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Mining Quantitative Associations
 Techniques can be categorized by how numerical attributes, such as age or salary are treated
 Static discretization based on predefined concept hierarchies (data cube methods)
 Dynamic discretization based on data distribution (quantitative rules, e.g., Agrawal & Srikant@SIGMOD96)
 Clustering: Distancebased association (e.g., Yang & Miller@SIGMOD97)
 One dimensional clustering then association
 Deviation: (such as Aumann and Lindell@KDD99)
 Sex = female => Wage: mean=$7/hr (overall mean = $9)
Static Discretization of Quantitative Attributes
 Discretized prior to mining using concept hierarchy.
 Numeric values are replaced by ranges
 In relational database, finding all frequent kpredicate sets will require k or k+1 table scans
 Data cube is well suited for mining
 The cells of an ndimensional
 cuboid correspond to the
 predicate sets
 Mining from data cubescan be much faster
Quantitative Association Rules Based on Statistical Inference Theory [Aumann and Lindell@DMKD’03]
 Finding extraordinary and therefore interesting phenomena, e.g.,
 (Sex = female) => Wage: mean=$7/hr (overall mean = $9)
 LHS: a subset of the population
 RHS: an extraordinary behavior of this subset
 The rule is accepted only if a statistical test (e.g., Ztest) confirms the inference with high confidence
 Subrule: highlights the extraordinary behavior of a subset of the pop. of the super rule
 E.g., (Sex = female) ^ (South = yes) => mean wage = $6.3/hr
 Two forms of rules
 Categorical => quantitative rules, or Quantitative => quantitative rules
 E.g., Education in [1418] (yrs) => mean wage = $11.64/hr
 Open problem: Efficient methods for LHS containing two or more quantitative attributes
Negative and Rare Patterns
 Rare patterns: Very low support but interesting
 E.g., buying Rolex watches
 Mining: Setting individualbased or special groupbased support threshold for valuable items
 Negative patterns
 Since it is unlikely that one buys Ford Expedition (an SUV car) and Toyota Prius (a hybrid car) together, Ford Expedition and Toyota Prius are likely negatively correlated patterns
 Negatively correlated patterns that are infrequent tend to be more interesting than those that are frequent
Defining Negative Correlated Patterns (I)
 Definition 1 (supportbased)
 If itemsets X and Y are both frequent but rarely occur together, i.e.,
sup(X U Y) < sup (X) * sup(Y)  Then X and Y are negatively correlated
 Problem: A store sold two needle 100 packages A and B, only one transaction containing both A and B.
 When there are in total 200 transactions, we have
s(A U B) = 0.005, s(A) * s(B) = 0.25, s(A U B) < s(A) * s(B)  When there are 105 transactions, we have
s(A U B) = 1/105, s(A) * s(B) = 1/103 * 1/103, s(A U B) > s(A) * s(B)  Where is the problem? —Null transactions, i.e., the supportbased definition is not nullinvariant!
Defining Negative Correlated Patterns (II)
 Definition 2 (negative itemsetbased)
 X is a negative itemset if (1) X = Ā U B, where B is a set of positive items, and Ā is a set of negative items, Ā≥ 1, and (2) s(X) ≥ μ
 Itemsets X is negatively correlated, if
\[ s(X)<\prod_{i=1}^{k} s(x_{i}), where, x_{i}\in X, s(x_{i})= support of x_{i} \]
 This definition suffers a similar nullinvariant problem
 Definition 3 (Kulzynski measurebased) If itemsets X and Y are frequent, but (P(XY) + P(YX))/2 < є, where є is a negative pattern threshold, then X and Y are negatively correlated.
 Ex. For the same needle package problem, when no matter there are 200 or 105 transactions, if є = 0.01, we have
 (P(AB) + P(BA))/2 = (0.01 + 0.01)/2 < є
Constraintbased (QueryDirected) Mining
 Finding all the patterns in a database autonomously? — unrealistic!
 The patterns could be too many but not focused!
 Data mining should be an interactive process
 User directs what to be mined using a data mining query language (or a graphical user interface)
 Constraintbased mining
 User flexibility: provides constraints on what to be mined
 Optimization: explores such constraints for efficient mining — constraintbased mining: constraintpushing, similar to push selection first in DB query processing
 Note: still find all the answers satisfying constraints, not finding some answers in “heuristic search”
Constraints in Data Mining
 Knowledge type constraint:
 classification, association, etc.
 Data constraint — using SQLlike queries
 find product pairs sold together in stores in Chicago this year
 Dimension/level constraint
 in relevance to region, price, brand, customer category
 Rule (or pattern) constraint
 small sales (price < $10) triggers big sales (sum > $200)
 Interestingness constraint
 strong rules: min_support ≥ 3%, min_confidence ≥ 60%
MetaRule Guided Mining
 Metarule can be in the rule form with partially instantiated predicates and constants
 P1(X, Y) ^ P2(X, W) => buys(X, “iPad”)
 The resulting rule derived can be
 age(X, “1525”) ^ profession(X, “student”) => buys(X, “iPad”)
 In general, it can be in the form of
 P1 ^ P2 ^ … ^ Pl => Q1 ^ Q2 ^ … ^ Qr
 Method to find metarules
 Find frequent (l+r) predicates (based on minsupport threshold)
 Push constants deeply when possible into the mining process (see the remaining discussions on constraintpush techniques)
 Use confidence, correlation, and other measures when possible
ConstraintBased Frequent Pattern Mining
 Pattern space pruning constraints
 Antimonotonic: If constraint c is violated, its further mining can be terminated
 Monotonic: If c is satisfied, no need to check c again
 Succinct: c must be satisfied, so one can start with the data sets satisfying c
 Convertible: c is not monotonic nor antimonotonic, but it can be converted into it if items in the transaction can be properly ordered
 Data space pruning constraint
 Data succinct: Data space can be pruned at the initial pattern mining process
 Data antimonotonic: If a transaction t does not satisfy c, t can be pruned from its further mining
Pattern Space Pruning with AntiMonotonicity Constraints

A constraint C is
antimonotone
if the super pattern satisfies C, all of its subpatterns do so too

In other words,
a
ntimonotonicity:
If an itemset S
violates
the constraint, so does any of its superset

Ex. 1.
sum(S.price)
≤
v
is antimonotone

Ex. 2. range(S.profit) ≤
15 is antimonotone

Itemset
ab
violates C

So does every superset of
ab

Ex. 3.
sum(S.Price)
≥
v
is not antimonotone

Ex. 4.
support count
is antimonotone: core property used in Apriori
Pattern Space Pruning with Monotonicity Constraints

A constraint C is
monotone
if the pattern satisfies C, we do not need to check C in subsequent mining

Alternatively, monotonicity:
If an itemset S
satisfies
the constraint, so does any of its superset

Ex. 1.
sum(S.Price)
≥
v
is monotone

Ex. 2.
min(S.Price) ≤
v
is monotone
 Ex. 3. C: range(S.profit) ≥
15

Itemset
ab
satisfies C

So does every superset of
ab
Data Space Pruning with Data Antimonotonicity

A constraint c is
data antimonotone
if for a pattern p cannot satisfy a transaction t under c, p’s superset cannot satisfy t under c either

The key for data antimonotone is
recursive data reduction

Ex. 1.
sum(S.Price)≥
v
is data antimonotone

Ex. 2.
min(S.Price) ≤
v
is data antimonotone

Ex. 3. C:
range(S.profit) ≥
25
is data antimonotone

Itemset {b, c}’s projected DB:

T10’: {d, f, h}, T20’: {d, f, g, h}, T30’: {d, f, g}

since C cannot satisfy T10’, T10’ can be pruned
Pattern Space Pruning with Succinctness
 Succinctness:
 Given A1, the set of items satisfying a succinctness constraint C, then any set S satisfying C is based on A1 , i.e., S contains a subset belonging to A1
 Idea: Without looking at the transaction database, whether an itemset S satisfies constraint C can be determined based on the selection of items
 min(S.Price) ≤ v is succinct
 sum(S.Price) ≥ v is not succinct
 Optimization: If C is succinct, C is precounting pushable
Constrained Apriori : Push a Succinct Constraint Deep
Constrained FPGrowth: Push a Succinct Constraint Deep
Constrained FPGrowth: Push a Data Antimonotonic Constraint Deep
Constrained FPGrowth: Push a Data Antimonotonic Constraint Deep
Convertible Constraints: Ordering Data in Transactions

Convert tough constraints into antimonotone or monotone by properly ordering items

Examine C: avg(
S
.profit) ≥ 25

Order items in valuedescending order

<
a, f, g, d, b, h, c, e
>

If an itemset
afb
violates C

So does
afbh, afb*

It becomes antimonotone!
Strongly Convertible Constraints

avg(X)
≥
25 is convertible antimonotone w.r.t. item value descending order R: <
a, f, g,
d, b, h, c, e
>

If an itemset
af
violates a constraint C, so does every itemset with
af
as prefix, such as
afd

avg(X)
≥
25 is convertible monotone w.r.t. item value ascending order R1: <
e, c, h, b, d, g, f, a
>

If an itemset
d
satisfies a constraint
C
, so does itemsets
df
and
dfa
, which having
d
as a prefix

Thus, avg(X)
≥ 25 is strongly convertible
Can Apriori Handle Convertible Constraints?

A convertible, not monotone nor antimonotone nor succinct constraint cannot be pushed deep into the an Apriori mining algorithm

Within the level wise framework, no direct pruning based on the constraint can be made

Itemset df violates constraint C: avg(X) >= 25

Since adf satisfies C, Apriori needs df to assemble adf, df cannot be pruned

But it can be pushed into frequentpattern growth framework!
Pattern Space Pruning w. Convertible Constraints

C: avg(X) >= 25, min_sup=2

List items in every transaction in value descending order R:

C is convertible antimonotone w.r.t. R

Scan TDB once

remove infrequent items

Itemsets a and f are good, …

Projectionbased mining

Imposing an appropriate order on item projection

Many tough constraints can be converted into (anti)monotone
Handling Multiple Constraints
 Different constraints may require different or even conflicting itemordering
 If there exists an order R s.t. both C1 and C2 are convertible w.r.t. R, then there is no conflict between the two convertible constraints
 If there exists conflict on order of items
 Try to satisfy one constraint first
 Then using the order for the other constraint to mine frequent itemsets in the corresponding projected database
ConstraintBased Mining — A General Picture
What Constraints Are Convertible?
Mining Colossal Frequent Patterns
 F. Zhu, X. Yan, J. Han, P. S. Yu, and H. Cheng, “Mining Colossal Frequent Patterns by Core Pattern Fusion”, ICDE'07.
 We have many algorithms, but can we mine large (i.e., colossal) patterns? ― such as just size around 50 to 100? Unfortunately, not!
 Why not? ― the curse of “downward closure” of frequent patterns
 The “downward closure” property
 Any subpattern of a frequent pattern is frequent.
 Example. If (a1, a2, …, a100) is frequent, then a1, a2, …, a100, (a1, a2), (a1, a3), …, (a1, a100), (a1, a2, a3), … are all frequent! There are about 2100 such frequent itemsets!
 No matter using breadthfirst search (e.g., Apriori) or depthfirst search (FPgrowth), we have to examine so many patterns
 Thus the downward closure property leads to explosion!
Colossal Patterns: A Motivating Example
Colossal Pattern Set: Small but Interesting
 It is often the case that only a small number of patterns are colossal, i.e., of large size
 Colossal patterns are usually attached with greater importance than those of small pattern sizes
Mining Colossal Patterns: Motivation and Philosophy
 Motivation: Many realworld tasks need mining colossal patterns
 Microarray analysis in bioinformatics (when support is low)
 Biological sequence patterns
 Biological/sociological/information graph pattern mining
 No hope for completeness
 If the mining of midsized patterns is explosive in size, there is no hope to find colossal patterns efficiently by insisting “complete set” mining philosophy
 Jumping out of the swamp of the midsized results
 What we may develop is a philosophy that may jump out of the swamp of midsized results that are explosive in size and jump to reach colossal patterns
 Striving for mining almost complete colossal patterns
 The key is to develop a mechanism that may quickly reach colossal patterns and discover most of them
Alas, A Show of Colossal Pattern Mining!
Methodology of PatternFusion Strategy
 PatternFusion traverses the tree in a boundedbreadth way
 Always pushes down a frontier of a boundedsize candidate pool
 Only a fixed number of patterns in the current candidate pool will be used as the starting nodes to go down in the pattern tree ― thus avoids the exponential search space
 PatternFusion identifies “shortcuts” whenever possible
 Pattern growth is not performed by singleitem addition but by leaps and bounded: agglomeration of multiple patterns in the pool
 These shortcuts will direct the search down the tree much more rapidly towards the colossal patterns
Observation: Colossal Patterns and Core Patterns
Robustness of Colossal Patterns
 Core Patterns
Intuitively, for a frequent pattern α, a subpattern β is a τcore pattern of α if β shares a similar support set with α, i.e.,
\[ \frac{D_{\alpha }}{D_{\beta }}\geq \tau, 0< \tau \leq 1 \]
where τ is called the core ratio
 Robustness of Colossal Patterns
A colossal pattern is robust in the sense that it tends to have much more core patterns than small patterns
Example: Core Patterns

A colossal pattern has far more core patterns than a smallsized pattern

A colossal pattern has far more core descendants of a smaller size c

A random draw from a complete set of pattern of size c would more likely to pick a core descendant of a colossal pattern

A colossal pattern can be generated by merging a set of core patterns
Robustness of Colossal Patterns
 (d,τ)robustness: A pattern α is (d, τ)robust if d is the maximum number of items that can be removed from α for the resulting pattern to remain a τcore pattern of α
 For a (d,τ)robust pattern α, it has Ω(2^d) core patterns
 A colossal patterns tend to have a large number of core patterns
 Pattern distance: For patterns α and β, the pattern distance of α and β is defined to be
\[ Dist(\alpha,\beta)=1\frac{D_{\alpha }\cap D_{\beta }}{D_{\alpha }\cup D_{\beta }} \]
 If two patterns α and β are both core patterns of a same pattern, they would be bounded by a “ball” of a radius specified by their core ratio τ
\[ Dist(\alpha,\beta)\leq 1\frac{1}{2/\tau 1}=r(\tau) \]
 Once we identify one core pattern, we will be able to find all the other core patterns by a bounding ball of radius r(τ)
Colossal Patterns Correspond to Dense Balls
 Due to their robustness, colossal patterns correspond to dense balls
 A random draw in the pattern space will hit somewhere in the ball with high probability
Idea of PatternFusion Algorithm
 Generate a complete set of frequent patterns up to a small size
 Randomly pick a pattern β, and β has a high probability to be a coredescendant of some colossal pattern α
 Identify all α’s descendants in this complete set, and merge all of them ― This would generate a much larger coredescendant of α
 In the same fashion, we select K patterns. This set of larger coredescendants will be the candidate pool for the next iteration
PatternFusion: The Algorithm
 Initialization (Initial pool): Use an existing algorithm to mine all frequent patterns up to a small size, e.g., 3
 Iteration (Iterative Pattern Fusion):
 At each iteration, k seed patterns are randomly picked from the current pattern pool
 For each seed pattern thus picked, we find all the patterns within a bounding ball centered at the seed pattern
 All these patterns found are fused together to generate a set of superpatterns. All the superpatterns thus generated form a new pool for the next iteration
 Termination: when the current pool contains no more than K patterns at the beginning of an iteration
Why Is PatternFusion Efficient?
 A boundedbreadth pattern tree traversal
 It avoids explosion in mining midsized ones
 Randomness comes to help to stay on the right path
 Ability to identify “shortcuts” and take “leaps”
 fuse small patterns together in one step to generate new patterns of significant sizes
 Efficiency
PatternFusion Leads to Good Approximation
 Gearing toward colossal patterns
 The larger the pattern, the greater the chance it will be generated
 Catching outliers
 The more distinct the pattern, the greater the chance it will be generated
Experimental Setting
 Synthetic data set
 Diagn an n x (n1) table where ith row has integers from 1 to n except i. Each row is taken as an itemset. min_support is n/2.
 Real data set
 Replace: A program trace data set collected from the “replace” program, widely used in software engineering research
 ALL: A popular gene expression data set, a clinical data on ALLAML leukemia (www.broad.mit.edu/tools/data.html).
 Each item is a column, representing the activitiy level of gene/protein in the same
 Frequent pattern would reveal important correlation between gene expression patterns and disease outcomes
Experiment Results on Diagn
 LCM run time increases exponentially with pattern size n
 PatternFusion finishes efficiently
 The approximation error of PatternFusion (with minsup 20) in comparison with the complete set) is rather close to uniform sampling (which randomly picks K patterns from the complete answer set)
Experimental Results on ALL
 ALL: A popular gene expression data set with 38 transactions, each with 866 columns
 There are 1736 items in total
 The table shows a high frequency threshold of 30
Experimental Results on REPLACE
 REPLACE
 A program trace data set, recording 4395 calls and transitions
 The data set contains 4395 transactions with 57 items in total
 With support threshold of 0.03, the largest patterns are of size 44
 They are all discovered by PatternFusion with different settings of K and τ, when started with an initial pool of 20948 patterns of size <=3
Experimental Results on REPLACE
 Approximation error when compared with the complete mining result
 Example. Out of the total 98 patterns of size >=42, when K=100, PatternFusion returns 80 of them
 A good approximation to the colossal patterns in the sense that any pattern in the complete set is on average at most 0.17 items away from one of these 80 patterns
Mining Compressed Patterns: δclustering

Why compressed patterns?

too many, but less meaningful

Pattern distance measure
\[ D(P_{1},P_{2})=1\frac{T(P_{1})\cap T(P_{2})}{T(P_{1})\cup T(P_{2})}e \]

δclustering: For each pattern P, find all patterns which can be expressed by P and their distance to P are within δ (δcover)

All patterns in the cluster can be represented by P

Xin et al., “Mining Compressed FrequentPattern Sets”, VLDB’05

Closed frequent pattern

Report P1, P2, P3, P4, P5

Emphasize too much on support

no compression

Maxpattern, P3: info loss

A desirable output: P2, P3, P4
RedundancyAward Topk Patterns
 Why redundancyaware topk patterns?
 Desired patterns: high significance & low redundancy
 Propose the MMS (Maximal Marginal Significance) for measuring the combined significance of a pattern set
 Xin et al., Extracting RedundancyAware TopK Patterns, KDD’06
How to Understand and Interpret Patterns?
Semantic Analysis with Context Models
 Task1: Model the context of a frequent pattern
 Based on the Context Model …
 Task2: Extract strongest context indicators
 Task3: Extract representative transactions
 Task4: Extract semantically similar patterns
Annotating DBLP Coauthorship & Title Pattern
Summary
 Roadmap: Many aspects & extensions on pattern mining
 Mining patterns in multilevel, multi dimensional space
 Mining rare and negative patterns
 Constraintbased pattern mining
 Specialized methods for mining highdimensional data and colossal patterns
 Mining compressed or approximate patterns
 Pattern exploration and understanding: Semantic annotation of frequent patterns
References
 Mining MultiLevel and Quantitative Rules
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 Mining Sequential and Structured Patterns
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 Mining Spatial, Multimedia, and Web Data
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R. Zaiane, M. Xin, J. Han, Discovering Web Access Patterns and Trends
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 Mining Frequent Patterns in TimeSeries Data
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 FP for Classification and Clustering
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References (cont')
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 Other Freq. Pattern Mining Applications
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 H. V. Jagadish, J. Madar, and R. Ng. Semantic Compression and Pattern Extraction with Fascicles. VLDB'99.
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