Induction of Decision Trees
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1 Induction of Decision Trees Blaž Zupan and Ivan Bratko magixfriuni-ljsi/predavanja/uisp
2 An Example Data Set and Decision Tree # Attribute Class Outlook Company Sailboat Sail? 1 sunny big small yes 2 sunny med small yes 3 sunny med big yes 4 sunny no small yes 5 sunny big big yes 6 rainy no small no 7 rainy med small yes 8 rainy big big yes 9 rainy no big no 10 rainy med big no sunny yes outlook rainy company no med big no sailboat yes small yes big no
3 Classification # Attribute Class Outlook Company Sailboat Sail? 1 sunny no big? 2 rainy big small? outlook sunny rainy yes company no med big no sailboat yes small big yes no
4 Induction of Decision Trees Data Set (Learning Set) Each example = Attributes + Class Induced description = Decision tree TDIDT Top Down Induction of Decision Trees Recursive Partitioning
5 Some TDIDT Systems ID3 (Quinlan 79) CART (Brieman et al 84) Assistant (Cestnik et al 87) C45 (Quinlan 93) See5 (Quinlan 97) Orange (Demšar, Zupan 98-03)
6 Analysis of Severe Trauma Patients Data <72 PH_ICU The worst ph value at ICU >733 Death 00 (0/15) Well 082 (9/11) APPT_WORST <787 >=787 Death 00 (0/7) Well 088 (14/16) The worst active partial thromboplastin time PH_ICU and APPT_WORST are exactly the two factors (theoretically) advocated to be the most important ones in the study by Rotondo et al, 1997
7 Breast Cancer Recurrence Degree of Malig < 3 >= 3 Tumor Size Involved Nodes < 15 >= 15 < 3 >= 3 Age no rec 125 recurr 39 no rec 30 recurr 18 recurr 27 no_rec 10 no rec 4 recurr 1 no rec 32 recurr 0 Tree induced by Assistant Professional Interesting: Accuracy of this tree compared to medical specialists
8 Prostate cancer recurrence Secondary Gleason Grade 1, No PSA Level Stage Yes T1c,T2a, T2b,T2c T1ab,T3 No Primary Gleason Grade No Yes 2,3 4 No Yes
9 TDIDT Algorithm Also known as ID3 (Quinlan) To construct decision tree T from learning set S: If all examples in S belong to some class C Then make leaf labeled C Otherwise select the most informative attribute A partition S according to A s values recursively construct subtrees T1, T2,, for the subsets of S
10 TDIDT Algorithm Resulting tree T is: A Attribute A v1 v2 vn A s values T1 T2 Tn Subtrees
11 Another Example # Attribute Class Outlook Temperature Humidity Windy Play 1 sunny hot high no N 2 sunny hot high yes N 3 overcast hot high no P 4 rainy moderate high no P 5 rainy cold normal no P 6 rainy cold normal yes N 7 overcast cold normal yes P 8 sunny moderate high no N 9 sunny cold normal no P 10 rainy moderate normal no P 11 sunny moderate normal yes P 12 overcast moderate high yes P 13 overcast hot normal no P 14 rainy moderate high yes N
12 Simple Tree Outlook sunny overcast rainy Humidity P Windy high normal yes no N P N P
13 Complicated Tree Temperature Outlook Windy cold moderate hot P sunny rainy N yes no P overcast Outlook sunny rainy P overcast Windy P N yes no Windy N P yes no Humidity P high normal Windy P N yes no Humidity P high normal Outlook N sunny rainy P overcast null
14 Attribute Selection Criteria Main principle Select attribute which partitions the learning set into subsets as pure as possible Various measures of purity Information-theoretic Gini index X 2 ReliefF Various improvements probability estimates normalization binarization, subsetting
15 Information-Theoretic Approach To classify an object, a certain information is needed I, information After we have learned the value of attribute A, we only need some remaining amount of information to classify the object Ires, residual information Gain Gain(A) = I Ires(A) The most informative attribute is the one that minimizes Ires, ie, maximizes Gain
16 Entropy The average amount of information I needed to classify an object is given by the entropy measure For a two-class problem: entropy p(c1)
17 Residual Information After applying attribute A, S is partitioned into subsets according to values v of A Ires is equal to weighted sum of the amounts of information for the subsets
18 Triangles and Squares # Attribute Shape Color Outline Dot 1 green dashed no triange 2 green dashed yes triange 3 yellow dashed no square 4 red dashed no square 5 red solid no square 6 red solid yes triange 7 green solid no square 8 green dashed no triange 9 yellow solid yes square 10 red solid no square 11 green solid yes square 12 yellow dashed yes square 13 yellow solid no square 14 red dashed yes triange
19 Triangles and Squares # Attribute Shape Color Outline Dot 1 green dashed no triange 2 green dashed yes triange 3 yellow dashed no square 4 red dashed no square 5 red solid no square 6 red solid yes triange 7 green solid no square 8 green dashed no triange 9 yellow solid yes square 10 red solid no square 11 green solid yes square 12 yellow dashed yes square 13 yellow solid no square 14 red dashed yes triange Data Set: A set of classified objects
20 Entropy 5 triangles 9 squares class probabilities entropy
21 red Entropy reduction by data set partitioning Color? yellow green
22 Entropija vrednosti atributa red Color? green yellow
23 Information Gain red Color? green yellow
24 Information Gain of The Attribute Attributes Gain(Color) = 0246 Gain(Outline) = 0151 Gain(Dot) = 0048 Heuristics: attribute with the highest gain is chosen This heuristics is local (local minimization of impurity)
25 red Color? green yellow Gain(Outline) = = 0971 bits Gain(Dot) = = 0020 bits
26 red Color? Gain(Outline) = = 0020 bits Gain(Dot) = = 0971 bits green yellow Outline? solid dashed
27 red yes Color? green Dot? no yellow Outline? dashed solid
28 Decision Tree Color red yellow green Dot square Outline yes no dashed solid triangle square triangle square
29 A Defect of Ires Ires favors attributes with many values Such attribute splits S to many subsets, and if these are small, they will tend to be pure anyway One way to rectify this is through a corrected measure of information gain ratio
30 Information Gain Ratio I(A) is amount of information needed to determine the value of an attribute A Information gain ratio
31 Information Gain Ratio red Color? green yellow
32 Information Gain and Information Gain Ratio A v(a) Gain(A) GainRatio(A) Color Outline Dot
33 Gini Index Another sensible measure of impurity (i and j are classes) After applying attribute A, the resulting Gini index is Gini can be interpreted as expected error rate
34 Gini Index
35 Gini Index for Color red Color? green yellow
36 Gain of Gini Index
37 Three Impurity Measures A Gain(A) GainRatio(A) GiniGain(A) Color Outline Dot These impurity measures assess the effect of a single attribute Criterion most informative that they define is local (and myopic ) It does not reliably predict the effect of several attributes applied jointly
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