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chap6_advanced_association_analysis.pptxChapter 6
Association Analysis: Advance Concepts
Introduction to Data Mining, 2nd Edition
by
Tan, Steinbach, Karpatne, Kumar
Data Mining
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Extensions of Association Analysis to Continuous and Categorical Attributes and Multi-level Rules
Data Mining Association Analysis: Advanced Concepts
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Continuous and Categorical Attributes
Example of Association Rule:
{Gender=Male, Age [21,30)} {No of hours online 10}
How to apply association analysis to non-asymmetric binary variables?
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Handling Categorical Attributes
Example: Internet Usage Data
{Level of Education=Graduate, Online Banking=Yes} {Privacy Concerns = Yes}
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Handling Categorical Attributes
Introduce a new “item” for each distinct attribute-value pair
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Handling Categorical Attributes
Some attributes can have many possible values
Many of their attribute values have very low support
Potential solution: Aggregate the low-support attribute values
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Handling Categorical Attributes
Distribution of attribute values can be highly skewed
Example: 85% of survey participants own a computer at home
Most records have Computer at home = Yes
Computation becomes expensive; many frequent itemsets involving the binary item (Computer at home = Yes)
Potential solution:
discard the highly frequent items
Use alternative measures such as h-confidence
Computational Complexity
Binarizing the data increases the number of items
But the width of the “transactions” remain the same as the number of original (non-binarized) attributes
Produce more frequent itemsets but maximum size of frequent itemset is limited to the number of original attributes
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Handling Continuous Attributes
Different methods:
Discretization-based
Statistics-based
Non-discretization based
minApriori
Different kinds of rules can be produced:
{Age[21,30), No of hours online[10,20)} {Chat Online =Yes}
{Age[21,30), Chat Online = Yes} No of hours online: =14, =4
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization-based Methods
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization-based Methods
Unsupervised:
Equal-width binning
Equal-depth binning
Cluster-based
Supervised discretization
100
150
100
100
0
0
0
100
150
0
0
0
0
20
10
20
0
0
9
8
7
6
5
4
3
2
1
Chat Online = No
Chat Online = Yes
bin1
bin3
bin2
Continuous attribute, v
<1 2 3> <4 5 6> <7 8 9>
<1 2 > <3 4 5 6 7 > < 8 9>
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization Issues
Interval width
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization Issues
Interval too wide (e.g., Bin size= 30)
May merge several disparate patterns
Patterns A and B are merged together
May lose some of the interesting patterns
Pattern C may not have enough confidence
Interval too narrow (e.g., Bin size = 2)
Pattern A is broken up into two smaller patterns
Can recover the pattern by merging adjacent subpatterns
Pattern B is broken up into smaller patterns
Cannot recover the pattern by merging adjacent subpatterns
Some windows may not meet support threshold
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization: all possible intervals
Execution time
If the range is partitioned into k intervals, there are O(k2) new items
If an interval [a,b) is frequent, then all intervals that subsume [a,b) must also be frequent
E.g.: if {Age [21,25), Chat Online=Yes} is frequent, then {Age [10,50), Chat Online=Yes} is also frequent
Improve efficiency:
Use maximum support to avoid intervals that are too wide
Number of intervals = k
Total number of Adjacent intervals = k(k-1)/2
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Discretization Issues
Redundant rules
R1: {Age [18,20), Age [10,12)} {Chat Online=Yes}
R2: {Age [18,23), Age [10,20)} {Chat Online=Yes}
If both rules have the same support and confidence, prune the more specific rule (R1)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Statistics-based Methods
Example:
{Income > 100K, Online Banking=Yes} Age: =34
Rule consequent consists of a continuous variable, characterized by their statistics
mean, median, standard deviation, etc.
Approach:
Withhold the target attribute from the rest of the data
Extract frequent itemsets from the rest of the attributes
Binarized the continuous attributes (except for the target attribute)
For each frequent itemset, compute the corresponding descriptive statistics of the target attribute
Frequent itemset becomes a rule by introducing the target variable as rule consequent
Apply statistical test to determine interestingness of the rule
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Statistics-based Methods
{Male, Income > 100K}
{Income < 30K, No hours [10,15)}
{Income > 100K, Online Banking = Yes}
….
Frequent Itemsets:
{Male, Income > 100K} Age: = 30
{Income < 40K, No hours [10,15)} Age: = 24
{Income > 100K,Online Banking = Yes} Age: = 34
….
Association Rules:
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Statistics-based Methods
How to determine whether an association rule interesting?
Compare the statistics for segment of population covered by the rule vs segment of population not covered by the rule:
A B: versus A B: ’
Statistical hypothesis testing:
Null hypothesis: H0: ’ = +
Alternative hypothesis: H1: ’ > +
Z has zero mean and variance 1 under null hypothesis
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Statistics-based Methods
Example:
r: Browser=Mozilla Buy=Yes Age: =23
Rule is interesting if difference between and ’ is more than 5 years (i.e., = 5)
For r, suppose n1 = 50, s1 = 3.5
For r’ (complement): n2 = 250, s2 = 6.5
For 1-sided test at 95% confidence level, critical Z-value for rejecting null hypothesis is 1.64.
Since Z is greater than 1.64, r is an interesting rule
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Min-Apriori
Example:
W1 and W2 tends to appear together in the same document
Document-term matrix:
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Min-Apriori
Data contains only continuous attributes of the same “type”
e.g., frequency of words in a document
Potential solution:
Convert into 0/1 matrix and then apply existing algorithms
lose word frequency information
Discretization does not apply as users want association among words not ranges of words
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Min-Apriori
How to determine the support of a word?
If we simply sum up its frequency, support count will be greater than total number of documents!
Normalize the word vectors – e.g., using L1 norms
Each word has a support equals to 1.0
Normalize
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Min-Apriori
New definition of support:
Example:
Sup(W1,W2,W3)
= 0 + 0 + 0 + 0 + 0.17
= 0.17
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Anti-monotone property of Support
Example:
Sup(W1) = 0.4 + 0 + 0.4 + 0 + 0.2 = 1
Sup(W1, W2) = 0.33 + 0 + 0.4 + 0 + 0.17 = 0.9
Sup(W1, W2, W3) = 0 + 0 + 0 + 0 + 0.17 = 0.17
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Concept Hierarchies
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multi-level Association Rules
Why should we incorporate concept hierarchy?
Rules at lower levels may not have enough support to appear in any frequent itemsets
Rules at lower levels of the hierarchy are overly specific
e.g., skim milk white bread, 2% milk wheat bread, skim milk wheat bread, etc. are indicative of association between milk and bread
Rules at higher level of hierarchy may be too generic
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multi-level Association Rules
How do support and confidence vary as we traverse the concept hierarchy?
If X is the parent item for both X1 and X2, then (X) ≤ (X1) + (X2)
If (X1 Y1) ≥ minsup, and X is parent of X1, Y is parent of Y1 then (X Y1) ≥ minsup, (X1 Y) ≥ minsup (X Y) ≥ minsup
If conf(X1 Y1) ≥ minconf, then conf(X1 Y) ≥ minconf
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multi-level Association Rules
Approach 1:
Extend current association rule formulation by augmenting each transaction with higher level items
Original Transaction: {skim milk, wheat bread}
Augmented Transaction: {skim milk, wheat bread, milk, bread, food}
Issues:
Items that reside at higher levels have much higher support counts
if support threshold is low, too many frequent patterns involving items from the higher levels
Increased dimensionality of the data
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multi-level Association Rules
Approach 2:
Generate frequent patterns at highest level first
Then, generate frequent patterns at the next highest level, and so on
Issues:
I/O requirements will increase dramatically because we need to perform more passes over the data
May miss some potentially interesting cross-level association patterns
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequential Patterns
Data Mining Association Analysis: Advanced Concepts
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Examples of Sequence
Sequence of different transactions by a customer at an online store:
< {Digital Camera,iPad} {memory card} {headphone,iPad cover} >
Sequence of initiating events causing the nuclear accident at 3-mile Island: (http://stellar-one.com/nuclear/staff_reports/summary_SOE_the_initiating_event.htm)
< {clogged resin} {outlet valve closure} {loss of feedwater} {condenser polisher outlet valve shut} {booster pumps trip} {main waterpump trips} {main turbine trips} {reactor pressure increases}>
Sequence of books checked out at a library:
<{Fellowship of the Ring} {The Two Towers} {Return of the King}>
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequential Pattern Discovery: Examples
In telecommunications alarm logs,
Inverter_Problem:
(Excessive_Line_Current) (Rectifier_Alarm) --> (Fire_Alarm)
In point-of-sale transaction sequences,
Computer Bookstore:
(Intro_To_Visual_C) (C++_Primer) --> (Perl_for_dummies,Tcl_Tk)
Athletic Apparel Store:
(Shoes) (Racket, Racketball) --> (Sports_Jacket)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequence Data
| Sequence Database | Sequence | Element (Transaction) | Event (Item) |
| Customer | Purchase history of a given customer | A set of items bought by a customer at time t | Books, diary products, CDs, etc |
| Web Data | Browsing activity of a particular Web visitor | A collection of files viewed by a Web visitor after a single mouse click | Home page, index page, contact info, etc |
| Event data | History of events generated by a given sensor | Events triggered by a sensor at time t | Types of alarms generated by sensors |
| Genome sequences | DNA sequence of a particular species | An element of the DNA sequence | Bases A,T,G,C |
Sequence
E1 E2
E1 E3
E2
E3 E4
E2
Element (Transaction)
Event (Item)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequence Data
Sequence ID
Timestamp
Events
A
10
2, 3, 5
A
20
6, 1
A
23
1
B
11
4, 5, 6
B
17
2
B
21
7, 8, 1, 2
B
28
1, 6
C
14
1, 8, 7
Sequence Database:
Sequence A:
Sequence B:
Sequence C:
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequence Data vs. Market-basket Data
Customer
Date
Items bought
A
10
2, 3, 5
A
20
1,6
A
23
1
B
11
4, 5, 6
B
17
2
B
21
1,2,7,8
B
28
1, 6
C
14
1,7,8
Sequence Database:
| Events |
| 2, 3, 5 |
| 1,6 |
| 1 |
| 4,5,6 |
| 2 |
| 1,2,7,8 |
| 1,6 |
| 1,7,8 |
Market- basket Data
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequence Data vs. Market-basket Data
Customer
Date
Items bought
A
10
2, 3, 5
A
20
1,6
A
23
1
B
11
4, 5, 6
B
17
2
B
21
1,2,7,8
B
28
1, 6
C
14
1,7,8
Sequence Database:
| Events |
| 2, 3, 5 |
| 1,6 |
| 1 |
| 4,5,6 |
| 2 |
| 1,2,7,8 |
| 1,6 |
| 1,7,8 |
Market- basket Data
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Formal Definition of a Sequence
A sequence is an ordered list of elements
s = < e1 e2 e3 … >
Each element contains a collection of events (items)
ei = {i1, i2, …, ik}
Length of a sequence, |s|, is given by the number of elements in the sequence
A k-sequence is a sequence that contains k events (items)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Formal Definition of a Subsequence
A sequence <a1 a2 … an> is contained in another sequence <b1 b2 … bm> (m ≥ n) if there exist integers i1 < i2 < … < in such that a1 bi1 , a2 bi2, …, an bin
Illustrative Example:
s: b1 b2 b3 b4 b5
t: a1 a2 a3
t is a subsequence of s if a1 b2, a2 b3, a3 b5.
| Data sequence | Subsequence | Contain? |
| < {2,4} {3,5,6} {8} > | < {2} {8} > | |
| < {1,2} {3,4} > | < {1} {2} > | |
| < {2,4} {2,4} {2,5} > | < {2} {4} > | |
| <{2,4} {2,5}, {4,5}> | < {2} {4} {5} > | |
| <{2,4} {2,5}, {4,5}> | < {2} {5} {5} > | |
| <{2,4} {2,5}, {4,5}> | < {2, 4, 5} > |
No
Yes
Yes
Yes
No
No
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequential Pattern Mining: Definition
The support of a subsequence w is defined as the fraction of data sequences that contain w
A sequential pattern is a frequent subsequence (i.e., a subsequence whose support is ≥ minsup)
Given:
a database of sequences
a user-specified minimum support threshold, minsup
Task:
Find all subsequences with support ≥ minsup
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequential Pattern Mining: Example
Minsup = 50%
Examples of Frequent Subsequences:
< {1,2} > s=60%
< {2,3} > s=60%
< {2,4}> s=80%
< {3} {5}> s=80%
< {1} {2} > s=80%
< {2} {2} > s=60%
< {1} {2,3} > s=60%
< {2} {2,3} > s=60%
< {1,2} {2,3} > s=60%
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Sequence Data vs. Market-basket Data
Customer
Date
Items bought
A
10
2, 3, 5
A
20
1,6
A
23
1
B
11
4, 5, 6
B
17
2
B
21
1,2,7,8
B
28
1, 6
C
14
1,7,8
Sequence Database:
| Events |
| 2, 3, 5 |
| 1,6 |
| 1 |
| 4,5,6 |
| 2 |
| 1,2,7,8 |
| 1,6 |
| 1,7,8 |
Market- basket Data
(1,8) -> (7)
{2} -> {1}
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Extracting Sequential Patterns
Given n events: i1, i2, i3, …, in
Candidate 1-subsequences:
<{i1}>, <{i2}>, <{i3}>, …, <{in}>
Candidate 2-subsequences:
<{i1, i2}>, <{i1, i3}>, …,
<{i1} {i1}>, <{i1} {i2}>, …, <{in} {in}>
Candidate 3-subsequences:
<{i1, i2 , i3}>, <{i1, i2 , i4}>, …,
<{i1, i2} {i1}>, <{i1, i2} {i2}>, …,
<{i1} {i1 , i2}>, <{i1} {i1 , i3}>, …,
<{i1} {i1} {i1}>, <{i1} {i1} {i2}>, …
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Extracting Sequential Patterns: Simple example
Given 2 events: a, b
Candidate 1-subsequences:
<{a}>, <{b}>.
Candidate 2-subsequences:
<{a} {a}>, <{a} {b}>, <{b} {a}>, <{b} {b}>, <{a, b}>.
Candidate 3-subsequences:
<{a} {a} {a}>, <{a} {a} {b}>, <{a} {b} {a}>, <{a} {b} {b}>,
<{b} {b} {b}>, <{b} {b} {a}>, <{b} {a} {b}>, <{b} {a} {a}>
<{a, b} {a}>, <{a, b} {b}>, <{a} {a, b}>, <{b} {a, b}>
()
(a)
(b)
(a,b)
Item-set patterns
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Generalized Sequential Pattern (GSP)
Step 1:
Make the first pass over the sequence database D to yield all the 1-element frequent sequences
Step 2:
Repeat until no new frequent sequences are found
Candidate Generation:
Merge pairs of frequent subsequences found in the (k-1)th pass to generate candidate sequences that contain k items
Candidate Pruning:
Prune candidate k-sequences that contain infrequent (k-1)-subsequences
Support Counting:
Make a new pass over the sequence database D to find the support for these candidate sequences
Candidate Elimination:
Eliminate candidate k-sequences whose actual support is less than minsup
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation
Base case (k=2):
Merging two frequent 1-sequences <{i1}> and <{i2}> will produce the following candidate 2-sequences: <{i1} {i1}>, <{i1} {i2}>, <{i2} {i2}>, <{i2} {i1}> and <{i1 i2}>.
General case (k>2):
A frequent (k-1)-sequence w1 is merged with another frequent (k-1)-sequence w2 to produce a candidate k-sequence if the subsequence obtained by removing an event from the first element in w1 is the same as the subsequence obtained by removing an event from the last element in w2
The resulting candidate after merging is given by extending the sequence w1 as follows-
If the last element of w2 has only one event, append it to w1
Otherwise add the event from the last element of w2 (which is absent in the last element of w1) to the last element of w1
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation Examples
Merging w1=<{1 2 3} {4 6}> and w2 =<{2 3} {4 6} {5}> produces the candidate sequence < {1 2 3} {4 6} {5}> because the last element of w2 has only one event
Merging w1=<{1} {2 3} {4}> and w2 =<{2 3} {4 5}> produces the candidate sequence < {1} {2 3} {4 5}> because the last element in w2 has more than one event
Merging w1=<{1 2 3} > and w2 =<{2 3 4} > produces the candidate sequence < {1 2 3 4}> because the last element in w2 has more than one event
We do not have to merge the sequences w1 =<{1} {2 6} {4}> and w2 =<{1} {2} {4 5}> to produce the candidate < {1} {2 6} {4 5}> because if the latter is a viable candidate, then it can be obtained by merging w1 with < {2 6} {4 5}>
11/19/2012 Introduction to Data Mining ‹#›
GSP Example
11/19/2012 Introduction to Data Mining ‹#›
GSP Example
11/19/2012 Introduction to Data Mining ‹#›
Timing Constraints (I)
{A B} {C} {D E}
<= ms
<= xg
>ng
xg: max-gap
ng: min-gap
ms: maximum span
| Data sequence, d | Sequential Pattern, s | d contains s? |
| < {2,4} {3,5,6} {4,7} {4,5} {8} > | < {6} {5} > | |
| < {1} {2} {3} {4} {5}> | < {1} {4} > | |
| < {1} {2,3} {3,4} {4,5}> | < {2} {3} {5} > | |
| < {1,2} {3} {2,3} {3,4} {2,4} {4,5}> | < {1,2} {5} > |
xg = 2, ng = 0, ms= 4
Yes
Yes
No
No
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Mining Sequential Patterns with Timing Constraints
Approach 1:
Mine sequential patterns without timing constraints
Postprocess the discovered patterns
Approach 2:
Modify GSP to directly prune candidates that violate timing constraints
Question:
Does Apriori principle still hold?
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Apriori Principle for Sequence Data
Suppose:
xg = 1 (max-gap)
ng = 0 (min-gap)
ms = 5 (maximum span)
minsup = 60%
<{2} {5}> support = 40%
Problem exists because of max-gap constraint
No such problem if max-gap is infinite
but
<{2} {3} {5}> support = 60%
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Contiguous Subsequences
s is a contiguous subsequence of w = <e1>< e2>…< ek> if any of the following conditions hold:
s is obtained from w by deleting an item from either e1 or ek
s is obtained from w by deleting an item from any element ei that contains at least 2 items
s is a contiguous subsequence of s’ and s’ is a contiguous subsequence of w (recursive definition)
Examples: s = < {1} {2} >
is a contiguous subsequence of < {1} {2 3}>, < {1 2} {2} {3}>, and < {3 4} {1 2} {2 3} {4} >
is not a contiguous subsequence of < {1} {3} {2}> and < {2} {1} {3} {2}>
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Modified Candidate Pruning Step
Without maxgap constraint:
A candidate k-sequence is pruned if at least one of its (k-1)-subsequences is infrequent
With maxgap constraint:
A candidate k-sequence is pruned if at least one of its contiguous (k-1)-subsequences is infrequent
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Timing Constraints (II)
{A B} {C} {D E}
<= ms
<= xg
>ng
<= ws
xg: max-gap
ng: min-gap
ws: window size
ms: maximum span
| Data sequence, d | Sequential Pattern, s | d contains s? |
| < {2,4} {3,5,6} {4,7} {4,5} {8} > | < {3,4,5}> | Yes |
| < {1} {2} {3} {4} {5}> | < {1,2} {3,4} > | No |
| < {1,2} {2,3} {3,4} {4,5}> | < {1,2} {3,4} > | Yes |
xg = 2, ng = 0, ws = 1, ms= 5
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Modified Support Counting Step
Given a candidate sequential pattern: <{a, c}>
Any data sequences that contain
<… {a c} … >, <… {a} … {c}…> ( where time({c}) – time({a}) ≤ ws) <…{c} … {a} …> (where time({a}) – time({c}) ≤ ws)
will contribute to the support count of candidate pattern
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Other Formulation
In some domains, we may have only one very long time series
Example:
monitoring network traffic events for attacks
monitoring telecommunication alarm signals
Goal is to find frequent sequences of events in the time series
This problem is also known as frequent episode mining
E1
E2
E1
E2
E1
E2
E3
E4
E3 E4
E1
E2
E2 E4
E3 E5
E2
E3 E5
E1
E2
E3 E1
Pattern: <E1> <E3>
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
General Support Counting Schemes
Assume:
xg = 2 (max-gap)
ng = 0 (min-gap)
ws = 0 (window size)
ms = 2 (maximum span)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Subgraph Mining
Data Mining Association Analysis: Advanced Concepts
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Frequent Subgraph Mining
Extends association analysis to finding frequent subgraphs
Useful for Web Mining, computational chemistry, bioinformatics, spatial data sets, etc
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Graph Definitions
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Representing Transactions as Graphs
Each transaction is a clique of items
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Representing Graphs as Transactions
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Challenges
Node may contain duplicate labels
Support and confidence
How to define them?
Additional constraints imposed by pattern structure
Support and confidence are not the only constraints
Assumption: frequent subgraphs must be connected
Apriori-like approach:
Use frequent k-subgraphs to generate frequent (k+1) subgraphs
What is k?
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Challenges…
Support:
number of graphs that contain a particular subgraph
Apriori principle still holds
Level-wise (Apriori-like) approach:
Vertex growing:
k is the number of vertices
Edge growing:
k is the number of edges
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Vertex Growing
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Edge Growing
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Apriori-like Algorithm
Find frequent 1-subgraphs
Repeat
Candidate generation
Use frequent (k-1)-subgraphs to generate candidate k-subgraph
Candidate pruning
Prune candidate subgraphs that contain infrequent (k-1)-subgraphs
Support counting
Count the support of each remaining candidate
Eliminate candidate k-subgraphs that are infrequent
In practice, it is not as easy. There are many other issues
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Example: Dataset
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Example
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation
In Apriori:
Merging two frequent k-itemsets will produce a candidate (k+1)-itemset
In frequent subgraph mining (vertex/edge growing)
Merging two frequent k-subgraphs may produce more than one candidate (k+1)-subgraph
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multiplicity of Candidates (Vertex Growing)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multiplicity of Candidates (Edge growing)
Case 1: identical vertex labels
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multiplicity of Candidates (Edge growing)
Case 2: Core contains identical labels
Core: The (k-1) subgraph that is common between the joint graphs
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Multiplicity of Candidates (Edge growing)
Case 3: Core multiplicity
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Topological Equivalence
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation by Edge Growing
Given:
Case 1: a c and b d
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation by Edge Growing
Case 2: a = c and b d
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation by Edge Growing
Case 3: a c and b = d
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Candidate Generation by Edge Growing
Case 4: a = c and b = d
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Graph Isomorphism
A graph is isomorphic if it is topologically equivalent to another graph
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Graph Isomorphism
Test for graph isomorphism is needed:
During candidate generation step, to determine whether a candidate has been generated
During candidate pruning step, to check whether its (k-1)-subgraphs are frequent
During candidate counting, to check whether a candidate is contained within another graph
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Graph Isomorphism
The same graph can be represented in many ways
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Graph Isomorphism
Use canonical labeling to handle isomorphism
Map each graph into an ordered string representation (known as its code) such that two isomorphic graphs will be mapped to the same canonical encoding
Example:
Lexicographically largest adjacency matrix
String: 011011
Canonical: 111100
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Example of Canonical Labeling (Kuramochi & Karypis, ICDM 2001)
Graph:
Adjacency matrix representation:
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Example of Canonical Labeling (Kuramochi & Karypis, ICDM 2001)
Order based on vertex degree:
Order based on vertex labels:
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Example of Canonical Labeling (Kuramochi & Karypis, ICDM 2001)
Find canonical label:
0 0 0 e1 e0 e0
0 0 0 e0 e1 e0
>
(Canonical Label)
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
20406010305070
20406010305070
104070
Age
Age
Age
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Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.4 | 0.4 | 0.0 | 0.0 | 0.2 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.0 | 0.0 | 0.2 | 0.4 | 0.4 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.4 | 0.6 | 0.0 | 0.0 | 0.0 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.0 | 0.0 | 0.5 | 0.0 | 0.5 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.2 | 0.2 | 0.2 | 0.0 | 0.4 |
Sheet2
Sheet3
TIDW1W2W3W4W5
D122001
D200122
D323000
D400101
D511102
Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.4 | 0.4 | 0.0 | 0.0 | 0.2 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.0 | 0.0 | 0.2 | 0.4 | 0.4 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.4 | 0.6 | 0.0 | 0.0 | 0.0 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.0 | 0.0 | 0.5 | 0.0 | 0.5 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.2 | 0.2 | 0.2 | 0.0 | 0.4 |
Sheet2
Sheet3
TIDW1W2W3W4W5
D10.400.330.000.000.17
D20.000.000.331.000.33
D30.400.500.000.000.00
D40.000.000.330.000.17
D50.200.170.330.000.33
Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.4 | 0.4 | 0.0 | 0.0 | 0.2 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.0 | 0.0 | 0.2 | 0.4 | 0.4 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.4 | 0.6 | 0.0 | 0.0 | 0.0 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.0 | 0.0 | 0.5 | 0.0 | 0.5 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.2 | 0.2 | 0.2 | 0.0 | 0.4 |
Sheet2
Sheet3
Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.40 | 0.33 | 0.00 | 0.00 | 0.17 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.00 | 0.00 | 0.33 | 1.00 | 0.33 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.40 | 0.50 | 0.00 | 0.00 | 0.00 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.00 | 0.00 | 0.33 | 0.00 | 0.17 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.20 | 0.17 | 0.33 | 0.00 | 0.33 |
Sheet2
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D50.200.170.330.000.33
Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.40 | 0.33 | 0.00 | 0.00 | 0.17 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.00 | 0.00 | 0.33 | 1.00 | 0.33 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.40 | 0.50 | 0.00 | 0.00 | 0.00 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.00 | 0.00 | 0.33 | 0.00 | 0.17 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.20 | 0.17 | 0.33 | 0.00 | 0.33 |
Sheet2
Sheet3
Sheet1
| TID | W1 | W2 | W3 | W4 | W5 | TID | W1 | W2 | W3 | W4 | W5 | ||
| D1 | 2 | 2 | 0 | 0 | 1 | D1 | 0.40 | 0.33 | 0.00 | 0.00 | 0.17 | ||
| D2 | 0 | 0 | 1 | 2 | 2 | D2 | 0.00 | 0.00 | 0.33 | 1.00 | 0.33 | ||
| D3 | 2 | 3 | 0 | 0 | 0 | D3 | 0.40 | 0.50 | 0.00 | 0.00 | 0.00 | ||
| D4 | 0 | 0 | 1 | 0 | 1 | D4 | 0.00 | 0.00 | 0.33 | 0.00 | 0.17 | ||
| D5 | 1 | 1 | 1 | 0 | 2 | D5 | 0.20 | 0.17 | 0.33 | 0.00 | 0.33 |
Sheet2
Sheet3
Food
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Home
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Wheat
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DVD
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Accessory
101520253035
2
3
5
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Transaction
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2{A,B,E}
3{B,C}
4{A,B,D,E}
5{B,C,D}
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TID = 1:
Sheet1
| Transaction Id | Items |
| 1 | {A,B,C,D} |
| 2 | {A,B,E} |
| 3 | {B,C} |
| 4 | {A,B,D,E} |
| 5 | {B,C,D} |
Sheet2
Sheet3
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A(1)
11101000
A(2)
11010100
A(3)
10110010
A(4)
01110001
B(5)
10001110
B(6)
01001101
B(7)
00101011
B(8)
00010111
A(2)
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B (6)
A(4)
B (7)
A(3)
B (5)
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A(2)
11100010
A(3)
01111000
A(4)
10110001
B(5)
00101011
B(6)
10000111
B(7)
01001110
B(8)
00011101
Sheet1
| A(1) | A(2) | A(3) | A(4) | B(5) | B(6) | B(7) | B(8) | ||
| A(1) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | |
| A(2) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | |
| A(3) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | |
| A(4) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | |
| B(5) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | |
| B(6) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | |
| B(7) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | |
| B(8) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | |
| A(1) | A(2) | A(3) | A(4) | B(5) | B(6) | B(7) | B(8) | ||
| A(1) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | |
| A(2) | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | |
| A(3) | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | |
| A(4) | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | |
| B(5) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | |
| B(6) | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |
| B(7) | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | |
| B(8) | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
Sheet2
Sheet3
Sheet1
| A(1) | A(2) | A(3) | A(4) | B(5) | B(6) | B(7) | B(8) | ||
| A(1) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | |
| A(2) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | |
| A(3) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | |
| A(4) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | |
| B(5) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | |
| B(6) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | |
| B(7) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | |
| B(8) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | |
| A(1) | A(2) | A(3) | A(4) | B(5) | B(6) | B(7) | B(8) | ||
| A(1) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | |
| A(2) | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | |
| A(3) | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | |
| A(4) | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | |
| B(5) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | |
| B(6) | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |
| B(7) | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | |
| B(8) | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
Sheet2
Sheet3
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