Sound Mathematical Foundation and Modification of TRICLUSTER

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Sound Mathematical Foundation and Modification of
TRICLUSTER

• Presented by
• Morshed Osmani

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Problem Statement

• We have a paper on clustering. A new method
  TriCluster is used to cluster gene expression
  data.
• The paper has a weak mathematical notation system
  (as indicated by Dr. Perrizo).
• Large search space.

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My Planned Research-Work

• Correct those mathematical notation
• Develop method for pruning large search space
• Implement the modified solution and get some
  simulation result
• Compare the modified algorithm with original
  papers algorithm

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Original Paper

• TRICLUSTER An Effective Algorithm for Mining
  Coherent Clusters in 3D Microarray Data
• Lizhuang Zhao, Mohammed J. Zaki
• Rensselaer Polytechnic Institute, New York
• ACM SIGMOD international conference on Management
  of data, 2005

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Introduction

• Traditional clustering algorithms work in the
  full dimensional space.
• Biclustering, on the other hand, does not have
  such a strict requirement. If some points are
  similar in several dimensions (a subspace), they
  will be clustered together in that subspace.
• Biclustering is able to identify the
  co-expression patterns of a subset of genes that
  might be relevant to a subset of the samples of
  interest.

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Introduction (Cont.)

• There has been a lot of interest in mining gene
  expression patterns across time. These approaches
  are also mainly two-dimensional, i.e., finding
  patterns along the gene-time dimensions.
• The paper deals with mining tri-clusters, i.e.,
  mining coherent clusters along the
  gene-sample-time (temporal) or gene-sample-region
  (spatial) dimensions.
• The authors claim TRICLUSTER is the first 3D
  microarray subspace clustering method.

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Related Work

• There has been work on mining gene expression
  patterns across time.
• There are many full-space and biclustering
  algorithms designed to work with microarray
  datasets, such as feature based clustering, graph
  based clustering and pattern based clustering.
• There is no previous method that mines
  tri-clusters.

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Challenges

• Biclustering itself is known to be a NP-hard
  problem. So heuristic methods or probabilistic
  approximations are used .
• Microarray data is inherently susceptible to
  noise, due to varying experimental conditions,
  thus it is essential that the methods be robust
  to noise.
• As we do not understand the complex gene
  regulation circuitry in the cell, clustering
  methods should allow overlapping clusters that
  share subsets of genes, samples or
  time-courses/spatial regions.
• Furthermore, the methods should be flexible
  enough to mine several (interesting) types of
  clusters, and should not be too sensitive to
  input parameters.

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Mathematical Notations Used
Let
be a set of n genes,
let
be a set of m biological samples (e.g.,
different tissues or experiments)
be a set of l experimental time points.
let
matrix
A three dimensional microarray dataset is a
real-valued
whose three dimensions correspond to genes,
samples and times respectively
A tricluster C is a submatrix of the dataset D,
provided certain conditions of homogeneity are
satisfied.
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Problems with Notation

• Used unconventional notation system which seems
  to be incorrect.
• May confuse the reader to comprehend the actual
  meaning.
• Wrong use of Cartesian product.
• May be solved by using mapping (function).

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The TRICLUSTER Algorithm

• 3D microarray datasets have more genes than
  samples, and perhaps an equal number of time
  points and samples, i.e.,
• Due to the symmetric property, TRICLUSTER always
  transposes the input 3D matrix such that
• the dimension with the largest cardinality (say
  G) is 1st dimension
• then make S as the 2nd and T as the 3rd
  dimension.

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Steps of TRICLUSTER

• TRICLUSTER has following main steps
• For each GxS time slice matrix, find the valid
  ratio-ranges for all pair of samples, and
  construct a range multigraph
• Mine the maximal biclusters from the range
  multigraph
• Construct a graph based on the mined biclusters
  (as vertices) and get the maximal TRICLUSTERs
• Optionally, delete or merge clusters if certain
  overlapping criteria are met.

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Future Direction

• Reduce the search space
• - How (yet to be determined)
• Implement the modified algorithm
• Compare the result with result from previous
  algorithm

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Need Suggestion

• Here dataset D is a data cube. What are the
  benefits we may receive from the data cube
  model in this case?
• How can this problem be integrated to DataDEX (if
  there is a possibility)?
• How can we reduce the search space?
• Any other comments or suggestions are welcome.