Probabilistic Interestingness Measures - An Introduction with Bayesian Belief Networks


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Probabilistic Interestingness Measures - An Introduction with Bayesian Belief Networks

  1. 1. A N I N T R O D U C T I O N W I T H B A Y E S I A N B E L I E F N E T W O R K SA D N A N M A S O O DS C I S . N O V A . E D U / ~ A D N A NA D N A N @ N O V A . E D UD O C T O R A L C A N D I D A T EN O V A S O U T H E A S T E R N U N I V E R S I T YProbabilistic InterestingnessMeasures
  2. 2. Introduction Interestingness measures play an important role in data miningregardless of the kind of patterns being mined. Good measuresshould select and rank patterns according to their potential interestto the user. Good measures should also reduce the time and spacecost of the mining process. (Geng & Hamilton, 2007) Measuring the interestingness of discovered patterns is an activeand important area of data mining research. Although much workhas been conducted in this area, so far there is no widespreadagreement on a formal definition of interestingness in this context.Based on the diversity of definitions presented to date,interestingness is perhaps best treated as a very broad concept,which emphasizes conciseness, coverage, reliability, peculiarity,diversity, novelty, surprisingness, utility, and actionability. (Geng &Hamilton, 2007)
  3. 3. Overview of interestingness measuresA Survey of Interestingness Measures for Knowledge Discovery (Ken McGarry, 2005)
  4. 4. Interestingness Measures & the RankingRef: A Survey of Interestingness Measures for Knowledge Discovery (Ken McGarry, 2005)
  5. 5. Usual Measures of Interest
  6. 6. Interestingness Measures and Expert basedQuality2 categories: Objective (D, M) Computed from data only Subjective (U) Hypothesis : goal, domain knowledge Hard to formalize (novelty)Quality Measures in Data Mining, (Fabrice Guillet & Howard J. Hamilton, 2007)
  7. 7. Interestingness Measure - Definitioni(XY) = f(n, nx, ny, nxy)General principles: Semantic and readability for the user Increasing value with the quality Sensibility to equiprobability (inclusion) Statistic Likelihood (confidence in the measure itself) Noise resistance, time stability Surprisingness, nuggets ?
  8. 8. PrincipleStatistics on data D (transactions) for each ruleR=XYInterestingness measure = i(R,D,H)Degree of satisfaction of the hypothesis H in Dindependently of U
  9. 9. Properties in the LiteratureProperties of i(XY) = f(n, nx, ny, nxy) [Piatetsky-Shapiro 1991] (strong rules): (P1) =0 if X and Y are independent (P2) increases with examples nxy (P3) decreases with premise nx (or conclusion ny)(?) [Major & Mangano 1993]: (P4) increases with nxy when confidence is constant (nxy/nx) [Freitas 1999]: (P5) asymmetry (i(XY)/=i(YX)) Small disjunctions (nuggets)[Tan et al. 2002], [Hilderman & Hamilton 2001] and [Gras et al. 2004]
  10. 10. Selected Properties Inclusion and equiprobability 0, interval of security Independence 0, interval of security Bounded maximum value Comparability, global threshold, inclusion Non linearity Noise Resistance, interval of security for independence andequiprobability Sensibility N (nuggets), dilation (likelihood) Frequency p(X)  cardinal nx Reinforcement by similar rules (contra-positive, negativerule,…)[Smyth & Goodman 1991][Kodratoff 2001][Gras et al 2001][Gras et al. 2004]
  11. 11. Interestingness Measure Classifying CriteriaThese interestigness measures can be categorized intothree classifications: objective, subjective, and semantics-based. Objective Measure: An objective measure is basedonly on the raw data. No knowledge about the user orapplication is required. Most objective measures arebased on theories in probability, statistics, orinformation theory. Conciseness, generality, reliability,peculiarity, and diversity depend only on the data andpatterns, and thus can be considered objective.
  12. 12. Interestingness Measure Classifying Criteria Subjective Measure: A subjective measure takes intoaccount both the data and the user of these data. To define asubjective measure, access to the user’s domain orbackground knowledge about the data is required. This accesscan be obtained by interacting with the user during the datamining process or by explicitly representing the user’sknowledge or expectations. In the latter case, the key issue isthe representation of the user’s knowledge, which has beenaddressed by various frameworks and procedures for datamining [Liu et al. 1997, 1999; Silberschatz and Tuzhilin 1995,1996; Sahar 1999]. Novelty and surprisingness depend on theuser of the patterns, as well as the data and patternsthemselves, and hence can be considered subjective.
  13. 13. Interestingness Measure Classifying Criteria Semantic Measure: A semantic measure considers the semantics andexplanations of the patterns. Because semantic measures involve domainknowledge from the user, some researchers consider them a special type ofsubjective measure [Yao et al. 2006]. Utility and actionability depend onthe semantics of the data, and thus can be considered semantic. Utility-based measures, where the relevant semantics are the utilities of thepatterns in the domain, are the most common type of semantic measure.To use a utility-based approach, the user must specify additionalknowledge about the domain. Unlike subjective measures, where thedomain knowledge is about the data itself and is usually represented in aformat similar to that of the discovered pattern, the domain knowledgerequired for semantic measures does not relate to the user’s knowledge orexpectations concerning the data. Instead, it represents a utility functionthat reflects the user’s goals. This function should be optimized in themined results. For example, a store manager might prefer association rulesthat relate to high-profit items over those with higher statisticalsignificance.
  14. 14. Probabilistic Interestingness MeasureRef: A Survey of Interestingness Measures for Knowledge Discovery (Ken McGarry, 2005)
  15. 15. Objective interestingness measures
  16. 16. Conciseness A pattern is concise if it contains relatively fewattribute-value pairs, while a set of patterns isconcise if it contains relatively few patterns. Aconcise pattern or set of patterns is relatively easy tounderstand and remember and thus is added moreeasily to the user’s knowledge (set of beliefs).Accordingly, much research has been conducted tofind a minimum set of patterns, using propertiessuch as monotonicity [Padmanabhan and Tuzhilin2000] and confidence invariance [Bastide et al.2000].
  17. 17. Generality/Coverage A pattern is general if it covers a relatively large subset of a dataset.Generality (or coverage) measures the comprehensiveness of a pattern, thatis, the fraction of all records in the dataset that matches the pattern. If apattern characterizes more information in the dataset, it tends to be moreinteresting [Agrawal and Srikant 1994; Webb and Brain 2002]. Frequentitemsets are the most studied general patterns in the data mining literature.An itemset is a set of items, such as some items from a grocery basket. Anitemset is frequent if its support, the fraction of records in the datasetcontaining the itemset, is above a given threshold [Agrawal and Srikant1994].The best known algorithm for finding frequent itemsets is the Apriorialgorithm [Agrawal and Srikant 1994]. Some generality measures can formthe bases for pruning strategies; for example, the support measure is usedin the Apriori algorithm as the basis for pruning itemsets. For classificationrules, Webb and Brain [2002] gave an empirical evaluation showing howgenerality affects classification results. Generality frequently coincides withconciseness because concise patterns tend to have greater coverage.
  18. 18. Reliability A pattern is reliable if the relationship described bythe pattern occurs in a high percentage of applicablecases. For example, a classification rule is reliable ifits predictions are highly accurate, and anassociation rule is reliable if it has high confidence.Many measures from probability, statistics, andinformation retrieval have been proposed to measurethe reliability of association rules [Ohsaki et al.2004; Tan et al. 2002].
  19. 19. Peculiarity A pattern is peculiar if it is far away from otherdiscovered patterns according to some distancemeasure. Peculiar patterns are generated frompeculiar data (or outliers), which are relatively few innumber and significantly different from the rest ofthe data [Knorr et al. 2000; Zhong et al. 2003].Peculiar patterns may be unknown to the user, henceinteresting.
  20. 20. Diversity A pattern is diverse if its elements differ significantly fromeach other, while a set of patterns is diverse if the patterns inthe set differ significantly from each other. Diversity is acommon factor for measuring the interestingness ofsummaries [Hilderman and Hamilton 2001]. According to asimple point of view, a summary can be considered diverse ifits probability distribution is far from the uniformdistribution. A diverse summary may be interesting becausein the absence of any relevant knowledge, a user commonlyassumes that the uniform distribution will hold in a summary.According to this reasoning, the more diverse the summary is,the more interesting it is. We are unaware of any existingresearch on using diversity to measure the interestingness ofclassification or association rules.
  21. 21. Novelty A pattern is novel to a person if he or she did not know itbefore and is not able to infer it from other known patterns.No known data mining system represents everything that auser knows, and thus, novelty cannot be measured explicitlywith reference to the user’s knowledge. Similarly, no knowndata mining system represents what the user does not know,and therefore, novelty cannot be measured explicitly withreference to the user’s ignorance. Instead, novelty is detectedby having the user either explicitly identify a pattern as novel[Sahar 1999] or notice that a pattern cannot be deduced fromand does not contradict previously discovered patterns. In thelatter case, the discovered patterns are being used as anapproximation to the user’s knowledge.
  22. 22. Surprisingness A pattern is surprising (or unexpected) if it contradicts aperson’s existing knowledge or expectations [Liu et al.1997, 1999; Silberschatz and Tuzhilin 1995, 1996]. Apattern that is an exception to a more general patternwhich has already been discovered can also beconsidered surprising [Bay and Pazzani 1999; Carvalhoand Freitas 2000]. Surprising patterns are interestingbecause they identify failings in previous knowledge andmay suggest an aspect of the data that needs furtherstudy. The difference between surprisingness and noveltyis that a novel pattern is new and not contradicted by anypattern already known to the user, while a surprisingpattern contradicts the user’s previous knowledge orexpectations.
  23. 23. Utility A pattern is of utility if its use by a personcontributes to reaching a goal. Different people mayhave divergent goals concerning the knowledge thatcan be extracted from a dataset. For example, oneperson may be interested in finding all sales withhigh profit in a transaction dataset, while anothermay be interested in finding all transactions withlarge increases in gross sales. This kind ofinterestingness is based on user-defined utilityfunctions in addition to the raw data [Chan et al.2003; Lu et al. 2001; Yao et al. 2004; Yao andHamilton 2006].
  24. 24. Actionability A pattern is actionable (or applicable) in somedomain if it enables decision making about futureactions in this domain [Ling et al. 2002;Wang et al.2002]. Actionability is sometimes associated with apattern selection strategy. So far, no general methodfor measuring actionability has been devised.Existing measures depend on the applications. Forexample, Ling et al. [2002], measured actionabilityas the cost of changing the customer’s currentcondition to match the objectives, whereas Wang etal. [2002], measured actionability as the profit thatan association rule can bring.
  25. 25. Objective Interestingness Measures Rule: XY Support: P(X∩Y) Confidence: P(Y|X) Lift(X,Y): P(X∪Y)/P(X)P(Y)
  26. 26. Objective interestingness measures Problems: nappies⇒babyfood nappies⇒beer We can reasonably expect that the sales of babyfood and nappies occur together frequently
  27. 27. Limits of SupportSupport: supp(XY) = freq(XUY) Generality of the rule Minimum support threshold (ex: 10%) Reduce the complexity Specific rule (low support) Valid rule (high confidence) High potential of novelty/surprise
  28. 28. Limits of ConfidenceConfidence: conf(XY) = P(Y|X) = freq(XUY)/freq(X) Validity/logical aspect of the rule (inclusion) Minimal confidence threshold (ex: 90%) Reduces the amount of extracted rules Interestingness /= validity No detection of independence Independence: X and Y are independent: P(Y|X) = P(Y) If P(Y) is high => nonsense rule with high supportEx: Couches  beer (supp=20%, conf=90%) if supp(beer)=90%[Guillaume et al. 1998], [Lallich et al. 2004]
  29. 29. Limits of the Pair Support-ConfidenceIn practice: High support threshold (10%) High confidence threshold (90%) Valid and general rules Common Sense but not novelty Efficient measures but insufficient to capture quality
  30. 30. Subjective interestingness measures Unexpected (What’s interesting?): Same condition, but different consequences Different conditions, but same consequence
  31. 31. Subjective interestingness measuresGeneral impressiongi(<S1, …, Sm>) [support, confidence]↓Reasonably precise conceptrpc(<S1, …, Sm → V1, …, Vg>) [support, confidence]↓Precise knowledgepk(<S1, …, Sm → V1, …, Vg>) [support, confidence]Analyzing the Subjective Interestingness of Association RulesBing Liu et al., 2000
  32. 32. Subjective interestingness measures Problems: Knowledge granularity Domain expert required? Vague expression
  33. 33. Objective Measures: Examples of Quality CriteriaCriteria of interestingness [Hussein 2000]: Objective: Generality : (ex: Support) Validity: (ex: Confidence) Reliability: (ex: High generality and validity) Subjective: Common Sense: reliable + known yet Actionability : utility for decision Novelty: previously unknown Surprise (Unexpectedness): contradiction ?
  34. 34. Association RulesAssociation rules [Agrawal et al. 1993]: Market-basket analysis Non supervised learning Algorithms + 2 measures (support and confidence)Problems: Enormous amount of rules (rough rules) Few semantic on support and confidence measures Need to help the user select the best rules
  35. 35. Association RulesSolutions: Redundancy reduction Structuring (classes, close rules) Improve quality measures Interactive decision aid (rule mining)
  36. 36. Association RulesInput : data p Boolean attributes (V0, V1, … Vp) (columns) n transactions (rows)Output : Association Rules: Implicative tendencies : X  Y X and Y (itemsets) ex: V0^V4^V8  V1 Negative examples 2 measures: Support: supp(XY) = freq(XUY) Confidence: conf(XY) = P(Y|X) = freq(XUY)/freq(X) Algorithm properties (monotony)Ex: Couches  beer (supp=20%, conf=90%)(NB: max nb of rules 3p)
  37. 37. Subjective Measures: CriteriaUser-oriented measures (U)Quality : interestingness: Unexpectedness [Silberschatz 1996] Unknown or contradictory rule Actionability (Usefulness) [Piatesky-shapiro 1994] Usefulness for decision making, gain Anticipation [Roddick 2001] Prediction on temporal dimension
  38. 38. Subjective Measures : CriteriaUnexpectedness and actionability: Unexpected + useful = high interestingness Expected + non-useful = ? Expected + useful = reinforcement Unexpected + non-useful = ?
  39. 39. Subjective Measures: Other Subjective Measures Projected Savings (KEFIR system’s interestingness)[Matheus & Piatetsky-Shapiro 1994] Fuzzy Matching Interestingness Measure [Lie et al. 1996] General Impression [Liu et al. 1997] Logical Contradiction [Padmanabhan & Tuzhilin’s 1997] Misclassification Costs [Frietas 1999] Vague Feelings (Fuzzy General Impressions) [Liu et al.2000] Anticipation [Roddick and rice 2001] Interestingness [Shekar & Natarajan’s 2001]
  40. 40. Subjective Measures: ClassificationInterestingness Measure Year Application Foundation Scope SubjectiveAspectsUser’s KnowledgeRepresentation1 Matheus and Piatetsky-Shapiro’s Projected Savings1994 Summaries Utilitarian SingleRuleUnexpectedness Pattern Deviation2 Klemettinen et al. RuleTemplates1994 AssociationRulesSyntactic SingleRuleUnexpectedness& ActionabilityRule Templates3 Silbershatz and Tuzhilin’sInterestingness1995 FormatIndependentProbabilistic Rule Set Unexpectedness Hard & Soft Beliefs4 Liu et al. Fuzzy MatchingInterestingness Measure1996 ClassificationrulesSyntacticDistanceSingleRuleUnexpectedness Fuzzy Rules5 Liu et al. GeneralImpressions1997 ClassificationRulesSyntactic SingleRuleUnexpectedness GI, RPK6 Padmanabhan and TuzhilinLogical Contradiction1997 AssociationRulesLogical, Statistic SingleRuleUnexpectedness Beliefs XY7 Freitas’ Attributes Costs 1999 AssociationRulesUtilitarian SingleRuleActionability Costs Values8 Freitas’ MisclassificationCosts1999 AssociationrulesUtilitarian Single rule Actionability Costs Values9 Liu et al. Vague Feelings(Fuzzy GeneralImpressions)2000 GeneralizedAssociationRulesSyntactic SingleRuleUnexpectedness GI, RPK, PK10 Roddick and Rice’sAnticipation2001 FormatIndependentProbabilistic SingleRuleTemporalDimensionProbability Graph11 Shekar and Natarajan’sInterestingness2002 AssociationRulesDistance SingleRuleUnexpectedness Fuzzy-graph basedtaxonomy
  41. 41. List Of Interestingness Measures (cont) Monodimensional e+, e- Support [Agrawal et al. 1996] Ralambrodrainy [Ralambrodrainy, 1991] Bidimensional - Inclusion Descriptive-Confirm [Yves Kodratoff, 1999] Sebag et Schoenauer [Sebag, Schoenauer, 1991] Examples neg examples ratio (*) Bidimensional – Inclusion – Conditional Probability Confidence [Agrawal et al. 1996] Wang index [Wang et al., 1988] Laplace (*) Bidimensional – Analogous Rules Descriptive Confirmed-Confidence [Yves Kodratoff, 1999] (*)
  42. 42. List Of Interestingness Measures (cont.) Tridimensional – Analogous Rules Causal Support [Kodratoff, 1999] Causal Confidence [Kodratoff, 1999] (*) Causal Confirmed-Confidence [Kodratoff, 1999] Least contradiction [Aze & Kodratoff 2004] (*) Tridimensional – Linear - Independent Pavillon index [Pavillon, 1991] Rule Interest [Piatetsky-Shapiro, 1991] (*) Pearl index [Pearl, 1988], [Acid et al., 1991] [Gammerman, Luo, 1991] Correlation [Pearson 1996] (*) Loevinger index [Loevinger, 1947] (*) Certainty factor [Tan & Kumar 2000] Rate of connection[Bernard et Charron 1996] Interest factor [Brin et al., 1997] Top spin(*) Cosine [Tan & Kumar 2000] (*) Kappa [Tan & Kumar 2000]
  43. 43. List Of Interestingness Measures (cont.) Tridimensional – Nonlinear – Independent Chi squared distance Logarithmic lift [Church & Hanks, 1990] (*) Predictive association [Tan & Kumar 2000] (Goodman & Kruskal) Conviction [Brin et al., 1997b] Odd’s ratio [Tan & Kumar 2000] Yule’Q [Tan & Kumar 2000] Yule’s Y [Tan & Kumar 2000] Jaccard [Tan & Kumar 2000] Klosgen [Tan & Kumar 2000] Interestingness [Gray & Orlowska, 1998] Mutual information ratio (Uncertainty) [Tan et al., 2002] J-measure [Smyth & Goodman 1991] [Goodman & Kruskal 1959] (*) Gini [Tan et al., 2002] General measure of rule interestingness [Jaroszewicz & Simovici, 2001] (*)
  44. 44. List Of Interestingness Measures (cont.) Quadridimensional – Linear – independent Lerman index of similarity[Lerman, 1981] Index of Involvement[Gras, 1996] Quadridimensional – likeliness (conditional probability?) ofdependence Probability of error of Chi2 (*) Intensity of Involvement [Gras, 1996] (*) Quadridimensional – Inclusion – dependent – analogous rules Entropic intensity of Involvement [Gras, 1996] (*) TIC [Blanchard et al., 2004] (*) Others Surprisingness (*) [Freitas, 1998] + rules of exception [Duval et al. 2004] + rule distance, similarity [Dong & Li 1998]
  45. 45. Belief Based Interestingness MeasureUsing a belief system is also the approach adopted byPadmanabhan and Tuzhilin for discovering exception rules thatcontradict belief rules.Consider a belief X → Y and a rule A → B, where both X and Aare conjunctions of atomic conditions and both Y and B aresingle atomic conditions on boolean attributes.A rule A → B is unexpected with respect to the belief X → Y onthe dataset D if the following conditions hold: 1. B and Y logically contradict each other. 2. X ∧ A holds on a statistically large subset of tuples in D. 3. A,X → B holds and since B and Y logically contradict eachother, it follows that A,X → ¬Y also holds.
  46. 46. Unexpectedness and the InterestingnessMeasuresSilberschatz and Tuzhilin used the term unexpectedness in thecontext of interestingness measures for patterns evaluation.They classify such measures into objective (data-driven) andsubjective (user-driven) measures. According to them, from thesubjective point of view, a pattern is interesting if it is: Actionable: the end-user can act on it to her/his advantage. Unexpected: the end-user is surprised by such findings.As pointed out by the authors, the actionability is subtle anddifficult to capture; they propose rather to capture it throughunexpectedness, arguing that unexpected patterns are those thatlead the expert of the domain to make some actions.
  47. 47. Example-unexpeted patternsunexpected patterns:
  48. 48. Example-actionable patternsAction:actionable patterns:
  49. 49. Interestingness Measures and Bayesian BeliefNetworkIn the framework presented by Silberschatz and Tuzhilin, evaluatingthe unexpectedness of a discovered pattern is done according to aBelief System that the user has: the more the pattern disagrees with abelief system, the more unexpected it is.There are two kinds of beliefs. On one hand, hard beliefs are thosebeliefs that are always true and that cannot be changed. In this case,detecting a contradicting pattern means that something is wrong withthe data used to find this pattern. On the other hand, soft beliefs arethose that the user is willing to change with a new evidence. Each softbelief is assigned with a degree specifying how the user is confident init. In their work, the authors proposed five approaches to affect suchdegrees: Bayesian, Dempster-Shafer, Frequency, Cyc’s and Statisticalapproaches.The authors (Silberschatz and Tuzhilin) claim that theBayesian one is the most appropriate for defining the degreeof beliefs even if any other approach they have defined canbe used.
  50. 50. Conclusion and Future Work Quality is a multidimensional concept Subjective (expert opionion) Interest = changes with the knowledge of the decision-maker Extract knowledge / objective decision-maker Objective (data and rules) Interest = on the Hypothetical Data: Inclusion, Independence,Imbalance, nuggets, robustness ... What is a good index? (ingredients of quality) The “hybrid” interestingness Such as paradox detection Detecting change over time Bayesian belief networks
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