Probation Talk: On Mining Microarray data by OrderPreserving Submatrix

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Probation TalkOn Mining Micro-array data by
Order-Preserving Submatrix

• Lin Cheung
• Kevin Y. Yip
• David W. Cheung
• Ben Kao
• Michael K. Ng

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Abstract

• Study the problem of pattern-based subspace
  clustering
• Unlike traditional clustering methods that focus
  on grouping objects with similar values on a set
  of dimensions
• clustering by pattern similarity finds objects
  that exhibit a coherent pattern of rises and
  falls in subspaces.
• Applications
• DNA micro-array data analysis
• Automatic recommendation systems
• Target marketing systems

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Outline

• Introduction
• Related Work
• Problem Definition
• Algorithm
• Experiment Results
• Future Plan

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Introduction

• Invention of DNA micro-array technologies has
  revolutionized the experimental study of gene
  expression.
• Thousands of genes are probed every day.
• Various gene expression data analysis techniques
  have been intensively studied.

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Introduction

• In DNA micro-array data analysis, gene expression
  data is organized as matrices.
• a row carries the information of a gene
• a column represents a sample for an experiment.
• The number in each cell records the expression
  value of a particular gene under a particular
  sample.
• In this presentation, we will use the terms
• object to mean a row (gene) of a dataset
• dimension (or attribute) to mean a column
  (sample) of a dataset

(columns)
(rows)
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Clustering

• Clustering
• One of the most popular methods of discovering
  useful biological insights from gene expression
  data.
• Group together data points that are close or
  similar to each other over all dimensions into
  clusters
• Traditional Distance-based clustering
• Highly depends on distance (or similarity)
  measure.
• Euclidean distance, Manhattan distance, etc

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Subspace clustering

• DNA micro-array data is typically
  high-dimensional
• Clustering over the global set of attributes
  often fails to extract all meaningful clusters
• Objects tend to exhibit strong similarity only
  over a (unknown) subset of the dimensions.
• Subspace clustering
• Group together data points that are close or
  similar to each other over a subset of the
  dimensions into clusters
• Discovery of clusters that are embedded in
  certain subspaces of high-dimensional data, such
  as gene expression data

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Pattern-based subspace clustering

• A special type of subspace clustering that uses
  pattern similarity as a measure of object
  distances.

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Pattern-based subspace clustering
Raw Data 3 rows and 10 columns
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Pattern-based subspace clustering

• In columns, b, c, f, h, i
• The expression values of the rows follow the same
  rise-and-fall pattern
• The 3 rows form a cluster in the subspace b, c,
  f, h, i.
• Traditional distance functions are sometimes
  inadequate in capturing correlations among rows.

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Pattern-based subspace clustering

• Rearrange the columns so that expression values
  are listed in ascending order
• Form a column sequence f, c, b, i, h
• all increasing under the new column sequence
• f, c, b, i, h is referred as an order-preserving
  pattern
• The length of a pattern is the number of columns
• A row is a supporting row if its values exhibit
  an increasing order with respect to the column
  sequence, where support refers to the number of
  supporting rows
• An order-preserving pattern together with the set
  of supporting rows form an Order Preserving
  Sub-matrix, or OPSM.

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Challenges

• Computationally challenging problem
• Complexity lies in the requirement of
  simultaneously determining both cluster members
  and relevant dimensions.
• Number of potential order-preserving patterns
  grows exponentially with respect to the number of
  attributes.
• A dataset D with n attributes has ?in!/(n-i)!
  potential order-preserving patterns
• There could be tens to hundreds of attributes in
  DNA micro-array datasets
• Number of potential OPSMs are huge
• a set of rows R and a set of columns C form a
  valid OPSM
• Any subset of those rows plus any subset of those
  columns form a valid OPSM too
• FOCUS
• Mine ALL maximal OPSMs only

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Related Work

• OPSM model, Ben-Dor et al. RECOMB 2002 3
• Proposed a partial model returns k good
  quality OPSMs

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Related Work

• d-pCluster, Wang et al, SIGMOD 2002, 11
• Only can discover shift-pattern and
  scaling-pattern

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Formal description of the OPSMproblem

• Consider a gene-expression dataset D, represented
  as a matrix.
• We use O and C to denote the set of rows and
  columns in D, respectively.
• We use di,j to denote the entry of D in row i and
  column j.

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Formal description of the OPSMproblem

• A cluster S is a submatrix of D formed by a
  subset of nS( 2) rows and a subset of mS( 2)
  columns of D.
• Rows and columns in S need not be contiguous in
  D. The rows in S are referenced by their row
  indices in D, each of which is a distinct integer
  in 1, 2, ..., nD. The set of row indices of S
  is denoted as RS.
• Columns in S are similarly referenced. The set of
  columns in S is denoted by CS.

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Formal description of the OPSMproblem

• The columns in S are enclosed in curly brackets,
  e.g., CS c1, c2, ..., cmS . A sequence CS of
  the columns in S is enclosed in angled brackets,
  e.g., CS lt c1, c2, ..., cmS gt. The columns in a
  sequence is totally ordered.
• For the basic OPSM problem, a cluster is a set of
  rows and a set of columns such that entries in
  every row are increasing w.r.t. a particular
  column sequence.
• A cluster S is thus written as S (RS, CS).

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Problem Definition

• Definition 1 A cluster S is an OPSM if there
  exists a sequence of columns such that in the
  cluster S, si,j si,j1 for all i ? 1, 2, ...,
  nS and all j ? 1, 2, ...,mS-1.
• A cluster S is a subcluster of a cluster S' if RS
  ? RS and CS ? CS. A cluster S is a proper
  subcluster of a cluster S' if S is a subcluster
  of S' and either RS ? RS or CS ? CS .
• Definition 2 An OPSM is a maximal OPSM if it is
  not a proper subcluster of any OPSM.

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Problem Definition

• (Maximal size-constrained OPSM problem) Given a
  data matrix D, a supporting row thresh old nmin,
  and a column threshold mmin, find all maximal
  OPSMs S in D such that nS nmin and mS mmin.

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Algorithm

• Devised an apriori-like algorithm to solve the
  problem
• Invented a new data structure
• Head-Tail Trees
• Efficient processing of
• identifying all pairs of column sequences with
  length k where the first k - 1 indices of the
  first clusters column sequence equal the last k
  - 1 indices of the second clusters column
  sequence, and
• Intersecting two sets of row indices

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Algorithm
Given a dataset D, nmin 2 and mmin 2
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Algorithm
The head tree for clusters with 2 columns
The tail tree for clusters with 2 columns
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Algorithm
The head tree for clusters with 3 columns
The tail tree for clusters with 3 columns
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Algorithm

• Property 1 (A priori property) A cluster S is an
  OPSM if and only if all proper subclusters of S
  are also OPSM.
• Property 2 (Transitivity) If S1 (RS1
  ,ltx1,x2,...,xi ,y1,y2,...,yj gt) and S2 (RS2 ,lt
  y1,y2,...,yj ,z1,z2,...,zk gt) are two maximal
  OPSMs and RS1 n RS2 contains nmin or more
  indices, then S (RS1 n RS2 , lt x1,x2,...,xi ,
  y1,y2,...,yj ,z1,z2,...,zk gt) is a maximal OPSM.

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Experiments

• Yeast Dataset
• Used in Y. Cheng and G.M. Church. Biclustering of
  expression data. ISMB, 2000.
• 2884 rows x 17 columns
• 54 gene groups
• Pick a gene group and put all the genes of the
  gene group in a cluster
• Find all dimension pairs along which the
  expression values of all the genes move in the
  same direction
• Merge the results in step 2 to find OPSMs with
  more than 2 dimensions
• Repeat 1

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Scalability with respect to number of columns
Number of rows 1000, Nmin 10, mmin 2
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Scalability with respect to number of rows
Number of columns 10, Nmin 10, mmin 2
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Future Work

• Uncertainty, experimental errors exist in
  micro-array datasets
• Mine error-tolerated OPSMs

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References

• 1 R. Agrawal and R. Srikant. Fast algorithms
  for mining association rules in large databases.
  VLDB, 1994.
• 2 Y. Cheng and G.M. Church. Biclustering of
  expression data. ISMB, 2000.
• 3 A. B.-D. et. al. Discovering local structure
  in gene expression data the order-preserving
  submatrix problem. RECOMB, 2002.
• 4 J. P. et. al. MaPle A fast algorithm for
  maximal patternbased clustering. ICDM, 2003.
• 5 J. P. B. et. al. Significance and statistical
  errors in the analysis of dna microarray data.
  PNAS, 2002.
• 6 Y. K. et. al. Spectral biclustering of
  microarray cancer data Co-clustering genes and
  conditions. Genome Research, 13(4)703716, 2003.
• 7 L. Lazzeroni and A. Owen. Plaid models for
  gene expression data. Statistica Sinica,
  126186, 2002.
• 8 J. Liu and W. Wang. Op-cluster Clustering by
  tendency in high dimensional space. ICDM, 2003.
• 9 J. Liu, J. Yang, and W. Wang. Biclustering in
  gene expression data by tendency. CSB, 2004.
• 10 H. Wang, F. Chu, W. Fan, P. Yu, and J. Pei.
  A fast algorithm for subspace clustering by
  pattern similarity. SSDBM, 2004.
• 11 H. Wang, W. Wang, J. Yang, and P. S. Yu.
  Clustering by pattern similarity in large data
  sets. SIGMOD, 2002.
• 12 J. Yang, H. Wang, W. Wang, and P. Yu.
  Enhanced biclustering on expression data. BIBE,
  2003.