?-Clusters Capturing Subspace Correlation in a Large Data Set

1
?-Clusters Capturing Subspace Correlation in a
Large Data Set

• Authors Yang Jiong, Wei Wang etc.(ICDE02)
• Presenter Xuehua Shen
• xshen_at_uiuc.edu

2
Presentation Layout

• Overview of Clustering
• Related Work of ?-Clusters
• ?-Clusters Model
• FLOC algorithm

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Clustering

• Clustering the process of grouping a set of
  objects into classes of similar objects
• Similar to one another within the same cluster
• Dissimilar to the objects in other clusters

4
Major Clustering Methods

• Partition algorithm
• Hierarchy algorithm
• Density-based
• Grid-based
• Model-based

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Similarity

• Clustering the process of grouping a set of
  objects into classes of similar objects
• But how to define similarity?

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Similarity cont.

• Traditional clustering model based on distance
  functions
• Some popular ones include Minkowski distance
• where i (xi1, xi2, , xip) and j (xj1, xj2,
  , xjp) are two p-dimensional data objects, and q
  is a positive integer
• But strong correlations may still exist among a
  set of objects even if they are far apart from
  each other as measured by the distance function

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Similarity cont.

• ?-Clusters model similar when exhibiting a
  coherent pattern on a subset of dimensions
• Can cluster objects which show shifting pattern
  or scaling pattern

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Similarity cont.

• Example of Coherent Pattern
• Shifting Pattern Scaling
  Pattern

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Subspace Clustering

• From high dimensional clustering (problematic)
  To subspace clustering
• Not restricted with fixed ordering of columns
  contrasted with pattern in time-series
  data
• Challenge curse of dimensionality!

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Subspace Clustering cont.

• Example of subspace clustering

CH11 CH1B CH1D CH2I CH2B
CTFC3 4392 284 4108 280 228
VPS8 401 281 120 275 298
EFB1 318 280 37 277 215
SSA1 401 292 109 580 238
FUN14 2857 285 2576 271 226
SP07 228 290 48 285 224
MDM10 538 272 266 277 236
CYS3 322 288 41 278 219
CH11 CH1D CH2B

VPS8 401 120 298
EFB1 318 37 215




CYS3 322 41 219
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Applications

• Microarray Data Analysis in Biology
• E-Commerce

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Microarray Data Analysis

• Matrix (Dense)
• Rows Genes
• Columns Various Samples
• experiment conditions or
  tissues
• Values in Matrix expression level
• relative abundance of the mRNA of a gene
  under
• a specific condition

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Microarray Data Analysis cont.

• From Scaling Pattern to Shifting Pattern
• Red Interested Gene, Green Controlled Gene
• Investigations show that several genes contribute
  
• to a disease, which motivates researchers to
• identify a subset of genes whose expression
  levels
• rise and fall coherently under a subset of
  conditions

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E-Commerce

• Example Rating of Movies (1 lowest rate, 10
  highest rate)
• Shifting Pattern
• If a new movies and 1st viewer rate 7 and 3rd
  viewer rate 9, 2nd viewer probably will like this
  movie too

Movie1 Movie2 Movie 3 Movie4
Viewer1 1 2 3 6
Viewer2 2 3 4 7
Viewer3 4 5 6 9
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Presentation Layout

• Overview of clustering
• Related Work of ?-Clusters
• ?-Clusters Model
• FLOC algorithm

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Related Work

• CLIQUE, ORCLUS, PROCLUS (subspace clustering)
• Cant capture neither the shifting pattern nor
  the scaling pattern
• Bicluster model proposed as a measure of
  coherence of genes and conditions in a submatrix
  of a DNA array

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Bicluster

• Model Mean squared residue score of submatrix
• a submatrix AIJ is called a ?-biCluster if
  H(I,J)??
• Algorithm A random algorithm to give an
  approximate answer

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Weakness of bicluster

• Missing Values
• Constraints

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Presentation Layout

• Overview
• Related Work
• ?-Clusters Model
• FLOC algorithm

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Occupancy Threshold

• A parameter to control the percentage of missing
  values in a submatrix
• Ji is the specified attributes for object i in
  ?-Clusters
• J is the number of attributes in the ?-Clusters
  

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Occupancy Threshold cont.

• Similar occupancy threshold for attribute j in
  ?-Clusters
• Example ?0.6

1 3
4 5
3 4
1 3 3
3 4 5
3 4 4
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Volume

• The volume of a ?-Clusters(I,J) is the number of
  specified entries dij in (I,J)
• Example
• volume is 339

1 3 3
3 4 5
3 4 4
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Base

• Object Base
• Attribute Base

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Base cont.

• ?-Clusters Base
• For perfect ?-Clusters

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Residue

• Entry Residue
• if dij is specified
• otherwise is 0

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Residue cont.

• ?-Clusters Residue
• r-residue ?-Clusters if ?-clusters residue is
  equal to or smaller than r

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Presentation Layout

• Overview of Clustering
• Related Work of ?-Clusters
• ?-Clusters Model
• FLOC algorithm(Flexible Overlapping Clustering)

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Flow Chart

• 
  Y
• 
  N

Generating initial clusters
Determine the best action For each row and
each column
Perform the best action sequentially
improved
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Initial Cluster

• Randomly Generate k initial cluster
• Different parameters ? makes different size
  cluster

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Choose best actions

• For every object or attribute, there are k
  actions which can be done,
• Choose the best action among the k candidates
  according to gain
• Gain is the difference between original residue
  and the residue assuming the action is done on
  the cluster

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Choose Best Actions cont.

• Even if gain is negative sometimes
• we do the action in order to get the global
  optimum

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Do the actions sequentially

• Generate the actions sequence
• 1) the same order in all iterations
• 2) random order sequence
• 3) weighted random order sequence

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Output the Best cluster

• After some iterations, no improvement of minimum
  residue, algorithm stops and k best cluster is
  output

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End

• Thank you!