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Data Mining Classification: Alternative Techniques

Data Mining Classification: Alternative Techniques
Data Mining Classification: Alternative Techniques © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1RuleBased Classifier  Classify records by using a collection of “if…then…” rules  Rule: (Condition)  y – where  Condition is a conjunctions of attributes  y is the class label – LHS: rule antecedent or condition – RHS: rule consequent – Examples of classification rules:  (Blood Type=Warm)  (Lay Eggs=Yes)  Birds  (Taxable Income 50K)  (Refund=Yes)  Evade=No © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rulebased Classifier (Example) Name Blood Type Give Birth Can Fly Live in Water Class human warm yes no no mammals python cold no no no reptiles salmon cold no no yes fishes whale warm yes no yes mammals frog cold no no sometimes amphibians komodo cold no no no reptiles bat warm yes yes no mammals pigeon warm no yes no birds cat warm yes no no mammals leopard shark cold yes no yes fishes turtle cold no no sometimes reptiles penguin warm no no sometimes birds porcupine warm yes no no mammals eel cold no no yes fishes salamander cold no no sometimes amphibians gila monster cold no no no reptiles platypus warm no no no mammals owl warm no yes no birds dolphin warm yes no yes mammals eagle warm no yes no birds R1: (Give Birth = no)  (Can Fly = yes)  Birds R2: (Give Birth = no)  (Live in Water = yes)  Fishes R3: (Give Birth = yes)  (Blood Type = warm)  Mammals R4: (Give Birth = no)  (Can Fly = no)  Reptiles R5: (Live in Water = sometimes)  Amphibians © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Application of RuleBased Classifier  A rule r covers an instance x if the attributes of the instance satisfy the condition of the rule R1: (Give Birth = no)  (Can Fly = yes)  Birds R2: (Give Birth = no)  (Live in Water = yes)  Fishes R3: (Give Birth = yes)  (Blood Type = warm)  Mammals R4: (Give Birth = no)  (Can Fly = no)  Reptiles R5: (Live in Water = sometimes)  Amphibians Name Blood Type Give Birth Can Fly Live in Water Class hawk warm no yes no grizzly bear warm yes no no The rule R1 covers a hawk = Bird The rule R3 covers the grizzly bear = Mammal © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Coverage and Accuracy Tid Refund Marital Taxable  Coverage of a rule: Class Status Income 1 Yes Single 125K No – Fraction of records 2 No Married 100K No that satisfy the 3 No Single 70K No antecedent of a rule 4 Yes Married 120K No 5 No Divorced 95K Yes  Accuracy of a rule: 6 No Married 60K No – Fraction of records 7 Yes Divorced 220K No that satisfy both the 8 No Single 85K Yes 9 No Married 75K No antecedent and 10 No Single 90K Yes consequent of a 10 (Status=Single)  No rule Coverage = 40, Accuracy = 50 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How does Rulebased Classifier Work R1: (Give Birth = no)  (Can Fly = yes)  Birds R2: (Give Birth = no)  (Live in Water = yes)  Fishes R3: (Give Birth = yes)  (Blood Type = warm)  Mammals R4: (Give Birth = no)  (Can Fly = no)  Reptiles R5: (Live in Water = sometimes)  Amphibians Name Blood Type Give Birth Can Fly Live in Water Class lemur warm yes no no turtle cold no no sometimes dogfish shark cold yes no yes A lemur triggers rule R3, so it is classified as a mammal A turtle triggers both R4 and R5 A dogfish shark triggers none of the rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Characteristics of RuleBased Classifier  Mutually exclusive rules – Classifier contains mutually exclusive rules if the rules are independent of each other – Every record is covered by at most one rule  Exhaustive rules – Classifier has exhaustive coverage if it accounts for every possible combination of attribute values – Each record is covered by at least one rule © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›From Decision Trees To Rules Classification Rules (Refund=Yes) == No Refund Yes No (Refund=No, Marital Status=Single,Divorced, Taxable Income80K) == No NO NO Marital (Refund=No, Marital Status=Single,Divorced, Status Single, Taxable Income80K) == Yes Married Divorced (Refund=No, Marital Status=Married) == No NO NO Taxable Income 80K 80K NO NO Y YE ES S Rules are mutually exclusive and exhaustive Rule set contains as much information as the tree © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rules Can Be Simplified Tid Refund Marital Taxable Cheat Status Income Refund 1 Yes Single 125K No Yes No 2 No Married 100K No NO NO Marital 3 No Single 70K No Status Single, Married 4 Yes 120K Married No Divorced 5 No Divorced 95K Yes NO NO Taxable 6 No Married 60K No Income 7 Yes Divorced 220K No 80K 80K 8 No Single 85K Yes NO NO Y YE ES S 9 No Married 75K No 10 No Single 90K Yes 10 Initial Rule: (Refund=No)  (Status=Married)  No Simplified Rule: (Status=Married)  No © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Effect of Rule Simplification  Rules are no longer mutually exclusive – A record may trigger more than one rule – Solution  Ordered rule set  Unordered rule set – use voting schemes  Rules are no longer exhaustive – A record may not trigger any rules – Solution  Use a default class © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Ordered Rule Set  Rules are rank ordered according to their priority – An ordered rule set is known as a decision list  When a test record is presented to the classifier – It is assigned to the class label of the highest ranked rule it has triggered – If none of the rules fired, it is assigned to the default class R1: (Give Birth = no)  (Can Fly = yes)  Birds R2: (Give Birth = no)  (Live in Water = yes)  Fishes R3: (Give Birth = yes)  (Blood Type = warm)  Mammals R4: (Give Birth = no)  (Can Fly = no)  Reptiles R5: (Live in Water = sometimes)  Amphibians Name Blood Type Give Birth Can Fly Live in Water Class turtle cold no no sometimes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Ordering Schemes  Rulebased ordering – Individual rules are ranked based on their quality  Classbased ordering – Rules that belong to the same class appear together Rulebased Ordering Classbased Ordering (Refund=Yes) == No (Refund=Yes) == No (Refund=No, Marital Status=Single,Divorced, (Refund=No, Marital Status=Single,Divorced, Taxable Income80K) == No Taxable Income80K) == No (Refund=No, Marital Status=Single,Divorced, (Refund=No, Marital Status=Married) == No Taxable Income80K) == Yes (Refund=No, Marital Status=Single,Divorced, (Refund=No, Marital Status=Married) == No Taxable Income80K) == Yes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Building Classification Rules  Direct Method:  Extract rules directly from data  e.g.: RIPPER, CN2, Holte’s 1R  Indirect Method:  Extract rules from other classification models (e.g. decision trees, neural networks, etc).  e.g: C4.5rules © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Direct Method: Sequential Covering 1. Start from an empty rule 2. Grow a rule using the LearnOneRule function 3. Remove training records covered by the rule 4. Repeat Step (2) and (3) until stopping criterion is met © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example of Sequential Covering (i) Original Data (ii) Step 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example of Sequential Covering… R1 R1 R2 (iii) Step 2 (iv) Step 3 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Aspects of Sequential Covering  Rule Growing  Instance Elimination  Rule Evaluation  Stopping Criterion  Rule Pruning © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Growing  Two common strategies Yes: 3 No: 4 Refund=No, Refund=No, Status=Single, Status=Single, Income=85K Income=90K (Class=Yes) (Class=Yes) Income Refund= Status = Status = Status = ... 80K No Single Divorced Married Refund=No, Status = Single (Class = Yes) Yes: 3 Yes: 2 Yes: 1 Yes: 0 Yes: 3 No: 4 No: 1 No: 0 No: 3 No: 1 (b) Specifictogeneral (a) Generaltospecific © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Growing (Examples)  CN2 Algorithm: – Start from an empty conjunct: – Add conjuncts that minimizes the entropy measure: A, A,B, … – Determine the rule consequent by taking majority class of instances covered by the rule  RIPPER Algorithm: – Start from an empty rule: = class – Add conjuncts that maximizes FOIL’s information gain measure:  R0: = class (initial rule)  R1: A = class (rule after adding conjunct)  Gain(R0, R1) = t log (p1/(p1+n1)) – log (p0/(p0 + n0))  where t: number of positive instances covered by both R0 and R1 p0: number of positive instances covered by R0 n0: number of negative instances covered by R0 p1: number of positive instances covered by R1 n1: number of negative instances covered by R1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Instance Elimination  Why do we need to eliminate instances R3 R2 – Otherwise, the next rule is R1 + + + + identical to previous rule + + + + + + + + +  Why do we remove + + + class = + + + + + + + positive instances + + + + + + + – Ensure that the next rule is different class =  Why do we remove negative instances – Prevent underestimating accuracy of rule – Compare rules R2 and R3 in the diagram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Rule Evaluation  Metrics: n c – Accuracy  n n1 c n : Number of instances – Laplace  covered by rule n k n : Number of instances c covered by rule k : Number of classes n kp c – Mestimate  p : Prior probability n k © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Stopping Criterion and Rule Pruning  Stopping criterion – Compute the gain – If gain is not significant, discard the new rule  Rule Pruning – Similar to postpruning of decision trees – Reduced Error Pruning:  Remove one of the conjuncts in the rule  Compare error rate on validation set before and after pruning  If error improves, prune the conjunct © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Summary of Direct Method  Grow a single rule  Remove Instances from rule  Prune the rule (if necessary)  Add rule to Current Rule Set  Repeat © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Direct Method: RIPPER  For 2class problem, choose one of the classes as positive class, and the other as negative class – Learn rules for positive class – Negative class will be default class  For multiclass problem – Order the classes according to increasing class prevalence (fraction of instances that belong to a particular class) – Learn the rule set for smallest class first, treat the rest as negative class – Repeat with next smallest class as positive class © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Direct Method: RIPPER  Growing a rule: – Start from empty rule – Add conjuncts as long as they improve FOIL’s information gain – Stop when rule no longer covers negative examples – Prune the rule immediately using incremental reduced error pruning – Measure for pruning: v = (pn)/(p+n)  p: number of positive examples covered by the rule in the validation set  n: number of negative examples covered by the rule in the validation set – Pruning method: delete any final sequence of conditions that maximizes v © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Direct Method: RIPPER  Building a Rule Set: – Use sequential covering algorithm  Finds the best rule that covers the current set of positive examples  Eliminate both positive and negative examples covered by the rule – Each time a rule is added to the rule set, compute the new description length  stop adding new rules when the new description length is d bits longer than the smallest description length obtained so far © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Direct Method: RIPPER  Optimize the rule set: – For each rule r in the rule set R  Consider 2 alternative rules: – Replacement rule (r): grow new rule from scratch – Revised rule(r’): add conjuncts to extend the rule r  Compare the rule set for r against the rule set for r and r’  Choose rule set that minimizes MDL principle – Repeat rule generation and rule optimization for the remaining positive examples © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Indirect Methods P No Yes Rule Set Q R No Yes No Yes r1: (P=No,Q=No) == r2: (P=No,Q=Yes) == + Q + + r3: (P=Yes,R=No) == + r4: (P=Yes,R=Yes,Q=No) == No Yes r5: (P=Yes,R=Yes,Q=Yes) == + + © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Indirect Method: C4.5rules  Extract rules from an unpruned decision tree  For each rule, r: A  y, – consider an alternative rule r’: A’  y where A’ is obtained by removing one of the conjuncts in A – Compare the pessimistic error rate for r against all r’s – Prune if one of the r’s has lower pessimistic error rate – Repeat until we can no longer improve generalization error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Indirect Method: C4.5rules  Instead of ordering the rules, order subsets of rules (class ordering) – Each subset is a collection of rules with the same rule consequent (class) – Compute description length of each subset  Description length = L(error) + g L(model)  g is a parameter that takes into account the presence of redundant attributes in a rule set (default value = 0.5) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example Name Give Birth Lay Eggs Can Fly Live in Water Have Legs Class human yes no no no yes mammals python no yes no no no reptiles salmon no yes no yes no fishes whale yes no no yes no mammals frog no yes no sometimes yes amphibians komodo no yes no no yes reptiles bat yes no yes no yes mammals pigeon no yes yes no yes birds cat yes no no no yes mammals leopard shark yes no no yes no fishes turtle no yes no sometimes yes reptiles penguin no yes no sometimes yes birds porcupine yes no no no yes mammals eel no yes no yes no fishes salamander no yes no sometimes yes amphibians gila monster no yes no no yes reptiles platypus no yes no no yes mammals owl no yes yes no yes birds dolphin yes no no yes no mammals eagle no yes yes no yes birds © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›C4.5 versus C4.5rules versus RIPPER C4.5rules: Give (Give Birth=No, Can Fly=Yes)  Birds Birth (Give Birth=No, Live in Water=Yes)  Fishes Yes No (Give Birth=Yes)  Mammals (Give Birth=No, Can Fly=No, Live in Water=No)  Reptiles Mammals Live In ( )  Amphibians Water RIPPER: Yes No (Live in Water=Yes)  Fishes (Have Legs=No)  Reptiles Sometimes (Give Birth=No, Can Fly=No, Live In Water=No) Can Fishes Amphibians  Reptiles Fly (Can Fly=Yes,Give Birth=No)  Birds No ()  Mammals Yes Birds Reptiles © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›C4.5 versus C4.5rules versus RIPPER C4.5 and C4.5rules: PREDICTED CLASS Amphibians Fishes Reptiles Birds Mammals ACTUAL Amphibians 2 0 0 0 0 CLASS Fishes 0 2 0 0 1 Reptiles 1 0 3 0 0 Birds 1 0 0 3 0 Mammals 0 0 1 0 6 RIPPER: PREDICTED CLASS Amphibians Fishes Reptiles Birds Mammals ACTUAL Amphibians 0 0 0 0 2 CLASS Fishes 0 3 0 0 0 Reptiles 0 0 3 0 1 Birds 0 0 1 2 1 Mammals 0 2 1 0 4 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Advantages of RuleBased Classifiers  As highly expressive as decision trees  Easy to interpret  Easy to generate  Can classify new instances rapidly  Performance comparable to decision trees © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›InstanceBased Classifiers Set of Stored Cases • Store the training records • Use training records to ……... Atr1 AtrN Class predict the class label of A unseen cases B B Unseen Case C ……... Atr1 AtrN A C B © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Instance Based Classifiers  Examples: – Rotelearner  Memorizes entire training data and performs classification only if attributes of record match one of the training examples exactly – Nearest neighbor  Uses k “closest” points (nearest neighbors) for performing classification © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest Neighbor Classifiers  Basic idea: – If it walks like a duck, quacks like a duck, then it’s probably a duck Compute Test Distance Record Training Choose k of the Records “nearest” records © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›NearestNeighbor Classifiers Unknown record  Requires three things – The set of stored records – Distance Metric to compute distance between records – The value of k, the number of nearest neighbors to retrieve  To classify an unknown record: – Compute distance to other training records – Identify k nearest neighbors – Use class labels of nearest neighbors to determine the class label of unknown record (e.g., by taking majority vote) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Definition of Nearest Neighbor X X X (a) 1nearest neighbor (b) 2nearest neighbor (c) 3nearest neighbor Knearest neighbors of a record x are data points that have the k smallest distance to x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›1 nearestneighbor Voronoi Diagram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest Neighbor Classification  Compute distance between two points: – Euclidean distance 2 d( p,q)( p q ) i i i  Determine the class from nearest neighbor list – take the majority vote of class labels among the knearest neighbors – Weigh the vote according to distance 2  weight factor, w = 1/d © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest Neighbor Classification…  Choosing the value of k: – If k is too small, sensitive to noise points – If k is too large, neighborhood may include points from other classes X © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest Neighbor Classification…  Scaling issues – Attributes may have to be scaled to prevent distance measures from being dominated by one of the attributes – Example:  height of a person may vary from 1.5m to 1.8m  weight of a person may vary from 90lb to 300lb  income of a person may vary from 10K to 1M © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest Neighbor Classification…  Problem with Euclidean measure: – High dimensional data  curse of dimensionality – Can produce counterintuitive results 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 vs 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 d = 1.4142 d = 1.4142  Solution: Normalize the vectors to unit length © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nearest neighbor Classification…  kNN classifiers are lazy learners – It does not build models explicitly – Unlike eager learners such as decision tree induction and rulebased systems – Classifying unknown records are relatively expensive © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: PEBLS  PEBLS: Parallel ExamplarBased Learning System (Cost Salzberg) – Works with both continuous and nominal features For nominal features, distance between two nominal values is computed using modified value difference metric (MVDM) – Each record is assigned a weight factor – Number of nearest neighbor, k = 1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: PEBLS Tid Refund Marital Taxable Distance between nominal attribute values: Cheat Status Income 1 Yes Single 125K No d(Single,Married) 2 No Married 100K No = 2/4 – 0/4 + 2/4 – 4/4 = 1 3 No Single 70K No d(Single,Divorced) 4 Yes Married 120K No = 2/4 – 1/2 + 2/4 – 1/2 = 0 Yes 5 No Divorced 95K d(Married,Divorced) 6 No Married 60K No No 7 Yes Divorced 220K = 0/4 – 1/2 + 4/4 – 1/2 = 1 8 No Single 85K Yes d(Refund=Yes,Refund=No) 9 No Married 75K No = 0/3 – 3/7 + 3/3 – 4/7 = 6/7 10 No Single 90K Yes 10 Marital Status Refund n n 1i 2i Class Class d(V ,V ) Yes No 1 2 Single Married Divorced n n i 1 2 Yes 0 3 Yes 2 0 1 No 2 4 1 No 3 4 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: PEBLS Tid Refund Marital Taxable Cheat Status Income X Yes Single 125K No Y No Married 100K No 10 Distance between record X and record Y: d 2 (X ,Y ) w w d(X ,Y )  X Y i i i1 Number of times X is used for prediction where: w X Number of times X predicts correctly w 1 if X makes accurate prediction most of the time X w 1 if X is not reliable for making predictions X © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bayes Classifier  A probabilistic framework for solving classification problems  Conditional Probability: P(A,C) P(C A) P(A) P(A,C) P(A C) P(C)  Bayes theorem: P(A C)P(C) P(C A) P(A) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example of Bayes Theorem  Given: – A doctor knows that meningitis causes stiff neck 50 of the time – Prior probability of any patient having meningitis is 1/50,000 – Prior probability of any patient having stiff neck is 1/20  If a patient has stiff neck, what’s the probability he/she has meningitis P(S M )P(M ) 0.51/ 50000 P(M S) 0.0002 P(S) 1/ 20 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bayesian Classifiers  Consider each attribute and class label as random variables  Given a record with attributes (A , A ,…,A ) 1 2 n – Goal is to predict class C – Specifically, we want to find the value of C that maximizes P(C A , A ,…,A ) 1 2 n  Can we estimate P(C A , A ,…,A ) directly from 1 2 n data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bayesian Classifiers  Approach: – compute the posterior probability P(C A , A , …, A ) for 1 2 n all values of C using the Bayes theorem P(A AA C)P(C) 1 2 n P(C A AA ) 1 2 n P(A AA ) 1 2 n – Choose value of C that maximizes P(C A , A , …, A ) 1 2 n – Equivalent to choosing value of C that maximizes P(A , A , …, A C) P(C) 1 2 n  How to estimate P(A , A , …, A C ) 1 2 n © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Naïve Bayes Classifier  Assume independence among attributes A when class is i given: – P(A , A , …, A C) = P(A C ) P(A C )… P(A C ) 1 2 n 1 j 2 j n j – Can estimate P(A C ) for all A and C . i j i j – New point is classified to C if P(C )  P(A C ) is j j i j maximal. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Estimate Probabilities from Data  Class: P(C) = N /N c Tid Refund Marital Taxable – e.g., P(No) = 7/10, Evade Status Income P(Yes) = 3/10 1 Yes Single 125K No 2 No Married 100K No  For discrete attributes: 3 No Single 70K No P(A C ) = A / N 4 Yes Married 120K No i k ik c k 5 No Divorced 95K Yes – where A is number of ik 6 No Married 60K No instances having attribute 7 Yes Divorced 220K No A and belongs to class C i k 8 No Single 85K Yes – Examples: 9 No Married 75K No 10 No Single 90K Yes P(Status=MarriedNo) = 4/7 10 P(Refund=YesYes)=0 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› categorical categorical continuous classHow to Estimate Probabilities from Data  For continuous attributes: – Discretize the range into bins  one ordinal attribute per bin k  violates independence assumption – Twoway split: (A v) or (A v)  choose only one of the two splits as new attribute – Probability density estimation:  Assume attribute follows a normal distribution  Use data to estimate parameters of distribution (e.g., mean and standard deviation)  Once probability distribution is known, can use it to estimate the conditional probability P(A c) i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Estimate Probabilities from Data Tid Refund Marital Taxable  Normal distribution: Evade Status Income 2 ( A ) i ij  2 1 2 ij 1 Yes Single 125K No P(A c ) e i j 2 2 No Married 100K No 2  ij 3 No Single 70K No 4 Yes Married 120K No – One for each (A ,c ) pair i i 5 No Divorced 95K Yes 6 No Married 60K No  For (Income, Class=No): 7 Yes Divorced 220K No – If Class=No 8 No Single 85K Yes  sample mean = 110 9 No Married 75K No  sample variance = 2975 10 No Single 90K Yes 10 2 (120110 )  1 2 ( 2975 ) P(Income120 No) e 0.0072 2 (54.54) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› categorical categorical continuous classExample of Naïve Bayes Classifier Given a Test Record: X (Refund No,Married,Income120K) naive Bayes Classifier:  P(XClass=No) = P(Refund=NoClass=No) P(Refund=YesNo) = 3/7 P(Refund=NoNo) = 4/7  P(Married Class=No) P(Refund=YesYes) = 0  P(Income=120K Class=No) P(Refund=NoYes) = 1 = 4/7  4/7  0.0072 = 0.0024 P(Marital Status=SingleNo) = 2/7 P(Marital Status=DivorcedNo)=1/7  P(XClass=Yes) = P(Refund=No Class=Yes) P(Marital Status=MarriedNo) = 4/7  P(Married Class=Yes) P(Marital Status=SingleYes) = 2/7 P(Marital Status=DivorcedYes)=1/7  P(Income=120K Class=Yes) 9 P(Marital Status=MarriedYes) = 0 = 1  0  1.2  10 = 0 For taxable income: Since P(XNo)P(No) P(XYes)P(Yes) If class=No: sample mean=110 sample variance=2975 Therefore P(NoX) P(YesX) If class=Yes: sample mean=90 = Class = No sample variance=25 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Naïve Bayes Classifier  If one of the conditional probability is zero, then the entire expression becomes zero  Probability estimation: N ic Original : P(A C) i N c c: number of classes N1 ic p: prior probability Laplace : P(A C) i N c c m: parameter N mp ic m estimate : P(A C) i N m c © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example of Naïve Bayes Classifier Name Give Birth Can Fly Live in Water Have Legs Class A: attributes human yes no no yes mammals python no no no no nonmammals M: mammals salmon no no yes no nonmammals whale yes no yes no mammals N: nonmammals frog no no sometimes yes nonmammals komodo no no no yes nonmammals 6 6 2 2 bat yes yes no yes mammals P(A M ) 0.06 pigeon no yes no yes nonmammals 7 7 7 7 cat yes no no yes mammals leopard shark yes no yes no nonmammals 1 10 3 4 turtle no no sometimes yes nonmammals P(A N) 0.0042 penguin no no sometimes yes nonmammals 13 13 13 13 porcupine yes no no yes mammals 7 eel no no yes no nonmammals P(A M )P(M ) 0.06 0.021 salamander no no sometimes yes nonmammals 20 gila monster no no no yes nonmammals platypus no no no yes mammals 13 owl no yes no yes nonmammals P(A N)P(N) 0.004 0.0027 dolphin yes no yes no mammals 20 eagle no yes no yes nonmammals P(AM)P(M) P(AN)P(N) Give Birth Can Fly Live in Water Have Legs Class yes no yes no = Mammals © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Naïve Bayes (Summary)  Robust to isolated noise points  Handle missing values by ignoring the instance during probability estimate calculations  Robust to irrelevant attributes  Independence assumption may not hold for some attributes – Use other techniques such as Bayesian Belief Networks (BBN) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Artificial Neural Networks (ANN) Black box X X X Y Input 1 2 3 1 0 0 0 X 1 1 0 1 1 Output 1 1 0 1 1 1 1 1 X Y 2 0 0 1 0 0 1 0 0 0 1 1 1 X 3 0 0 0 0 Output Y is 1 if at least two of the three inputs are equal to 1. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Artificial Neural Networks (ANN) Input nodes Black box X X X Y 1 2 3 Output 1 0 0 0 X node 1 1 0 1 1 0.3 1 1 0 1 0.3 1 1 1 1 X  Y 2 0 0 1 0 0 1 0 0 0 1 1 1 X 0.3 3 t=0.4 0 0 0 0 Y I(0.3X 0.3X 0.3X 0.4 0) 1 2 3 1 if z is true  where I(z)  0 otherwise  © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Artificial Neural Networks (ANN) Input  Model is an assembly of nodes Black box interconnected nodes Output X and weighted links node 1 w 1 w 2 X  Y 2 w 3  Output node sums up X each of its input value 3 t according to the weights of its links Perceptron Model or Y I( w X t)  i i  Compare output node i against some threshold t Y sign( w X t)  i i i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›General Structure of ANN x x x x x 1 2 3 4 5 Input Layer Input Neuron i Output I w 1 i1 Activation w i2 I O function O 2 S i i i w g(S ) Hidden i3 i I Layer 3 threshold, t Output Training ANN means learning Layer the weights of the neurons y © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Algorithm for learning ANN  Initialize the weights (w , w , …, w ) 0 1 k  Adjust the weights in such a way that the output of ANN is consistent with class labels of training examples 2 – Objective function: E Y f (w , X )  i i i i – Find the weights w’s that minimize the above i objective function  e.g., backpropagation algorithm (see lecture notes) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines  Find a linear hyperplane (decision boundary) that will separate the data © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 1  One Possible Solution © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 2  Another possible solution © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 2  Other possible solutions © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 1 B 2  Which one is better B1 or B2  How do you define better © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 1 B 2 b 21 b 22 margin b 11 b 12  Find hyperplane maximizes the margin = B1 is better than B2 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines B 1  w x b 0   w x b1 w x b1 b 11  b 12 2 1 if w x b 1  Margin  f (x)   2 w 1 if w x b1  © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines 2  We want to maximize: Margin   2 w  2 w L(w) – Which is equivalent to minimizing: 2 – But subjected to the following constraints:  1 if w x b 1   i f (x )   i 1 if w x b1  i  This is a constrained optimization problem – Numerical approaches to solve it (e.g., quadratic programming) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines  What if the problem is not linearly separable © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Support Vector Machines  What if the problem is not linearly separable – Introduce slack variables  2 N  Need to minimize: w  k L(w) C   i 2  i1  Subject to:  1 if w x b 1   i i f (x )   i 1 if w x b1  i i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nonlinear Support Vector Machines  What if decision boundary is not linear © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Nonlinear Support Vector Machines  Transform data into higher dimensional space © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Ensemble Methods  Construct a set of classifiers from the training data  Predict class label of previously unseen records by aggregating predictions made by multiple classifiers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›General Idea Original D Training data Step 1: .... Create Multiple D D D D 2 1 t1 t Data Sets Step 2: Build Multiple C C C C 1 2 t 1 t Classifiers Step 3: Combine C Classifiers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Why does it work  Suppose there are 25 base classifiers – Each classifier has error rate,  = 0.35 – Assume classifiers are independent – Probability that the ensemble classifier makes a wrong prediction: 25 25  i 25i  (1 ) 0.06   i i13  © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Examples of Ensemble Methods  How to generate an ensemble of classifiers – Bagging – Boosting © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bagging  Sampling with replacement Original Data 1 2 3 4 5 6 7 8 9 10 Bagging (Round 1) 7 8 10 8 2 5 10 10 5 9 Bagging (Round 2) 1 4 9 1 2 3 2 7 3 2 Bagging (Round 3) 1 8 5 10 5 5 9 6 3 7  Build classifier on each bootstrap sample n  Each sample has probability (1 – 1/n) of being selected © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Boosting  An iterative procedure to adaptively change distribution of training data by focusing more on previously misclassified records – Initially, all N records are assigned equal weights – Unlike bagging, weights may change at the end of boosting round © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Boosting  Records that are wrongly classified will have their weights increased  Records that are classified correctly will have their weights decreased Original Data 1 2 3 4 5 6 7 8 9 10 Boosting (Round 1) 7 3 2 8 7 9 4 10 6 3 Boosting (Round 2) 5 4 9 4 2 5 1 7 4 2 Boosting (Round 3) 4 4 8 10 4 5 4 6 3 4 • Example 4 is hard to classify • Its weight is increased, therefore it is more likely to be chosen again in subsequent rounds © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: AdaBoost  Base classifiers: C , C , …, C 1 2 T  Error rate: N 1  wC (x ) y  i j i j j N j1  Importance of a classifier:  1 1 i   ln i  2  i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Example: AdaBoost  Weight update:  ( j) j  exp if C (x ) y w  ( j1) j i i i w  i j Z exp if C (x ) y  j j i i  where Z is the normalization factor j  If any intermediate rounds produce error rate higher than 50, the weights are reverted back to 1/n and the resampling procedure is repeated  Classification: T C (x) arg maxC (x) y  j j y j1 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Illustrating AdaBoost Initial weights for each data point Data points for training 0.1 0.1 0.1 Original + + + + + Data B1 0.0094 0.0094 0.4623 Boosting + + + Round 1  = 1.9459 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Illustrating AdaBoost B1 0.0094 0.0094 0.4623 Boosting + + + Round 1  = 1.9459 B2 0.0009 0.0422 0.3037 Boosting + + Round 2  = 2.9323 B3 0.0038 0.0276 0.1819 Boosting + + + + + + + + + +  = 3.8744 Round 3 + + + + + Overall © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›
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