Biclustering of Expression Data

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Biclustering of Expression Data

• by Yizong Cheng and Geoge M. Church
• Presented by Bojun Yan
• March 25, 2004

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outline

• MicroArray and its relative research
• 1.1 MicroArray Gene Expression Data
• 1.2 Main research about MicroArray
• 2. Why Bicluster?
• 2.1 Preceding research and its faults
• 2.2 The concept of Bicluster
• 2.3 Similarity measure
• 3. The hardness of Bicluster

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• 4. Methods proposed by this paper
• 4.1 Relative Works and papers goal
• 4.2 Definition of mean squared residue
  score
• 4.3 Some special matrices scores
• 4.4 Some Theorems deduced by authors
• 4.5 Algorithms proposed by this paper
• 5. Experiment
• 5.1 Data preparation
• 5.2 Determining Algorithm Parameters
• 5.3 Final Algorithm
• 5.4 Results and Display

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1. MicroArray and its relative Research

• 1.1 MicroArray Gene expression data
• Being generated by DNA chips and other
  microarray technique, Row---Genes,
  Column---Conditions or Samples
• 1.2 Main Research about MicroArray
• (1) Gene Clustering Finding the genes
  having similar functions
• (2) Conditions Clustering Helpful to case
  analysis
• (3) Classification Tumor Classification,
  Cancer prediction
• (4) Gene Selection Find the genes relative
  to some disease
• (5) Gen Network Explore the regulatory
  interaction between the genes
• 1.3 Paper Target Biclustering

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2. Why Bicluster?

• 2.1 Preceding research and its faults
• Goal Discover the regulatory patterns or
  condition similarities
• Methods Based on Euclidean distance or the dot
  product between the vectors (equally weighted)
• (1) Group genes (row)
• (2) Group conditions (column)
• Result Partition the genes or conditions into
  mutually exclusive groups or hierarchies

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• Faults obscuring some other similarity groups
  while discovering some similarity groups
• 2.2 The concept of Bicluster
• Clustering the genes(rows) and
  conditions(columns) simultaneously---subspace
  clustering
• 2.3 Similarity Measure
• (1)Based on Distance Metric, such as Minkowski
  distances
• (2)Cosine Measure

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• (3)Pearson Correlation
• (4)Extended Jaccard Similarity
• (5)Mean Sqare Residue (proposed by this
  paper)
• A measure of the coherence of the genes
  and conditions in the bicluster
• Symmetric function of the genes and
  conditions
• Group genes and conditions simultaneously

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3. Hardness of the bicluster

• The problem of finding a maximum bicluster with a
  score lower than a threshold includes the problem
  of finding a maximum biclique in a bipartite
  graph as a special case
• Finding the largest constant square submatrix is
  proven to be NP-hard (Johnson, 1987)
• The problem of finding a minimum set of
  biclusters, either mutually exclusive or
  overlapping, to cover all the elements in a data
  matrix has been shown to be NP-hard(Orlin,1977)

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4. Methods proposed by this paper

• 4.1 Relative Works and the papers goal
• Relative Works
• Divisive algorithm partitioning data into sets
  with approximately constant values, proposed by
  Morgan and Sonquist(1963) and Hartigan(1972)
• Hartigan mentioned that the criterion for
  partitioning may be a two-way analysis of
  variance model, similar to the mean squared
  residue scoring proposed in this article
• Mirkin(1996) presents a node addition algorithm.

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• biclustering has been used by Mirkin(1996),
  which means simultaneous clustering of both row
  and column sets in a data matrix.
• The term direct clustering(Hartigan 1972),and
  box clustering(Mirkin,1996) have the same
  meaning.
• (2) The Papers Goal and criterion
• Goal Finding of a set of genes showing
  strikingly similar up-regulation and
  down-regulation under a set of conditions.
• Criterion A low mean squared residue score plus
  a large variation from the constant as a
  criterion for identifying these genes and
  conditions
• Overlapping Biclusters should be allowed to
  overlap in expression data analysis

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4.2 Definition of mean squared residue score
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• The row variance
• It is an accompanying score to reject trival
  or constant biclusters.

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4.3 Scores of some special matrice

• A special case for a perfect score( a zero mean
  squared residue score) is a constant bicluster of
  elements of a single value
• For the matrix aijij, i,jgt0, no submatrix of a
  size larger than a single cell has a score lower
  than 0.5
• A KK matrix of all 0s except one 1 has the score
  
• Equation
• A matrix with elements randomly and uniformly
  generated in the range of a,b has an expected
  score of (b-a)(b-a)/12. For example the range is
  0,800, the expected score is 53,333.

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• Some characteristic of mean square residue score
• (1)Adding a constant number to the matrix
  will not affect the H(I,J) score
• (2)Multiplying a constant number will affect
  the score (by the square of the constant)
• (3)Both will not affect the ranking of the
  biclusters in a matrix

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4.4 Theorems deduced by authors
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Comments on Algorithm 0

• Algorithm 0, although a polynomial-time one, will
  not be efficient enough for a quick analysis of
  most expression data matrices.
• The complexity of Algorithm 0 is o((nm)nm)

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Comments on Algorithm 1

• In each iterate, a complete recalculation for
  step1 and step 2 is needed
• The time complexity of Algorithm 1 is o(nm)
• Higher efficiency than Algorithm 0, but not the
  best.

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Comments on Algorithm 2

• Need to properly select parameter agt1
• Without updating the score after the removal of
  each node
• The time complexity of Algorithm 2 is
  o(lognlongm)
• One may miss some large d-bicluster

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Comments on Algorithm 3

• The time complexity is o(mn)
• The resulting d-bicluster may still not be
  maximal because of two reasons
• (1)Lemma 3 only gives a sufficient
  condition for adding rows and conditions
• (2)By adding rows and columns, the score
  may decrease to the point it much smaller than d

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5. Experiment 5.1 Data preparation

• Datasets and Parameters
• (1)Yeast data,o-value300, n100
• (2)Human data, o-value1200,n100
• Missing Data Replacement
• Replace the missing data using the random
  number underlying the uinform distriubiton
• Biclusters is Compared to the Cluster results
  from
• (1)Travazoie et al. (1999)
• (2)Alizadeh et al. (2000)

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5.2 Determining Algorithm Parameters

• Thinking about the clusters from the papres
  Travazoie et al. (1999) and Alizadeh et al.
  (2000)
• For yeast data, d 300, a1.2
• For human gene data, d 1200, a1.2
• The number of biclusters is n100
• Masking discovered Biclusters Each time a
  bicluster was discovered, the elements in it will
  be replaced by random number because the
  algorithms are deterministic

5.3 Final Algorithm
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Biclusters for Yeast data
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Biclusters for Yeast data
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Biclusters for Yeast data
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Biclusters for Yeast data
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Biclusters for human data
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Biclusters for human data