ID3 (R: a set of non-categorical attributes, C: the categorical attribute, S: a training set) returns a decision tree; begin If S is empty, return a single node with value Failure; If S consists of records all with the same value for the categorical attribute, return a single node with that value; If R is empty, then return a single node with as value the most frequent of the values of the categorical attribute that are found in records of S; [note that then there will be errors, that is, records that will be improperly classified]; Let D be the attribute with largest Gain( D,S) among attributes in R; Let {dj| j=1,2, .., m} be the values of attribute D; Let {Sj| j=1,2, .., m} be the subsets of S consisting respectively of records with value dj for attribute D; Return a tree with root labeled D and arcs labeled d1, d2, .., dm going respectively to the trees ID3(R-{D}, C, S1), ID3(R-{D}, C, S2), .., ID3(R-{D}, C, Sm); end ID3;
- 23. Pros of ID3 Algorithm● Builds decision tree in min. steps ○ The most important point while tree induction is collecting enough reliable associated data over specific properties. ○ Asking right questions determines tree induction.● Each level benefits from previous level choices● Whole dataset is scanned to create tree
- 24. Cons of ID3 Algorithm● Tree can not be updated when new data is classified incorrectly, instead a new tree must be generated.● Only one attribute at a time is tested for making a decision.● Can not work with defective data● Can not work with numerical attributes
- 25. An Example - The Simpsons Person Hair Length Weight Age Class Homer 0 250 36 M Marge 10 150 34 F Bart 2 90 10 M Lisa 6 78 8 M Maggie 4 20 1 F Abe 1 170 70 F Selma 8 160 41 F Otto 10 180 38 M Krusty 6 200 45 M
- 26. Information Gain over Hair LengthE(4F, 5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 ==> General Information Gain Hair Length <= 5 Yes No E(1F,3M) = -(1/4)log2(1/4) - (3/4)log2(3/4) = 0.9710 E(3F,2M) = -(3/5)log2(3/5) - (2/5)log2(2/5) =0.8113Gain(Hair Length <= 5) = 0.9911 – (4/9 * 0.9710 + 5/9 * 0.8113) = 0.0911
- 27. Information Gain over WeightE(4F, 5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 ==> General Information Gain Weight <= 160 Yes NoE(4F,1M) = -(4/5)log2(4/5) - (1/5)log2(1/5) = 0.7219 E(0F,4M) = -(0/4)log2(0/4) - (4/4)log2(4/4) = 0Gain(Weight<= 160) = 0.9911 – (5/9 * 0.7219 + 4/9 * 0 ) = 0.5900
- 28. Information Gain over AgeE(4F, 5M) = -(4/9)log2(4/9) - (5/9)log2(5/9) = 0.9911 ==> General Information Gain Age <= 40 Yes NoE(3F,3M) = -(3/6)log2(3/6) - (3/6)log2(3/6) = 1 E(1F,2M) = -(1/3)log2(1/3) - (2/3)log2(2/3)= 0.9188Gain(Age z= 40) = 0.9911 – (6/9 * 1 + 3/9 * 0.9183 ) = 0.0183
- 29. Results Attribute Information Gain (IG) Hair Length <= 5 0.0911 Weight <= 160 0.5900 Age <= 40 0.0183● As seen in the results, weight is the best attribute to classify these group.
- 30. Constructed Decision Tree Weight <= 160 Yes No Hair Length <= 5 Yes No Female Male
- 31. Entropy over Nominal Values● If an attribute has nominal values: ○ First calculate information gain for each attribute value ○ Then calculate attribute information gain
- 32. Example IIIG= -(5/15)log2(5/15)-(10/15)log2(10/15) = ~0.918
- 33. Example II (Cont.) Information Gain over Engine ● Engine: 6 small, 5 medium, 4 large ● 3 values for attribute engine, so we need 3 entropy calculations ● small: 5 no, 1 yes ○ IGsmall = -(5/6)log2(5/6)-(1/6)log2(1/6) = ~0.65 ● medium: 3 no, 2 yes ○ IGmedium = -(3/5)log2(3/5)-(2/5)log2(2/5) = ~0.97 ● large: 2 no, 2 yes ○ IGlarge = 1 (evenly distributed subset)=> IGEngine = IE(S) – [(6/15)*IGsmall + (5/15)*IGmedium +(4/15)*Ilarge] = IGEngine = 0.918 – 0.85 = 0.068
- 34. Example II (Cont.) Information Gain over SC/Turbo● SC/Turbo: 4 yes, 11 no● 2 values for attribute SC/Turbo, so we need 2 entropy calculations● yes: 2 yes, 2 no ○ IGturbo = 1 (evenly distributed subset)● no: 3 yes, 8 no ○ IGnoturbo = -(3/11)log2(3/11)-(8/11)log2(8/11) = ~0.84 IGturbo = IE(S) – [(4/15)*IGturbo + (11/15)*IGnoturbo] IGturbo = 0.918 – 0.886 = 0.032
- 35. Example II (Cont.) Information Gain over Weight● Weight: 6 Average, 4 Light, 5 Heavy● 3 values for attribute weight, so we need 3 entropy calculations● average: 3 no, 3 yes ○ IGaverage = 1 (evenly distributed subset)● light: 3 no, 1 yes ○ IGlight = -(3/4)log2(3/4)-(1/4)log2(1/4) = ~0.81● heavy: 4 no, 1 yes ○ IGheavy = -(4/5)log2(4/5)-(1/5)log2(1/5) = ~0.72 IGWeight = IE(S) – [(6/15)*IGaverage + (4/15)*IGlight + (5/15)*IGheavy] IGWeight = 0.918 – 0.856 = 0.062
- 36. Example II (Cont.) Information Gain over Full Eco● Fuel Economy: 2 good, 3 average, 10 bad● 3 values for attribute Fuel Eco, so we need 3 entropy calculations● good: 0 yes, 2 no ○ IGgood = 0 (no variability)● average: 0 yes, 3 no ○ IGaverage = 0 (no variability)● bad: 5 yes, 5 no ○ IGbad = 1 (evenly distributed subset) We can omit calculations for good and average since they alwaysend up not fast. IGFuelEco = IE(S) – [(10/15)*IGbad] IGFuelEco = 0.918 – 0.667 = 0.251
- 37. Example II (Cont.)● Results: IGEngine 0.068 ■ Root of the tree IGturbo 0.032 IGWeight 0.062 IGFuelEco 0.251
- 38. Example II (Cont.)● Since we selected the Fuel Eco attribute for our Root Node, it is removed from the table for future calculations. General Information Gain = 1 (Evenly distributed set)
- 39. Example II (Cont.) Information Gain over Engine● Engine: 1 small, 5 medium, 4 large● 3 values for attribute engine, so we need 3 entropy calculations● small: 1 yes, 0 no ○ IGsmall = 0 (no variability)● medium: 2 yes, 3 no ○ IGmedium = -(2/5)log2(2/5)-(3/5)log2(3/5) = ~0.97● large: 2 no, 2 yes ○ IGlarge = 1 (evenly distributed subset) IGEngine = IE(SFuelEco) – (5/10)*IGmedium + (4/10)*IGlarge] IGEngine = 1 – 0.885 = 0.115
- 40. Example II (Cont.) Information Gain over SC/Turbo● SC/Turbo: 3 yes, 7 no● 2 values for attribute SC/Turbo, so we need 2 entropy calculations● yes: 2 yes, 1 no ○ IGturbo = -(2/3)log2(2/3)-(1/3)log2(1/3) = ~0.84● no: 3 yes, 4 no ○ IGnoturbo = -(3/7)log2(3/7)-(4/7)log2(4/7) = ~0.84 IGturbo = IE(SFuelEco) – [(3/10)*IGturbo + (7/10)*IGnoturbo] IGturbo = 1 – 0.965 = 0.035
- 41. Example II (Cont.) Information Gain over Weight● Weight: 3 average, 5 heavy, 2 light● 3 values for attribute weight, so we need 3 entropy calculations● average: 3 yes, 0 no ○ IGaverage = 0 (no variability)● heavy: 1 yes, 4 no ○ IGheavy = -(1/5)log2(1/5)-(4/5)log2(4/5) = ~0.72● light: 1 yes, 1 no ○ IlGight = 1 (evenly distributed subset) IGEngine = IE(SFuel Eco) – [(5/10)*IGheavy+(2/10)*IGlight] IGEngine = 1 – 0.561 = 0.439
- 42. Example II (Cont.)● Results:IGEngine 0.115IGturbo 0.035IGWeight 0.439Weight has the highest gain, and is thus thebest choice.
- 43. Example II (Cont.)Since there are only two items for SC/Turbo whereWeight = Light, and the result is consistent, we cansimplify the weight = Light path.
- 44. Example II (Cont.)● Updated Table: (Weight = Heavy)● All cars with large engines in this table are not fast.● Due to inconsistent patterns in the data, there is no way to proceed since medium size engines may lead to either fast or not fast.
- 45. ROC Analysis● Receiver Operating Characteristic● The limitations of diagnostic “accuracy” as a measure of decision performance require introduction of the concepts of the “sensitivity” and “specificity” of a diagnostic test. These measures and the related indices, “true positive rate” and “false positive rate”, are more meaningful than “accuracy”.● ROC curve is shown to be a complete description of this decision threshold effect, indicating all possible combinations of the relative frequencies of the various kinds of correct and incorrect decisions.
- 46. ROC Analysis (Cont.)● Combinations of correct & incorrect decisions:Actual Value Prediction Outcome Descriptionp p True Positive Rate (TPR)p n False Negative Rate (FNR)n p False Positive Rate (FPR)n n True Negative Rate (TNR)● TPR is equivalent with sensitivity.● FPR is equivalent with 1 - specificity.● Best possible prediction would be 100% sensitivity and 100% specificity (which means FPR = 0%).
- 47. ROC Space● A ROC space is defined by FPR and TPR as x and y axes respectively, which depicts relative trade-offs between true positive (benefits) and false positive (costs).● Since TPR is equivalent with sensitivity and FPR is equal to 1 − specificity, the ROC graph is sometimes called the sensitivity vs (1 − specificity) plot.● Each prediction result one point in the ROC space.
- 48. Calculations● Sensitivity ○ TPR = TP / P = TP / (TP + FN)● Specificity ○ FPR = FP / N = FP / (FP + TN)● Accuracy ○ ACC = (TP + TN) / (P + N)
- 49. A ROC Space Example● Let A, B, C, D to be predictions over 100 negative and 100 positive instance:Prediction/ TP FP FN TN TPR FPR ACCCombination A 63 28 37 72 0.63 0.28 0.68 B 77 77 23 23 0.77 0.77 0.50 C 24 88 76 12 0.24 0.88 0.18 D 76 12 24 88 0.76 0.12 0.82
- 50. A ROC Space Example (Cont.)
- 51. References1. Data Mining Course Lectures, Ass. Prof. Nilüfer Yurtay2. Quinlan, J.R. 1986, Machine Learning, 1, 813. http://www.cse.unsw.edu. au/~billw/cs9414/notes/ml/06prop/id3/id3.html4. J. Han, M. Kamber, J. Pie, Data Mining Concepts and Techniques, 3rd Edition, Elsevier, 2011.5. http://www.cise.ufl.edu/~ddd/cap6635/Fall- 97/Short-papers/2.htm6. C. E. Metz, Basic Principles of ROC Analysis, Seminars in Nuclear Medicine, Volume 8, Issue 4, P 283-298

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