msc bd 5002/it 5210: knowledge discovery and data...

11

MSC BD 5002/IT 5210: Knowledge Discovery and Data

Mining

Acknowledgement: Slides modified by Dr. Lei Chen based on the slides provided by Jiawei Han, Micheline Kamber, and

Jian Pei©2012 Han, Kamber & Pei. All rights reserved.

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Chapter 3: Data Preprocessing

n Data Preprocessing: An Overview

n Data Quality

n Major Tasks in Data Preprocessing

n Data Cleaning

n Data Integration

n Data Reduction

n Data Transformation and Data Discretization

n Summary

3

Data Quality: Why Preprocess the Data?

n Measures for data quality: A multidimensional view

n Accuracy: correct or wrong, accurate or not

n Completeness: not recorded, unavailable, …

n Consistency: some modified but some not, dangling, …

n Timeliness: timely update?

n Believability: how trustable the data are correct?

n Interpretability: how easily the data can be understood?

4

Major Tasks in Data Preprocessing

n Data cleaningn Fill in missing values, smooth noisy data, identify or remove

outliers, and resolve inconsistenciesn Data integration

n Integration of multiple databases, data cubes, or filesn Data reduction

n Dimensionality reductionn Numerosity reductionn Data compression

n Data transformation and data discretizationn Normalization n Concept hierarchy generation

55



n Data Quality


n Data Cleaning

n Data Integration

n Data Reduction


n Summary

6

Data Cleaning

n Data in the Real World Is Dirty: Lots of potentially incorrect data, e.g., instrument faulty, human or computer error, transmission errorn incomplete: lacking attribute values, lacking certain attributes of

interest, or containing only aggregate datan e.g., Occupation=“ ” (missing data)

n noisy: containing noise, errors, or outliersn e.g., Salary=“−10” (an error)

n inconsistent: containing discrepancies in codes or names, e.g.,n Age=“42”, Birthday=“03/07/2010”n Was rating “1, 2, 3”, now rating “A, B, C”n discrepancy between duplicate records

n Intentional (e.g., disguised missing data)n Jan. 1 as everyone’s birthday?

7

Incomplete (Missing) Data

n Data is not always availablen E.g., many tuples have no recorded value for several

attributes, such as customer income in sales datan Missing data may be due to

n equipment malfunctionn inconsistent with other recorded data and thus deletedn data not entered due to misunderstandingn certain data may not be considered important at the

time of entryn not register history or changes of the data

n Missing data may need to be inferred

8

How to Handle Missing Data?

n Ignore the tuple: usually done when class label is missing (when doing classification)—not effective when the % of missing values per attribute varies considerably

n Fill in the missing value manually: tedious + infeasible?n Fill in it automatically with

n a global constant : e.g., “unknown”, a new class?! n the attribute meann the attribute mean for all samples belonging to the

same class: smartern the most probable value: inference-based such as

Bayesian formula or decision tree

9

Noisy Data

n Noise: random error or variance in a measured variablen Incorrect attribute values may be due to

n faulty data collection instrumentsn data entry problemsn data transmission problemsn technology limitationn inconsistency in naming convention

n Other data problems which require data cleaningn duplicate recordsn incomplete datan inconsistent data

10

How to Handle Noisy Data?

n Binningn first sort data and partition into (equal-frequency) binsn then one can smooth by bin means, smooth by bin

median, smooth by bin boundaries, etc.n Regression

n smooth by fitting the data into regression functionsn Clustering

n detect and remove outliersn Combined computer and human inspection

n detect suspicious values and check by human (e.g., deal with possible outliers)

1111



n Data Quality


n Data Cleaning

n Data Integration

n Data Reduction


n Summary

1212

Data Integration

n Data integration: n Combines data from multiple sources into a coherent store

n Schema integration: e.g., A.cust-id º B.cust-#n Integrate metadata from different sources

n Entity identification problem: n Identify real world entities from multiple data sources, e.g., Bill

Clinton = William Clintonn Detecting and resolving data value conflicts

n For the same real world entity, attribute values from different sources are different

n Possible reasons: different representations, different scales, e.g., metric vs. British units

EntityResolutionforBigData

Lise GetoorUniversityofMaryland

CollegePark,MD

Ashwin MachanavajjhalaDukeUniversityDurham,NC

http://www.cs.umd.edu/~getoor/Tutorials/ER_KDD2013.pdfhttp://goo.gl/7tKiiL

13

WhatisEntityResolution?Problemofidentifyingandlinking/groupingdifferent

manifestationsofthesamerealworldobject.

Examplesofmanifestationsandobjects:• Differentwaysofaddressing(names,emailaddresses,FaceBook

accounts)thesamepersonintext.• Webpageswithdifferingdescriptionsofthesamebusiness.• Differentphotosofthesameobject.• …

14

WhatisEntityResolution?

15

Ironically,EntityResolutionhasmanyduplicatenames

Doubles

DuplicatedetectionRecordlinkage

Deduplication

ObjectidentificationObjectconsolidation

Coreference resolution

Entityclustering

Referencereconciliation

ReferencematchingHouseholding

Householdmatching

Fuzzymatch

Approximatematch

Merge/purge

Hardeningsoftdatabases

Identityuncertainty

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ERMotivatingExamples• LinkingCensusRecords• PublicHealth• Websearch• Comparisonshopping• Counter-terrorism• KnowledgeGraphConstruction• …

17

TraditionalChallengesinER• Name/Attributeambiguity

ThomasCruise

MichaelJordan

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TraditionalChallengesinER• Name/Attributeambiguity• Errorsduetodataentry

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TraditionalChallengesinER• Name/Attributeambiguity• Errorsduetodataentry• MissingValues

[Gilletal;Univ ofOxford2003]20

TraditionalChallengesinER• Name/Attributeambiguity• Errorsduetodataentry• MissingValues• ChangingAttributes

• Dataformatting

• Abbreviations/DataTruncation21

Big-DataERChallenges

22

Big-DataERChallenges• LargerandmoreDatasets

– Needefficientparalleltechniques

• MoreHeterogeneity– Unstructured,UncleanandIncompletedata.Diversedatatypes.– Nolongerjustmatchingnameswithnames,butAmazonprofileswith

browsinghistoryonGoogleandfriendsnetworkinFacebook.

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Big-DataERChallenges• LargerandmoreDatasets

– Needefficientparalleltechniques

• MoreHeterogeneity– Unstructured,UncleanandIncompletedata.Diversedatatypes.

• Morelinked– Needtoinferrelationshipsinadditionto“equality”

• Multi-Relational– Dealwithstructureofentities(AreWalmart andWalmart Pharmacy

thesame?)

• Multi-domain– Customizablemethodsthatspanacrossdomains

• Multipleapplications(websearchversuscomparisonshopping)– Servediverseapplicationwithdifferentaccuracyrequirements

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ERReferences• Book/SurveyArticles

– DataQualityandRecordLinkageTechniques[T.Herzog,F.Scheuren,W.Winkler,Springer,’07]

– DuplicateRecordDetection[A.Elmagrid,P.Ipeirotis,V.Verykios,TKDE‘07]– AnIntroductiontoDuplicateDetection[F.Naumann,M.Herschel,M&P

synthesislectures2010]– EvaluationofEntityResolutionApproachedonReal-worldMatchProblems

[H.Kopke,A.Thor,E.Rahm,PVLDB2010]– DataMatching[P.Christen,Springer2012]

• Tutorials– RecordLinkage:SimilaritymeasuresandAlgorithms

[N.Koudas,S.Sarawagi,D.Srivatsava SIGMOD‘06]– Datafusion--Resolvingdataconflictsforintegration

[X.Dong,F.Naumann VLDB‘09]– EntityResolution:Theory,PracticeandOpenChallenges

http://goo.gl/Ui38o [L.Getoor,A.Machanavajjhala AAAI‘12]25

Notation• R:setofrecords/mentions(typed)• H:setofrelations/hyperedges (typed)• M:setofmatches(recordpairsthatcorrespondtosameentity)

• N:setofnon-matches(recordpairscorrespondingtodifferententities)• E:setofentities• L:setoflinks

• True(Mtrue,Ntrue,Etrue,Ltrue):accordingtorealworldvs Predicted(Mpred,Npred,Epred,Lpred ):byalgorithm

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RelationshipbetweenMtrue andMpred

• Mtrue (SameAs ,Equivalence)• Mpred (Similarrepresentationsandsimilarattributes)

MtrueRxR Mpred

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Metrics• Pairwise metrics

– Precision/Recall,F1– #ofpredictedmatchingpairs

• Clusterlevelmetrics– purity,completeness,complexity– Precision/Recall/F1:Cluster-level,closestcluster,MUC,B3 ,RandIndex

– Generalizedmergedistance[Menestrina etal,PVLDB10]

• Littleworkthatevaluatescorrectpredictionoflinks

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TypicalAssumptionsMade• Eachrecord/mentionisassociatedwithasinglerealworldentity.

• Inrecordlinkage,noduplicatesinthesamesource• Iftworecords/mentionsareidentical,thentheyaretruematches

(,)ε Mtrue29

ERversusClassificationFindingmatchesvs non-matchesisaclassificationproblem

• Imbalanced:typicallyO(R)matches,O(R^2)non-matches

• Instancesarepairsofrecords.PairsarenotIID

(,)ε Mtrue

(,)ε Mtrue

(,)ε MtrueAND

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ERvs (Multi-relational)ClusteringComputingentitiesfromrecordsisaclusteringproblem

• Intypicalclusteringalgorithms(k-means,LDA,etc.)numberofclustersisaconstantorsublinearinR.

• InER:numberofclustersislinearinR,andaverageclustersizeisaconstant.Significantfractionofclustersaresingletons.

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MOTIVATING EXAMPLE: BIBLIOGRAPHIC DOMAIN

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En##es & Rela#ons in Bibliographic Domain

Wrote

Paper Title # of Authors Topic Word1 Word 2 … WordN

Cites

Author Name Research Area

Author Mention NameString

Paper Mention TitleString

Institution Name

Institute Mention NameString

Venue Name

Venue Mention NameString

WorksAt

AppearsIn

: co-‐occurrence rela#onships : resolu#on rela#onships

: en#ty rela#onships

47

DATA PREPARATION & MATCH FEATURES

PART 2-‐a

48

Normaliza#on •  Schema normaliza#on

–  Schema Matching – e.g., contact number and phone number –  Compound a]ributes – full address vs str,city,state,zip –  Nested a]ributes

•  List of features in one dataset (air condi#oning, parking) vs each feature a boolean a]ribute

–  Set valued a]ributes •  Set of phones vs primary/secondary phone

–  Record segmenta#on from text

•  Data normaliza#on –  OOen convert to all lower/all upper; remove whitespace –  detec#ng and correc#ng values that contain known typographical errors or

varia#ons, –  expanding abbrevia#ons and replacing them with standard forms; replacing

nicknames with their proper name forms –  Usually done based on dic#onaries (e.g., commercial dic#onaries, postal addresses,

etc.) 49

Normaliza#on •  Schema normaliza#on

–  Schema Matching – e.g., contact number and phone number –  Compound a]ributes – full address vs str,city,state,zip –  Nested a]ributes

•  List of features in one dataset (air condi#oning, parking) vs each feature a boolean a]ribute

–  Set valued a]ributes •  Set of phones vs primary/secondary phone

–  Record segmenta#on from text

•  Data normaliza#on –  OOen convert to all lower/all upper; remove whitespace –  detec#ng and correc#ng values that contain known typographical errors or

varia#ons, –  expanding abbrevia#ons and replacing them with standard forms; replacing

nicknames with their proper name forms –  Usually done based on dic#onaries (e.g., commercial dic#onaries, postal addresses,

etc.) 50

Matching Features •  For two references x and y, compute a “comparison” vector of

similarity scores of component a]ribute. –  [ 1st-‐author-‐match-‐score,

paper-‐match-‐score, venue-‐match-‐score, year-‐match-‐score, …. ]

•  Similarity scores –  Boolean (match or not-‐match) –  Real values based on distance func#ons

52

Summary of Matching Features

•  Equality on a boolean predicate •  Edit distance

–  Levenstein, Smith-‐Waterman, Affine

•  Set similarity –  Jaccard, Dice

•  Vector Based –  Cosine similarity, TFIDF

•  Useful packages: –  SecondString, h]p://secondstring.sourceforge.net/ –  Simmetrics: h]p://sourceforge.net/projects/simmetrics/ –  LingPipe, h]p://alias-‐i.com/lingpipe/index.html

•  Alignment-‐based or Two-‐#ered –  Jaro-‐Winkler, SoO-‐TFIDF, Monge-‐Elkan

•  Phone#c Similarity –  Soundex

•  Transla#on-‐based •  Numeric distance between values •  Domain-‐specific

Good for Names

Good for Text like reviews/ tweets Useful for

abbreviaKons, alternate names.

Handle Typographical errors

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Rela#onal Matching Features •  Rela#onal features are oOen set-‐based

–  Set of coauthors for a paper –  Set of ci#es in a country –  Set of products manufactured by manufacturer

•  Can use set similarity func#ons men#oned earlier –  Common Neighbors: Intersec#on size –  Jaccard’s Coefficient: Normalize by union size –  Adar Coefficient: Weighted set similarity

•  Can reason about similarity in sets of values –  Average or Max –  Other aggregates

54

PAIRWISE MATCHING PART 2-‐b

55

Pairwise Match Score Problem: Given a vector of component-‐wise similari#es for a pair of

records (x,y), compute P(x and y match). Solu#ons: 1.  Weighted sum or average of component-‐wise similarity scores.

Threshold determines match or non-‐match. –  0.5*1st-‐author-‐match-‐score + 0.2*venue-‐match-‐score + 0.3*paper-‐match-‐score. –  Hard to pick weights.

•  Match on last name match more predic<ve than login name. •  Match on “Smith” less predic<ve than match on “Getoor” or “Machanavajjhala”.

–  Hard to tune a threshold.

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Pairwise Match Score Problem: Given a vector of component-‐wise similari#es for a pair of

records (x,y), compute P(x and y match). Solu#ons: 1.  Weighted sum or average of component-‐wise similarity scores.

Threshold determines match or non-‐match. 2.  Formulate rules about what cons#tutes a match.

–  (1st-‐author-‐match-‐score > 0.7 AND venue-‐match-‐score > 0.8) OR (paper-‐match-‐score > 0.9 AND venue-‐match-‐score > 0.9)

–  Manually formula#ng the right set of rules is hard.

57

Fellegi & Sunter Model [FS, Science ‘69] •  Record pair: r = (x,y) in A x B

•  γ = γ(r) is a comparison vector –  E.g., γ = [“Is x.name = y.name?”, “Is x.address = y.address?” …] –  Assume binary vector for simplicity

•  M : set of matching pairs of records •  U : set of non-‐matching pairs of records

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•  r = (x,y) is record pair, γ is comparison vector, M matches, U non-‐matches

•  Linkage decisions are based on:

•  Linkage Rule: L(tl, tu)

Fellegi & Sunter Model [FS, Science ‘69]

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tl tU

MATCH NON-‐MATCH UNCERTAIN

Error due to a Linkage Rule •  Type I Error: r = (x,y) in U, but the linkage rule calls it a match

•  Type II Error: r = (x,y) in M, but the linkage rule calls it a non-‐match

61

Op#mal Linkage Rule •  L* = (tl*, tu*) is an op#mal decision rule for comparison space Γ

with error bounds μ and λ, if

–  L* meets the type I and type II requirements

–  L* has the least condi#onal probabili#es of not making a decision. That is for all other decision rules L (with error bounds μ and λ),

62

Finding the Op#mal Linkage Rule •  Suppose there are N comparison vectors •  Sort them in decreasing order of m(γ) / u(γ)

•  Pick the largest n and n’ such that:

63

Using Fellegi Sunter in Prac#ce •  Γ is usually high dimensional (compu#ng m(γ) and u(γ) is

inefficient) –  Use condi#onal independence of features in γ given match or non-‐match –  Naïve Bayes assump#on

•  Compu#ng P(γ | r ε M) requires some knowledge of matches. –  Supervised learning (assume a training set is provided) –  EM-‐based techniques can used to learn the parameters jointly while

iden#fying matches.

64

ML Pairwise Approaches •  Supervised machine learning algorithms

–  Decision trees •  [Cochinwala et al, IS01]

–  Support vector machines •  [Bilenko & Mooney, KDD03]; [Christen, KDD08]

–  Ensembles of classifiers •  [Chen et al., SIGMOD09]

–  Condi#onal Random Fields (CRF) •  [Gupta & Sarawagi, VLDB09]

–  … and many others.

•  Issues: –  Training set generaKon –  Imbalanced classes – many more nega#ves than posi#ves (even aOer

elimina#ng obvious non-‐matches … using Blocking) –  Misclassifica#on cost

65

Crea#ng a Training Set is a key issue •  Construc#ng a training set is hard – since most pairs of records are “easy non-‐matches”. –  100 records from 100 ci#es. –  Only 106 pairs out of total 108 (1%) come from the same city

•  Some pairs are hard to judge even by humans –  Inherently ambiguous

•  E.g., Paris Hilton (person or business) – Missing a]ributes

•  Starbucks, Toronto vs Starbucks, Queen Street ,Toronto

66

Avoiding Training Set Genera#on •  Unsupervised / Semi-‐supervised Techniques

–  EM based techniques to learn parameters •  [Winkler ‘06, Herzog et al ’07]

–  Genera#ve Models •  [Ravikumar & Cohen, UAI04]

•  Ac#ve Learning –  Commi]ee of Classifiers

•  [Sarawagi et al KDD ’00, Tajeda et al IS ‘01] –  Provably op#mizing precision/recall

•  [Arasu et al SIGMOD ‘10, Bellare et al KDD ‘12] –  Crowdsourcing

•  [Wang et al VLDB ‘12, Marcus et al VLDB ’12, …]

67

Commi]ee of Classifiers [Tejada et al, IS ‘01]

68

Ac#ve Learning with Provable Guarantees

•  Most ac#ve learning techniques minimize 0-‐1 loss [Beygelzimer et al NIPS 2010].

•  However, ER is very imbalanced: –  Number of non-‐matches > 100 * number of matches. –  Classifying all pairs as “non-‐matches” has low 0-‐1 loss (< 1%).

•  Hence, need ac#ve learning techniques that maximize precision/recall.

69


•  Monotonicity of Precision [Arasu et al SIGMOD ‘10]

There is a larger fracKon of matches in C1 than in C2.

Algorithm searches for the opKmal classifier using binary search on each dimension

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[Bellare et al KDD ‘12]

1.  Precision Constrained à Weighted 0-‐1 Loss Problem (using a Lagrange Mul#plier λ).

2.  Given a fixed value for λ, weighted 0-‐1 Loss can be op#mized by (one call to) a blackbox ac#ve learning classifier.

3.  Right value of λ is computed by searching over all op#mal classifiers. –  Classifiers are embedded in a 2-‐d plane (precision/recall) –  Search is along the convex hull of the embedded classifiers

O (log2 n) calls to a blackbox 0-‐1 loss acKve learning algorithm.

ExponenKally smaller label complexity than [Arasu et al SIGMOD ‘10] (in the worst case).

71

Crowdsourcing •  Growing interest in integra#ng human computa#on in declara#ve

workflow engines. –  ER is an important problem (e.g., for evalua#ng fuzzy joins) –  [Wang et al VLDB ‘12, Marcus et al VLDB ’12, …]

•  Opportunity: u#lize crowdsourcing for crea#ng training sets, or for ac#ve learning.

•  Key open issue: Handling errors in human judgments –  In an experiment on Amazon Mechanical Turk:

•  Pairwise matching judgment, each given to 5 different people –  Majority of workers agreed on truth on only 90% of pairwise judgments.

72

Summary of Single-‐En#ty ER Algorithms

•  Many algorithms for independent classifica#on of pairs of records as match/non-‐match

•  ML based classifica#on & Fellegi-‐Sunter –  Pro: Advanced state of the art –  Con: Building high fidelity training sets is a hard problem

•  Ac#ve Learning & Crowdsourcing for ER are ac#ve areas of research.

73

3232

Handling Redundancy in Data Integration

n Redundant data occur often when integration of multiple databasesn Object identification: The same attribute or object

may have different names in different databasesn Derivable data: One attribute may be a “derived”

attribute in another table, e.g., annual revenuen Redundant attributes may be able to be detected by

correlation analysis and covariance analysisn Careful integration of the data from multiple sources may

help reduce/avoid redundancies and inconsistencies and improve mining speed and quality

33

Correlation Analysis (Nominal Data)

n Χ2 (chi-square) test

n The larger the Χ2 value, the more likely the variables are related

n The cells that contribute the most to the Χ2 value are those whose actual count is very different from the expected count

n Correlation does not imply causalityn # of hospitals and # of car-theft in a city are correlatedn Both are causally linked to the third variable: population

å -=

ExpectedExpectedObserved 2

2 )(c

34

Chi-Square Calculation: An Example

n Χ2 (chi-square) calculation (numbers in parenthesis are expected counts calculated based on the data distribution in the two categories)

n It shows that like_science_fiction and play_chess are correlated in the group

93.507840

)8401000(360

)360200(210

)21050(90

)90250( 22222 =

-+

-+

-+

-=c

Play chess Not play chess Sum (row)Like science fiction 250(90) 200(360) 450

Not like science fiction 50(210) 1000(840) 1050

Sum(col.) 300 1200 1500

35

Correlation Analysis (Numeric Data)

n Correlation coefficient (also called Pearson’s product moment coefficient)

where n is the number of tuples, and are the respective means of A and B, σA and σB are the respective standard deviation of A and B, and Σ(aibi) is the sum of the AB cross-product.

n If rA,B > 0, A and B are positively correlated (A’s values increase as B’s). The higher, the stronger correlation.

n rA,B = 0: independent; rAB < 0: negatively correlated

BA

n

i ii

BA

n

i iiBA n

BAnban

BbAar

ssss )1()(

)1())((

11, -

-=

-

--= åå ==

A B

36

Visually Evaluating Correlation

Scatter plots showing the similarity from –1 to 1.

37

Correlation (viewed as linear relationship)n Correlation measures the linear relationship

between objectsn To compute correlation, we standardize data

objects, A and B, and then take their dot product

)(/))((' AstdAmeanaa kk -=

)(/))((' BstdBmeanbb kk -=

''),( BABAncorrelatio •=

38

Covariance (Numeric Data)

n Covariance is similar to correlation

where n is the number of tuples, and are the respective mean or expected values of A and B, σA and σB are the respective standard deviation of A and B.

n Positive covariance: If CovA,B > 0, then A and B both tend to be larger than their expected values.

n Negative covariance: If CovA,B < 0 then if A is larger than its expected value, B is likely to be smaller than its expected value.

n Independence: CovA,B = 0 but the converse is not true:n Some pairs of random variables may have a covariance of 0 but are not

independent. Only under some additional assumptions (e.g., the data follow multivariate normal distributions) does a covariance of 0 imply independence

A B

Correlation coefficient:

Co-Variance: An Example

n It can be simplified in computation as

n Suppose two stocks A and B have the following values in one week: (2, 5), (3, 8), (5, 10), (4, 11), (6, 14).

n Question: If the stocks are affected by the same industry trends, will their prices rise or fall together?

n E(A) = (2 + 3 + 5 + 4 + 6)/ 5 = 20/5 = 4

n E(B) = (5 + 8 + 10 + 11 + 14) /5 = 48/5 = 9.6

n Cov(A,B) = (2×5+3×8+5×10+4×11+6×14)/5 − 4 × 9.6 = 4

n Thus, A and B rise together since Cov(A, B) > 0.

4040



n Data Quality


n Data Cleaning

n Data Integration

n Data Reduction


n Summary

41

Data Reduction Strategies

n Data reduction: Obtain a reduced representation of the data set that is much smaller in volume but yet produces the same (or almost the same) analytical results

n Why data reduction? — A database/data warehouse may store terabytes of data. Complex data analysis may take a very long time to run on the complete data set.

n Data reduction strategiesn Dimensionality reduction, e.g., remove unimportant attributes

n Wavelet transformsn Principal Components Analysis (PCA)n Feature subset selection, feature creation

n Numerosity reduction (some simply call it: Data Reduction)n Regression and Log-Linear Modelsn Histograms, clustering, samplingn Data cube aggregation

n Data compression

42

Data Reduction 1: Dimensionality Reductionn Curse of dimensionality

n When dimensionality increases, data becomes increasingly sparsen Density and distance between points, which is critical to clustering, outlier

analysis, becomes less meaningfuln The possible combinations of subspaces will grow exponentially

n Dimensionality reductionn Avoid the curse of dimensionalityn Help eliminate irrelevant features and reduce noisen Reduce time and space required in data miningn Allow easier visualization

n Dimensionality reduction techniquesn Wavelet transformsn Principal Component Analysisn Supervised and nonlinear techniques (e.g., feature selection)

43

Mapping Data to a New Space

Two Sine Waves Two Sine Waves + Noise Frequency

n Fourier transformn Wavelet transform

44

What Is Wavelet Transform?

n Decomposes a signal into different frequency subbandsn Applicable to n-

dimensional signalsn Data are transformed to

preserve relative distance between objects at different levels of resolution

n Allow natural clusters to become more distinguishable

n Used for image compression

45

Wavelet Transformation

n Discrete wavelet transform (DWT) for linear signal processing, multi-resolution analysis

n Compressed approximation: store only a small fraction of the strongest of the wavelet coefficients

n Similar to discrete Fourier transform (DFT), but better lossy compression, localized in space

n Method:n Length, L, must be an integer power of 2 (padding with 0’s, when

necessary)n Each transform has 2 functions: smoothing, differencen Applies to pairs of data, resulting in two set of data of length L/2n Applies two functions recursively, until reaches the desired length

Haar2 Daubechie4

Introduction toWavelets

http://www.math.tau.ac.il/~turkel/notes/wavelets.pdf

Background

Fourier Analysisn Breaks down a signal into constituent

sinusoids of different frequencies

In other words: Transform the view of the signal from time-base to frequency-base.

What’s wrong with Fourier?

n FT is only the function of , frequency.n By using Fourier Transform , we loose the time information :

WHEN did a particular event take place ?n Fourier Transform can not locate drift, trends, abrupt

changes, beginning and ends of events, etc.n Fourier Transform is unable to pick out local frequency

content. It has a “hard time” representing functions that are oscillatory or discontinuous (Gibbs phenomena)

n Frequency domain Jn Temporal domain L

F f t e d tj t( ) ( ) ,wp

w= -

-¥

¥

ò12

w

Wavelet Analysis

What is Wavelet Analysis ?

n And…what is a wavelet…?

n A wavelet is a waveform of effectively limited duration that has an average value of zero.

Wavelet's properties

n Short time localized waves with zero integral value.

n Possibility of time shifting.n Flexibility.

Wavelet Analysis Detailn Wavelet transform decomposes a

signal into a set of basis functions (wavelets)

n Wavelets are obtained from a single prototype wavelet Ψ(t) called motherwavelet by dilations and shifting:

n where a is the scaling parameter and b is the shifting parameter

)(1)(, abt

atba

-= yy

Scalingn Wavelet analysis produces a time-scale

view of the signal.n Scaling means stretching or

compressing of the signal.n scale factor (a) for sine waves:

f t a

f t a

f t a

t

t

t

( )

( )

( )

sin( )

sin( )

sin( )

= =

= =

= =

;

;

;

1

2 12

4 14

Scaling

n Scale factor works exactly the same with wavelets:

f t a

f t a

f t a

t

t

t

( )

( )

( )

( )

( )

( )

= =

= =

= =

Y

Y

Y

;

;

;

1

2 12

4 14

Wavelet function

( ) ( )aby

abx

abbayx

yxyx--Y=Y ,1, ,,

n b – shift coefficient

n a – scale coefficient

n 2D function

( ) ( )abx

xba-Y=Y

a1

,

Haar transformn A Haar wavelet is the simplest type of wavelet. n In discrete form, Haar wavelets are related to a

mathematical operation called the Haar transform. n The Haar transform serves as a prototype for all other

wavelet transforms.

Haar transformn Like all wavelet transforms, the Haar transform

decomposes a discrete signal into two subsignals of half its length.

n One subsignal is a running average or trend; the other subsignal is a running difference or fluctuation.

Haar Transform Example1. Find the average of each pair of samples. Fill the first

half of the array with averages. (first trend subsignal)

1 3 5 71. Iteration

(1+3)/2 = 2 (5+7)/2 = 6

Original Signal

62

Haar Wavelet Transform1. Find the average of each pair of samples. Fill the first half

of the array with averages. 2. Find the difference between the average and the

samples. Fill the second half of the array with differences. (first fluctuation subsignal)

1 3 5 71. Iteration -1-162

1-2 = -1 5-6= -1

Original Signal

Haar Wavelet Transform1. Find the average of each pair of samples. Fill the first half

of the array with averages. 2. Find the difference between the average and the

samples. Fill the second half of the array with differences.3. Repeat the process on the first half of the array.

Original Signal 1 3 5 71. Iteration

2. Iteration -1

-1-1

-1

6

-2

2

4

A more complicated example

66

Wavelet Decomposition

n Wavelets: A math tool for space-efficient hierarchical decomposition of functions

n S = [2, 2, 0, 2, 3, 5, 4, 4] can be transformed to S^ = [23/4, -11/4, 1/2, 0, 0, -1, -1, 0]

n Compression: many small detail coefficients can be replaced by 0’s, and only the significant coefficients are retained

67

Haar Wavelet Coefficients Coefficient “Supports”

2 2 0 2 3 5 4 4

-1.25

2.75

0.5 0

0 -1 0-1

+

-+

+

+ + +

+

+

- -

- - - -

+-+

+ -+ -

+-+-

-++--1

-1

0.5

0

2.75

-1.25

0

0Original frequency distribution

Hierarchical decomposition structure (a.k.a. “error tree”)

68

Why Wavelet Transform?

n Use hat-shape filtersn Emphasize region where points clustern Suppress weaker information in their boundaries

n Effective removal of outliersn Insensitive to noise, insensitive to input order

n Multi-resolutionn Detect arbitrary shaped clusters at different scales

n Efficientn Complexity O(N)

n Only applicable to low dimensional data

69

x2

x1

e

Principal Component Analysis (PCA)

n Find a projection that captures the largest amount of variation in datan The original data are projected onto a much smaller space, resulting

in dimensionality reduction. We find the eigenvectors of the covariance matrix, and these eigenvectors define the new space

What is PCA?n Principal components analysis (PCA):

Dimensionality Reduction and Feature Construction¨PCA is used to reduce dimensions of data

without much loss of information. ¨Used in machine learning and in signal

processing and image compression (among other things).

Goal of PCA§ We wish to explain/summarize the underlying

variance-covariance structure of a large set of variables through a few linear combinations of these variables.

Background for PCAn Suppose attributes are A1 and A2, and we

have n training examples. x’s denote values of A1 and y’s denote values of A2 over the training examples.

n Variance of an attribute A1:

)1(

)()var( 1

2

1 -

-=å=

n

xxA

n

ii

n Covariance of two attributes A1 and A2 :

¨ If covariance is positive, both dimensions increase together.

¨ If negative, as one increases, the other decreases. ¨ Zero: independent of each other.

)1(

))((),cov( 121 -

--=å=

n

yyxxAA

n

iii

Background for PCA

n Covariance matrix¨Suppose we have n attributes, A1, ..., An. ¨Covariance matrix:

),cov( where),( ,, jijijinn AAccC ==´

Background for PCA

Covariance matrix÷÷ø

öççè

æ=

÷÷ø

öççè

æ=

÷÷ø

öççè

æ

3705.1045.1047.47

)var(5.1045.104)var(

),cov(),cov(),cov(),cov(

MH

MMHMMHHH

Example

n Eigenvectors: ¨ Let M be an n´n matrix.

n v is an eigenvector of M if M ´ v = lvn l is called the eigenvalue associated with v

¨ For any eigenvector v of M and scalar a,

¨ Thus you can always choose eigenvectors of length 1:

¨ If M has any eigenvectors, it has n of them, and they are orthogonal to one another.

¨ Thus eigenvectors can be used as a new basis for a n-dimensional vector space.

vvM aa l=´

1... 221 =++ nvv

Background for PCA

A 2D Numerical Example

PCA Example –STEP 1

n Subtract the meanq From each of the data dimensions, all the x values

have x subtracted and y values have y subtracted from them. This produces a data set whose mean is zero.

q Subtracting the mean makes variance and covariance calculation easier by simplifying their equations. The variance and co-variance values are not affected by the mean value.


§ Given original data set S = {x1, ..., xk}, produce new set by subtracting the mean of attribute Ai from each xi.

Mean: 1.81 1.91 Mean: 0 0


n Calculate the covariance matrix

cov = .616555556 .615444444.615444444 .716555556

n since the non-diagonal elements in this covariance matrix are positive, we should expect that both the x and y variable increase together.

x yx

y


n Calculate the (unit )eigenvectors and eigenvalues of the covariance matrix

eigenvalues = .04908339891.28402771

eigenvectors = -.735178656 -.677873399.677873399 -.735178656

PCA Example –STEP 3•eigenvectors are plotted as diagonal dotted lines on the plot. •Note they are perpendicular to each other.•Note one of the eigenvectors goes through the middle of the points, like drawing a line of best fit. •The second eigenvector gives us the other, less important, pattern in the data, that all the points follow the main line, but are off to the side of the main line by some amount.


n Reduce dimensionality and form feature vectorthe eigenvector with the highest eigenvalue is the principle component of the data set.

In our example, the eigenvector with the larges eigenvalue was the one that pointed down the middle of the data.

Once eigenvectors are found from the covariance matrix, the next step is to order them by eigenvalue, highest to lowest. This gives you the components in order of significance.


Now, if you like, you can decide to ignore the components of lesser significance.

You do lose some information, but if the eigenvalues are small, you don’t lose much

n n dimensions in your data n calculate n eigenvectors and eigenvaluesn choose only the first p eigenvectorsn final data set has only p dimensions.


n Order eigenvectors by eigenvalue, highest to lowest.

n In general, you get n components.To reduce dimensionality to p, ignore n-p components at the bottom of the list.

0490833989.677873399.735178956.

28402771.1735178956.677873399.

2

1

=÷÷ø

öççè

æ-=

=÷÷ø

öççè

æ--

=

l

l

v

v


n Feature VectorFeatureVector = (eig1 eig2 eig3 … eign)

We can either form a feature vector with both of the eigenvectors:

-.677873399 -.735178656 -.735178656 .677873399

or, we can choose to leave out the smaller, less significant component and only have a single column:

- .677873399 - .735178656


n Deriving the new dataFinalData = RowFeatureVector x RowZeroMeanDataRowFeatureVector is the matrix with the eigenvectors in the columns transposed so that the eigenvectors are now in the rows, with the most significant eigenvector at the topRowZeroMeanData is the mean-adjusted data transposed, ie. the data items are in each column, with each row holding a separate dimension.


n Deriving the new dataFinalData = RowFeatureVector x RowZeroMeanData (RowDataAdjust)

( )735178956.677873399.2

677873399.735178956.735178956.677873399.

1

--=

÷÷ø

öççè

æ-

--=

VectorRowFeature

VectorRowFeature

÷÷ø

öççè

æ---------

=01.131.81.31.79.09.129.99.21.149.71.31.81.19.49.29.109.39.31.169.

ustRowDataAdj

PCA Example –STEP 5FinalData transpose: dimensions along columnsx y

-.827970186 -.1751153071.77758033 .142857227-.992197494 .384374989-.274210416 .130417207-1.67580142 -.209498461-.912949103 .175282444.0991094375 -.3498246981.14457216 .0464172582.438046137 .01776462971.22382056 -.162675287

Reconstruction of original Data

n If we reduced the dimensionality, obviously, when reconstructing the data we would lose those dimensions we chose to discard. In our example let us assume that we considered only the x dimension…

n We did:TransformedData = RowFeatureVector ´

RowDataAdjustso we can do RowDataAdjust = RowFeatureVector -1 ´

TransformedData = RowFeatureVector T ´ TransformedData and RowDataOriginal = RowDataAdjust + OriginalMean

Reconstruction of original Datax

-.827970186 1.77758033 -.992197494 -.274210416 -1.67580142 -.912949103 .0991094375 1.14457216 .438046137 1.22382056

94

n Given N data vectors from n-dimensions, find k ≤ n orthogonal vectors (principal components) that can be best used to represent data n Normalize input data: Each attribute falls within the same rangen Compute k orthonormal (unit) vectors, i.e., principal componentsn Each input data (vector) is a linear combination of the k principal

component vectorsn The principal components are sorted in order of decreasing

“significance” or strengthn Since the components are sorted, the size of the data can be

reduced by eliminating the weak components, i.e., those with low variance (i.e., using the strongest principal components, it is possible to reconstruct a good approximation of the original data)

n Works for numeric data only

Principal Component Analysis (Steps)

95

Attribute Subset Selection

n Another way to reduce dimensionality of datan Redundant attributes

n Duplicate much or all of the information contained in one or more other attributes

n E.g., purchase price of a product and the amount of sales tax paid

n Irrelevant attributesn Contain no information that is useful for the data

mining task at handn E.g., students' ID is often irrelevant to the task of

predicting students' GPA

96

Heuristic Search in Attribute Selection

n There are 2d possible attribute combinations of d attributesn Typical heuristic attribute selection methods:

n Best single attribute under the attribute independence assumption: choose by significance tests

n Best step-wise feature selection:n The best single-attribute is picked firstn Then next best attribute condition to the first, ...

n Step-wise attribute elimination:n Repeatedly eliminate the worst attribute

n Best combined attribute selection and eliminationn Optimal branch and bound:

n Use attribute elimination and backtracking

97

Attribute Creation (Feature Generation)

n Create new attributes (features) that can capture the important information in a data set more effectively than the original ones

n Three general methodologiesn Attribute extraction

n Domain-specificn Mapping data to new space (see: data reduction)

n E.g., Fourier transformation, wavelet transformation, manifold approaches (not covered)

n Attribute construction n Combining features (see: discriminative frequent

patterns in Chapter 7)n Data discretization

98

Data Reduction 2: Numerosity Reductionn Reduce data volume by choosing alternative, smaller

forms of data representationn Parametric methods (e.g., regression)

n Assume the data fits some model, estimate model parameters, store only the parameters, and discard the data (except possible outliers)

n Ex.: Log-linear models—obtain value at a point in m-D space as the product on appropriate marginal subspaces

n Non-parametric methodsn Do not assume modelsn Major families: histograms, clustering, sampling, …

99

Parametric Data Reduction: Regression and Log-Linear Models

n Linear regressionn Data modeled to fit a straight linen Often uses the least-square method to fit the line

n Multiple regressionn Allows a response variable Y to be modeled as a

linear function of multidimensional feature vectorn Log-linear model

n Approximates discrete multidimensional probability distributions

100

Regression Analysis

n Regression analysis: A collective name for techniques for the modeling and analysis of numerical data consisting of values of a dependent variable (also called response variable or measurement) and of one or more independent variables (aka. explanatory variables or predictors)

n The parameters are estimated so as to give a "best fit" of the data

n Most commonly the best fit is evaluated by using the least squares method, but other criteria have also been used

n Used for prediction (including forecasting of time-series data), inference, hypothesis testing, and modeling of causal relationships

y

x

y = x + 1

X1

Y1

Y1’

101

n Linear regression: Y = w X + bn Two regression coefficients, w and b, specify the line and are to be

estimated by using the data at handn Using the least squares criterion to the known values of Y1, Y2, …,

X1, X2, ….n Multiple regression: Y = b0 + b1 X1 + b2 X2

n Many nonlinear functions can be transformed into the aboven Log-linear models:

n Approximate discrete multidimensional probability distributionsn Estimate the probability of each point (tuple) in a multi-dimensional

space for a set of discretized attributes, based on a smaller subset of dimensional combinations

n Useful for dimensionality reduction and data smoothing

Regress Analysis and Log-Linear Models

102

Histogram Analysis

n Divide data into buckets and store average (sum) for each bucket

n Partitioning rules:n Equal-width: equal bucket

rangen Equal-frequency (or equal-

depth)

0510152025303540

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

103

Clustering

n Partition data set into clusters based on similarity, and store cluster representation (e.g., centroid and diameter) only

n Can be very effective if data is clustered but not if data is “smeared”

n Can have hierarchical clustering and be stored in multi-dimensional index tree structures

n There are many choices of clustering definitions and clustering algorithms

n Cluster analysis will be studied in depth in Chapter 10

104

Sampling

n Sampling: obtaining a small sample s to represent the whole data set N

n Allow a mining algorithm to run in complexity that is potentially sub-linear to the size of the data

n Key principle: Choose a representative subset of the datan Simple random sampling may have very poor

performance in the presence of skewn Develop adaptive sampling methods, e.g., stratified

sampling: n Note: Sampling may not reduce database I/Os (page at a

time)

105

Types of Sampling

n Simple random samplingn There is an equal probability of selecting any particular

itemn Sampling without replacement

n Once an object is selected, it is removed from the population

n Sampling with replacementn A selected object is not removed from the population

n Stratified sampling: n Partition the data set, and draw samples from each

partition (proportionally, i.e., approximately the same percentage of the data)

n Used in conjunction with skewed data

106

Sampling: With or without Replacement

Raw Data

107

Sampling: Cluster or Stratified Sampling

Raw Data Cluster/Stratified Sample

108

Data Cube Aggregation

n The lowest level of a data cube (base cuboid)n The aggregated data for an individual entity of interestn E.g., a customer in a phone calling data warehouse

n Multiple levels of aggregation in data cubesn Further reduce the size of data to deal with

n Reference appropriate levelsn Use the smallest representation which is enough to

solve the taskn Queries regarding aggregated information should be

answered using data cube, when possible

109

Data Reduction 3: Data Compression

n String compressionn There are extensive theories and well-tuned algorithmsn Typically lossless, but only limited manipulation is

possible without expansionn Audio/video compression

n Typically lossy compression, with progressive refinementn Sometimes small fragments of signal can be

reconstructed without reconstructing the wholen Time sequence is not audio

n Typically short and vary slowly with timen Dimensionality and numerosity reduction may also be

considered as forms of data compression

110

Data Compression

Original Data Compressed Data

lossless

Original DataApproximated

111



n Data Quality


n Data Cleaning

n Data Integration

n Data Reduction


n Summary

112

Data Transformation

n A function that maps the entire set of values of a given attribute to a new set of replacement values s.t. each old value can be identified with one of the new values

n Methodsn Smoothing: Remove noise from datan Attribute/feature construction

n New attributes constructed from the given onesn Aggregation: Summarization, data cube constructionn Normalization: Scaled to fall within a smaller, specified range

n min-max normalizationn z-score normalizationn normalization by decimal scaling

n Discretization: Concept hierarchy climbing

113

Normalization

n Min-max normalization: to [new_minA, new_maxA]

n Ex. Let income range $12,000 to $98,000 normalized to [0.0, 1.0]. Then $73,000 is mapped to

n Z-score normalization (μ: mean, σ: standard deviation):

n Ex. Let μ = 54,000, σ = 16,000. Thenn Normalization by decimal scaling

716.00)00.1(000,12000,98000,12600,73

=+---

AAA

AA

A minnewminnewmaxnewminmaxminvv _)__(' +--

-=

A

Avvsµ-

='

j

vv10

'= Where j is the smallest integer such that Max(|ν’|) < 1

225.1000,16

000,54600,73=

-

114

Discretization

n Three types of attributesn Nominal—values from an unordered set, e.g., color, professionn Ordinal—values from an ordered set, e.g., military or academic

rank n Numeric—real numbers, e.g., integer or real numbers

n Discretization: Divide the range of a continuous attribute into intervalsn Interval labels can then be used to replace actual data values n Reduce data size by discretizationn Supervised vs. unsupervisedn Split (top-down) vs. merge (bottom-up)n Discretization can be performed recursively on an attributen Prepare for further analysis, e.g., classification

115

Data Discretization Methods

n Typical methods: All the methods can be applied recursivelyn Binning

n Top-down split, unsupervisedn Histogram analysis

n Top-down split, unsupervisedn Clustering analysis (unsupervised, top-down split or

bottom-up merge)n Decision-tree analysis (supervised, top-down split)n Correlation (e.g., c2) analysis (unsupervised, bottom-up

merge)

116

Simple Discretization: Binning

n Equal-width (distance) partitioningn Divides the range into N intervals of equal size: uniform gridn if A and B are the lowest and highest values of the attribute, the

width of intervals will be: W = (B –A)/N.n The most straightforward, but outliers may dominate presentationn Skewed data is not handled well

n Equal-depth (frequency) partitioningn Divides the range into N intervals, each containing approximately

same number of samplesn Good data scalingn Managing categorical attributes can be tricky

117

Binning Methods for Data Smoothing

q Sorted data for price (in dollars): 4, 8, 9, 15, 21, 21, 24, 25, 26, 28, 29, 34

* Partition into equal-frequency (equi-depth) bins:- Bin 1: 4, 8, 9, 15- Bin 2: 21, 21, 24, 25- Bin 3: 26, 28, 29, 34

* Smoothing by bin means:- Bin 1: 9, 9, 9, 9- Bin 2: 23, 23, 23, 23- Bin 3: 29, 29, 29, 29

* Smoothing by bin boundaries:- Bin 1: 4, 4, 4, 15- Bin 2: 21, 21, 25, 25- Bin 3: 26, 26, 26, 34

118

Discretization Without Using Class Labels(Binning vs. Clustering)

Data Equal interval width (binning)

Equal frequency (binning) K-means clustering leads to better results

119

Discretization by Classification & Correlation Analysis

n Classification (e.g., decision tree analysis)

n Supervised: Given class labels, e.g., cancerous vs. benign

n Using entropy to determine split point (discretization point)

n Top-down, recursive split

n Details to be covered in Chapter 7

n Correlation analysis (e.g., Chi-merge: χ2-based discretization)

n Supervised: use class information

n Bottom-up merge: find the best neighboring intervals (thosehaving similar distributions of classes, i.e., low χ2 values) to merge

n Merge performed recursively, until a predefined stopping condition

120

Concept Hierarchy Generation

n Concept hierarchy organizes concepts (i.e., attribute values) hierarchically and is usually associated with each dimension in a data warehouse

n Concept hierarchies facilitate drilling and rolling in data warehouses to view data in multiple granularity

n Concept hierarchy formation: Recursively reduce the data by collecting and replacing low level concepts (such as numeric values for age) by higher level concepts (such as youth, adult, or senior)

n Concept hierarchies can be explicitly specified by domain experts and/or data warehouse designers

n Concept hierarchy can be automatically formed for both numeric and nominal data. For numeric data, use discretization methods shown.

121

Concept Hierarchy Generation for Nominal Data

n Specification of a partial/total ordering of attributes explicitly at the schema level by users or expertsn street < city < state < country

n Specification of a hierarchy for a set of values by explicit data groupingn {Urbana, Champaign, Chicago} < Illinois

n Specification of only a partial set of attributesn E.g., only street < city, not others

n Automatic generation of hierarchies (or attribute levels) by the analysis of the number of distinct valuesn E.g., for a set of attributes: {street, city, state, country}

122

Automatic Concept Hierarchy Generation

n Somehierarchiescanbeautomaticallygeneratedbasedontheanalysisofthenumberofdistinctvaluesperattributeinthedatasetn Theattributewiththemostdistinctvaluesisplacedatthelowestlevelofthehierarchy

n Exceptions,e.g.,weekday,month,quarter,year

country

province_or_ state

city

street

15 distinct values

365 distinct values

3567 distinct values

674,339 distinct values

123



n Data Quality


n Data Cleaning

n Data Integration

n Data Reduction


n Summary

124

Summaryn Data quality: accuracy, completeness, consistency, timeliness,

believability, interpretabilityn Data cleaning: e.g. missing/noisy values, outliersn Data integration from multiple sources:

n Entity identification problemn Remove redundanciesn Detect inconsistencies

n Data reductionn Dimensionality reductionn Numerosity reductionn Data compression

n Data transformation and data discretizationn Normalizationn Concept hierarchy generation

125

References

n D.P.BallouandG.K.Tayi.Enhancingdataqualityindatawarehouseenvironments.Comm.ofACM,42:73-78,1999

n A.Bruce,D.Donoho,andH.-Y.Gao.Waveletanalysis.IEEESpectrum,Oct1996n T.DasuandT.Johnson.ExploratoryDataMiningandDataCleaning.JohnWiley,2003n J.DevoreandR.Peck.Statistics:TheExplorationandAnalysisofData.DuxburyPress,1997.n H.Galhardas,D.Florescu,D.Shasha,E.Simon,andC.-A.Saita.Declarativedatacleaning:

Language,model,andalgorithms.VLDB'01n M.HuaandJ.Pei.Cleaningdisguisedmissingdata:Aheuristicapproach.KDD'07n H.V.Jagadish,etal.,SpecialIssueonDataReductionTechniques.BulletinoftheTechnical

CommitteeonDataEngineering,20(4),Dec.1997n H.LiuandH.Motoda(eds.).FeatureExtraction,Construction,andSelection:ADataMining

Perspective.KluwerAcademic,1998n J.E.Olson.DataQuality:TheAccuracyDimension.MorganKaufmann,2003n D.Pyle.DataPreparationforDataMining.MorganKaufmann,1999n V.RamanandJ.Hellerstein.PottersWheel:AnInteractiveFrameworkforDataCleaningand

Transformation,VLDB’2001n T.Redman.DataQuality:TheFieldGuide.DigitalPress(Elsevier),2001n R.Wang,V.Storey,andC.Firth.Aframeworkforanalysisofdataqualityresearch.IEEETrans.

KnowledgeandDataEngineering,7:623-640,1995

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