TAR: Temporal Association Rules on Evolving Numerical Attributes

1
TAR Temporal Association Rules on Evolving
Numerical Attributes

• Wei Wang, Jiong Yang, and Richard Muntz
• Speaker Sarah Chan
• CSIS DB Seminar
• May 7, 2003

2
Presentation Outline

• Introduction
• Problem Definition
• Mining Algorithms
• Performance Evaluation
• Conclusions

3
Introduction

• Association rule mining
• X ? Y (itemsets)
• Existence of X implies existence of Y
• Earlier work focused on binary attributes and
  intra-transaction relationships
• E.g. ham ? bread means A customer who buys
  ham is likely to buy bread as well

4
Introduction

• Cannot describe relationships such as
• If price of item A falls below 1, then monthly
  sales of item B rise by a margin between 10K and
  20K.
• People between 35 and 45 with salary between 80K
  and 120K are likely to buy a house whose price is
  between 300K and 400K within 2 years of marriage.
• Goal to mine ARs involving numerical attributes
  and temporal evolution

5
Problem Definition

• Each object has a set of numerical attributes
• Database a sequence of snapshots S1, S2, .. St
  of objects
• Evolution temporal changes of values of some
  attribute of some object
• E.g. Evolution of salary attr. with 3 snapshots
• (salary ? 40000,45000) ? (salary ?
  47500,55000) ? (salary ? 60000,70000)

6
Problem Definition

• TARs (on evolving numerical attributes) ARs that
  capture correlations among attr. evolutions
• Scope of paper only consider correlations of
  simultaneous evolutions (i.e. attr. evolutions
  over same set of snapshots)

7
Mining Quantitative ARs

• Srikant and Agrawal (SIGMOD96)
• Divide domain of each quantitative attr. into
  intervals
• Combine intervals as long as their support is
  less than max-sup threshold
• A set of items original and combined intervals
• Apply traditional AR mining algorithm

8
Mining Quantitative ARs

• BitOp (Lent et al., ICDE97)
• Rule form
• A ? B ? C
• quantitative categorical
• Partition attribute domain
• into 2-D grids
• For each value of attr. C
• Examine data in each grid cell to see if AR
  applies
• Represent result by a bit in a 2-D bitmap
• Combine ARs with adjacent LHS attr. values to
  form a clustered AR
• Smoothing to cover small holes in a big cluster

a1
a2
a3
a4
a5
a6
b1
x
x
x
b2
x
x
x
x
x
x
x
x
x
x
x
b3
x
x
x
x
x
x
x
x
x
b4
9
Mining TARs

• SR algorithm (based on Srikant et al., 1996)
• Map numerical attribute evolutions to binary
  attrs.
• Apply any traditional AR mining algorithm
• Transform binary attr. values in rules to
  numerical ranges
• Complexity
• For a numerical attr. quantized to b intervals
• Need O(b2) items to represent all possible
  sub-ranges
• For t snapshots, need O(b2t) items to encode all
  possible evolutions
• Huge number of items, very inefficient

10
Mining TARs

• LE algorithm (based on BitOp)
• Quantize domains
• Map each possible evolution of RHS attr. into an
  item
• For each rule form, generate clustered rules for
  each possible value of each possible RHS attr.
• Complexity
• For a RHS attr. quantized to b intervals,
  consider its evolution over t snapshots
• There could be b2t distinct evolutions
• Total no. of possible evolutions increases
  exponentially with no. of attrs. and no. of
  snapshots

11
Mining TARs

• TAR algorithm

12
The Model Evolution and Its Space

• Given attr. Ai and m snapshots
• Evolution E(Ai ) (Ai ? l1, u1) ? (Ai ? l2,
  u2) ? ? (Ai ? lm, um)
• Length of evolution m
• Evolution space of Ai m dimensional space (jth
  dimension associated with value of Ai at jth
  snapshot)

13
The Model Evolution and Its Space

• E.g. E1 (salary ? 40000,45000) ? (salary ?
  47500,55000) ? (salary ? 60000,70000)

14
The Model Evolution Conj. and Its Space

• Given n attrs A1, A2, , An (length m)
• Evolution conjunction E(A1) ? E(A2) ? ? E(An)
• Evolution space n x m dimensional space (each
  dimension associated with value of one attr. at
  one snapshot)

15
The Model TAR

• TAR X ? Y (evolution conjunctions)
• Symmetric relationship
• Assumption Y only contains evolution of one
  attr.
• E(A1)?E(A2)??E(Ak-1)?E(Ak1)??E(An) ? E(Ak)

16
The Model Window

• Window
• Subsequence of m consecutive snapshots
• For t available snapshots S1, S2, , St, there
  are t-m1 windows of width m

17
The Model Object History

• Object history of an object o over a window W
• The sequence of changes of o over W
• Follows an evolution E(Ai) iff, for each snapshot
  in the window, the value of Ai in the object
  history falls into corr. interval specified in
  E(Ai)
• Follows an evolution conjunction E(A1) ? E(A2) ?
  ? E(An) iff it follows every evolution in it
• o satisfies the TAR X ? Y iff, it has an object
  history that follows X and Y

18
The Model TAR as Hypercube

• Each object history can be mapped to a point in
  evolution space of involved attributes
• TAR a hypercube in this space, which contains
  the set of object histories satisfying the rule
• Support, density strength thresholds
  constraints on number distribution of object
  histories in hypercube

19
The Model Rule Set

• Rule set ltrmin, rmaxgt set of all rules r s.t. r
  is a specialization of rmax and a generalization
  of rmin
• Each rule set can summarize a large no. of valid
  rules

20
Mining TARs TAR algorithm

• Find density-based (subspace) clusters
• Find all valid rule sets

21
Mining TARs TAR algorithm

• Find density-based (subspace) clusters
• Create base intervals for each attribute
• Form base cubes from base intervals n1, m1
• Bottom-up clustering algorithm
• Density of an evolution cube object history
  concentration of the sparsest base cube in it
• The Apriori property holds on density
• Find all valid rule sets

22
Mining TARs TAR algorithm

• Find density-based (subspace) clusters
• Find all valid rule sets
• Make use of the strength and support metrics
• For rule X ? Y,
• strength Sup(X ?Y) / (Sup(X) x Sup(Y))
• Strength is used to prune search space

23
Pruning with the Strength Threshold

• Property 1
• For any rule r, ? a base rule bri
• which is a specialization of r and
• with strength ? that of r.
• Implication
• Only have to examine rules which are
  generalizations of BR (set of base rules) whose
  strength ? thres.

24
Pruning with the Strength Threshold

• Property 2
• For any two rules r and r where
• r is a specialization of r, and
• strength of r lt strength of r,
• ? another base rule bri which is
• a specialization of r but not r and
• strength of bri gt strength of r.
• Implication
• Can skip rules which are generalizations of r
  but which do not contain any other base rule in
  BR.

25
Finding Rule Sets from Each Cluster

• Find BR
• For each subset of BR, explore
• corr. search region from rule r
• (min. bounding box of rules in BR)
• If strength of r lt thres., ignore region
• min-rule
• If sup of r ? thres., min-rule ? r
• If sup of r lt thres., search for its valid
  generalizations within region.
• Stop when strength lt thres.
• max-rule
• Search similarly until a rule is found s.t. all
  of its generalizations either violate strength
  requirement or another base rule is included
• There can be multiple max-rules for a min-rule

26
Performance Evaluation

• 300MHz CPU with 128MB memory
• Three synthetic datasets
• 100,000 objects with 5 attributes
• 100 snapshots
• Embedded 500 rules of length 5 or less
• User-specified thresholds
• Density 2 (2 times the average density)
• Support 5
• Strength 1.3

27
Performance Evaluation
Recall

• Precision 100 for all algorithms

28
Performance Evaluation

• Observations
• TAR is faster than SR and LE
• Strength is used to prune the search space in TAR
• Search a smaller set of candidate rules
• Response time of TAR increases at a slower pace
  w.r.t. number of base intervals

29
Performance Evaluation

• Real dataset
• 20,000 objects (persons)
• 5 attributes age, title, salary, family status
  (single, married, head of household), distance
  between persons house and a major city
• 10 snapshots (one per year)
• No. of base intervals 100 support 3, density 2,
  strength 1.3

30
Performance Evaluation

• Performance of TAR alg. on real dataset
• Time taken 260s to mine 347 rule sets
• Examples of TARs
• People receiving a salary raise tend to move
  further away from city center.
• If people with a salary in the range 70K and 100K
  get a raise, the range of the raise will likely
  be from 7K to 15K.

31
Conclusions

• A TAR model is proposed to represent correlations
  among numerical attribute evolutions.
• A novel approach to mine TARs by first
  discovering clusters and then efficiently
  constructing rule sets is introduced.
• Empirical evaluation shows TAR algorithm
  outperforms alternative algs. by a large margin.

32
References

• W. Wang, J. Yang, and R. Muntz. TAR Temporal
  association rules on evolving numerical
  attributes, ICDE01.
• R. Srikant and R. Agrawal. Mining quantitative
  association rules in large relational tables,
  SIGMOD96.
• B. Lent, A. Swami, and J. Widom. Clustering
  association rules, ICDE97.
• R. Agrawal, J. Gehrke, D. Gunopulos, and P.
  Raghavan. Automatic subspace clustering of high
  dimensional data for data mining application,
  SIGMOD98.