Data Mining
Association Rules: Advanced Concepts and Algorithms
Lecture Notes for Chapter 7
Introduction to Data Mining
by
Tan, Steinbach, Kumar

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 *

Continuous and Categorical Attributes
Example of Association Rule:
{Number of Pages [5,10)  (Browser=Mozilla)}  {Buy = No}
How to apply association analysis formulation to non-asymmetric binary variables?

Session Id

Country

Session Length (sec)

Number of Web Pages viewed

Gender

Browser Type

Buy

1

USA

982

8

Male

IE

No

2

China

811

10

Female

Netscape

No

3

USA

2125

45

Female

Mozilla

Yes

4

Germany

596

4

Male

IE

Yes

5

Australia

123

9

Male

Mozilla

No

…

…

…

…

…

…

…

10

Handling Categorical Attributes
Transform categorical attribute into asymmetric binary variables
Introduce a new “item” for each distinct attribute-value pair
Example: replace Browser Type attribute with
Browser Type = Internet Explorer
Browser Type = Mozilla
Browser Type = Mozilla

Handling Categorical Attributes
Potential Issues
What if attribute has many possible values
Example: attribute country has more than 200 possible values
Many of the attribute values may have very low support
Potential solution: Aggregate the low-support attribute values

What if distribution of attribute values is highly skewed
Example: 95% of the visitors have Buy = No
Most of the items will be associated with (Buy=No) item
Potential solution: drop the highly frequent items

Handling Continuous Attributes
Different kinds of rules:
Age[21,35)  Salary[70k,120k)  Buy
Salary[70k,120k)  Buy  Age: =28, =4
Different methods:
Discretization-based
Statistics-based
Non-discretization based
minApriori

Discretization Issues
Size of the discretized intervals affect support & confidence

If intervals too small
may not have enough support
If intervals too large
may not have enough confidence
Potential solution: use all possible intervals
{Refund = No, (Income = $51,250)}  {Cheat = No}
{Refund = No, (60K  Income  80K)}  {Cheat = No}
{Refund = No, (0K  Income  1B)}  {Cheat = No}

Approach by Srikant & Agrawal
Preprocess the data
Discretize attribute using equi-depth partitioning
Use partial completeness measure to determine number of partitions
Merge adjacent intervals as long as support is less than max-support
Apply existing association rule mining algorithms
Determine interesting rules in the output

Approach by Srikant & Agrawal
Discretization will lose information

Use partial completeness measure to determine how much information is lost
C: frequent itemsets obtained by considering all ranges of attribute values
P: frequent itemsets obtained by considering all ranges over the partitions
P is K-complete w.r.t C if P  C,and X  C,  X’  P such that:
1. X’ is a generalization of X and support (X’)  K  support(X) (K  1)
2. Y  X,  Y’  X’ such that support (Y’)  K  support(Y)

Given K (partial completeness level), can determine number of intervals (N)
X
Approximated X

Statistics-based Methods
Example:
Browser=Mozilla  Buy=Yes  Age: =23
Rule consequent consists of a continuous variable, characterized by their statistics
mean, median, standard deviation, etc.
Approach:
Withhold the target variable from the rest of the data
Apply existing frequent itemset generation on the rest of the data
For each frequent itemset, compute the descriptive statistics for the corresponding target variable
Frequent itemset becomes a rule by introducing the target variable as rule consequent
Apply statistical test to determine interestingness of the rule

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

Min-Apriori (Han et al)
Example:
W1 and W2 tends to appear together in the same document
Document-term matrix:

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

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

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

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

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

Formal Definition of a Sequence
A sequence is an ordered list of elements (transactions)

s = < e1 e2 e3 … >
Each element contains a collection of events (items)

ei = {i1, i2, …, ik}
Each element is attributed to a specific time or location

Length of a sequence, |s|, is given by the number of elements of the sequence

A k-sequence is a sequence that contains k events (items)

Examples of Sequence
Web sequence:

< {Homepage} {Electronics} {Digital Cameras} {Canon Digital Camera} {Shopping Cart} {Order Confirmation} {Return to Shopping} >

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}>

Sequential Pattern Mining: Definition
Given:
a database of sequences
a user-specified minimum support threshold, minsup
Task:
Find all subsequences with support ≥ minsup

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-1} {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}>, …

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

Candidate Generation
Base case (k=2):
Merging two frequent 1-sequences <{i1}> and <{i2}> will produce two candidate 2-sequences: <{i1} {i2}> 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 the first event in w1 is the same as the subsequence obtained by removing the last event in w2
The resulting candidate after merging is given by the sequence w1 extended with the last event of w2.
If the last two events in w2 belong to the same element, then the last event in w2 becomes part of the last element in w1
Otherwise, the last event in w2 becomes a separate element appended to the end of w1

Candidate Generation Examples
Merging the sequences
w1=<{1} {2 3} {4}> and w2 =<{2 3} {4 5}>
will produce the candidate sequence < {1} {2 3} {4 5}> because the last two events in w2 (4 and 5) belong to the same element
Merging the sequences
w1=<{1} {2 3} {4}> and w2 =<{2 3} {4} {5}>
will produce the candidate sequence < {1} {2 3} {4} {5}> because the last two events in w2 (4 and 5) do not belong to the same element
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
< {1} {2 6} {5}>

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?

Contiguous Subsequences
s is a contiguous subsequence of
w = < 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 more than 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}>

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

Modified Support Counting Step
Given a candidate 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

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?

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

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

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

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

Session
Id
Country Session
Length
(sec)
Number of
Web Pages
viewed
Gender
Browser
Type
Buy
1 USA 982 8 Male IE No
2 China 811 10 Female Netscape No
3 USA 2125 45 Female Mozilla Yes
4 Germany 596 4 Male IE Yes
5 Australia 123 9 Male Mozilla No
… … … … … … …
10

2
2
2
1
2
1
‘
n
s
n
s
Z
+
D
–
–
=
m
m
TIDW1W2W3W4W5
D122001
D200122
D323000
D400101
D511102