CACTUS

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CACTUS Clustering Categorical Data Using
Summaries

• By Venkatesh Ganti, Johannes Gehrke and Raghu
  Ramakrishnan
• RongEn Li
• School of Informatics, Edinburgh

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Overview

• Introduction and motivation
• Existing tools for clustering categorical data
  STIRR and ROCK
• Definition of a cluster over categorical data
• The algorithm CACTUS
• Experiments and results
• Summary

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Introduction and motivation

• Numeric data, 1,2,3,4,5,
• Categorical data, LFD, PMR, DME
• Usually small number of attribute values in their
  domains. Large domains are typically hard to
  infer useful information
• Use relations! Relations contain different
  attributes, but the cross product of domain
  attributes can be large.
• CACTUS a fast summarisation-based algorithm
  which uses summary information to find well-fined
  clusters.

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Existing tools for clustering categorical data

• STIRR
• Each attribute value is represented as a weighted
  vertex in a graph.
• Multiple copies b1,,bm (basins) of weighted
  vertices are maintained. They can have different
  weights.
• Starting Step a set of weights on all vertices
  in all basins.
• Iterative Step Increment the weight in basin bi
  on vertex tj, for each vertices tuple tltt1, ,
  tngt in bi, using a function combining the weights
  of vertices other than tj in bi.
• At fixed point the large positive weights and
  small negative weights across the basins isolate
  two groups of attribute values on each attribute.
• ROCK
• Starts with each tuple in its own cluster.
• Merges close clusters until a required number
  (user specified) of clusters remains. Closeness
  defined by a similarity function.
• Use STIRR to compare with CACTUS.

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Definitions Interval region, support and
belonging

• A1,,An is a set of categorical attributes with
  domains D1,,Dn respectively. D is a set tuples
  where each tuple t ? 1,,n.
• Interval region SS1 X X Sn if Si subset of Di
  for all i ? 1,,n. Equivalent to intervals in
  numeric data
• The support of a value pair sD(ai,aj)t?Dt.Aia
  i t.Ajaj/D. The support of a region sD(S)
  is the number of tuples in D contained in S
• Belonging A tuple tltt.A1,t.Angt ? D belongs to
  a region S if for all t ? 1,,n, t.Ai ? Si.

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Definitions expected support, strongly connected

• The expected support under attribute-independence
  assumption,
• Of a region EsD(S) DS1XXSn/D1XXDn
  
• Of a pair ai and aj EsD(ai,aj)
  aD/DiXDj
• a is normally set to 2 or 3
• Strongly Connected
• ai and aj if sD(ai,aj)gtEsD(ai,aj),
  sD(ai,aj)sD(ai,aj) Otherwise, 0.
• ai ? Si w.r.t Sj for all x ? Sj, ai and x are
  strongly connected.
• Si and Sj if each ai ? Si is strongly connected
  with each aj ? Sj and if each aj ? Sj is strongly
  connected with each ai ? Si.

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Definitions Cluster, Cluster-projection,
sub-cluster and subspace cluster

• CltC1,..Cngt is a cluster over A1,,An if
• 1. Ci and Cj are strongly connected
• 2. There exists on Ci such that Ci is a proper
  superset of Ci and Ci and Ci are strongly
  connected
• 3. sD(C) of C is gt a the expected support of C
  under attribute-independence assumption
• Ci is a cluster-projection of C on Ai.
• C is a sub-cluster if it only satisfies 1 and 3.
• A cluster C over a subset of all attributes S
  proper subset of A1,,An is a subspace cluster
  on S.

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Definitions similarity, inter-attribute
summaries, intra-attribute summaries

• Similarity ?j(ai,a2) x ? Dj sD(a1,x)gt0 and
  sD(a2,x)gt0
• Inter-attribute summary
• ?ij(ai,aj, sD(ai,aj) ai ? Di, aj ? Dj, and
  sD(ai,aj)gt0
• Strongly connected attribute values pairs where
  each pair has attribute values from different
  attributes
• Intra-attribute summary
• ?ij(ai,aj, ?jD(ai,aj) ai ? Di, aj ? Dj, and
  ?jD(ai,aj)gt0
• Similarities between attribute values of the same
  attribute

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CACTUS Vs STIRR clusters found by CACTUS
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CACTUS Vs STIRR clusters found by STIRR
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CACTUS CAtegorical ClusTering Using Summaries

• Central idea data summary (inter- intra-
  attribute summary) is sufficient enough to find
  candidate clusters which can then be validated.
• A three-phase clustering algorithm
• Summarisation
• Clustering
• Validation

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Summarisation Phase

• Assumption the inter- intra- attribute summary
  of any pair of attributes fits easily into main
  memory.
• Inter-attribute Summaries
• Use a counter set to 0 initially for each pair
  (ai,aj) ? Di x Dj.
• Scan the dataset, increment the counter for each
  pair.
• After the scan, compute sD(ai,aj) and reset the
  counters of those whose s lt EsD(ai,aj). Store
  those values pairs.
• Intra-attribute Summaries
• Scan the dataset and find those tuples (T1,T2) of
  one domain such that T1.a is strongly connected
  with T1.b and T2.a is strongly connected with
  T2.b.
• Very fast operation, hence only compute them when
  needed

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Clustering Phase

• A two-step operation
• Step 1. analyse each attribute to compute all
  cluster-projections on it
• Step 2. Synthesise candidate clusters on sets of
  attributes from the cluster-projections on
  individual attributes

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Clustering Phase continued

• Step1 Compute cluster-projections on attributes
• Step A. Find all cluster-projections on Ai of
  cluster over (Ai,Aj).
• Step B. Compute all the cluster-projections on Ai
  of cluster over A1,,An by intersecting sets of
  cluster-projects from Step A.
• Step A is NP-Hard! Solution use distinguishing
  sets.
• Distinguishing sets identify different
  cluster-projections.
• Construct distinguishing sets on Ai and extend
  w.r.t Aj some of the candidate distinguishing
  sets on Ai.
• Detailed steps are too long for this
  presentation, sorry!
• StepB intersection of Cluster-projection
• Intersection joint S1?S2 s there exist s1?S1
  and s2?S2 such that ss1?s2 and sgt1
• Apply intersection joint to all sets of attribute
  values on Ai.
• Step2 Try to augment ck with a cluster
  projection ck1 on attribute Ak1. If new cluster
  ltci,ck1gt is a sub-cluster on (Ai,Ak1), i ?
  1,,k, then add ck1 ltc1,ck1gt to the final
  cluster.

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Validation Phase

• Use a required threshold to recognise false
  candidates which do not have enough support
  because some of the 2-clusters combined to form a
  candidate cluster may be due to different sets of
  tuples.

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Experiments and Results

• To compare with STIRR
• Use 1 million tuples, 10 attributes and 100
  attribute values for each attribute.
• CACTUS discovers a broader class of clusters than
  STIRR.

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Experiments and Results
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Conclusion

• The authors formalised the definition of a
  cluster in categorical data
• CACTUS is a fast and efficient algorithm for
  clustering in categorical data.
• I am sorry that I did not show some part of the
  algorithm due to time constraint.

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Question Time