Clustering analysis of microarray gene expression data

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Clustering analysis of microarray gene expression
data
Ping Zhang November 19th, 2008
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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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What is a DNA Microarray?
DNA microarray technology allows measuring
expressions for tens of thousands of genes at a
time
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Scanning/Signal Detection
Cy3 channel
Cy5 channel
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Data-flow schema of microarray data analysis
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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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Gene expression profiles
Expression (relatively levels to reference point
at 0)
Time/Condition
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Similarity between Profiles
expression

• Similarity measure
• Euclidean distance
• Correlation coefficient
• Trend
• Correlation coefficient
• often works better.

0
time
Expression profile
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Pearson Correlation Coefficient

• Compares scaled profiles!
• Can detect inverse relationships
• Most commonly used

nnumber of conditions xaverage expression of
gene x in all n conditions yaverage expression
of gene y in all n conditions sxstandard
deviation of x Systandard deviation of y
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Correlation Pitfalls
Correlation0.97
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Correlation coefficient
Gene Y
Gene X
S(X,Y) 0
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Euclidean Distance

• Scaled versus unscaled
• Cannot detect inverse relation ships

For Gene X(x1, x2,xn) and Gene Y(y1, y2,yn)
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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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Data-Mining through Clustering

• Assumptions for clustering analysis
• Expression level of a gene reflects the genes
  activity.
• Genes involved in same biological process exhibit
  
• statistical relationship in their expression
  profiles.

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Idea of Clustering

• Clustering group objects into clusters so that
• objects in each cluster have similar features
• objects of different clusters have dissimilar
  features

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Methods of Clustering

• discriminant analysis (Fisher,1931)

• K-means (Lloyd,1948)

• hierarchical clustering

• self-organizing maps (Kohonen, 1980)

• support vector machines (Vapnik, 1985)

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Issues in Cluster Analysis

• A lot of clustering algorithms
• A lot of distance/similarity metrics
• Which clustering algorithm runs faster and uses
  less memory?
• How many clusters after all?
• Are the clusters stable?
• Are the clusters meaningful?

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Which Clustering Method Should I Use?

• What is the biological question?
• Do I have a preconceived notion of how many
  clusters there should be?
• How strict do I want to be? Spilt or Join?
• Can a gene be in multiple clusters?
• Hard or soft boundaries between clusters

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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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K-means clustering for expression profiles
Step 1 Transform n (genes) m (experiments)
matrix into n(genes) n(genes) distance matrix
To transform the nm matrix into nn matrix, use
a similarity (distance) metric.
Step 2 Cluster genes based on a k-means
clustering algorithm
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K-means algorithm
The most popular algorithm for clustering
What is so attractive?

• Simple

• Fast

• Mathematically correct

• Invariant to dimension

• Easy to implement

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K-Means Clustering

• Basic Ideas using cluster centre (means) to
  represent cluster
• Assigning data elements to the closet cluster
  (centre).
• Goal Minimize square error (intra-class
  dissimilarity)
• There is no hierarchy.
• Must supply the number of clusters (k) into which
  the data are to be grouped.

2
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K-means Clustering Procedure (1)
Initialization 1 Specify the number of cluster k
-- for example, k 4
Expression matrix
Each point is called gene
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K-means Clustering Procedure (2)
Initialization 2 Genes are randomly assigned to
one of k clusters
or choose random starting centers
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K-means Clustering Procedure (3)
Calculate the mean of each cluster
(6,7)
(3,4)
(3,2)
(1,2)
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K-means Clustering Procedure (4)
Each gene is reassigned to the nearest cluster
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K-means Clustering Procedure (5)
Iterate until the means are converged
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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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Hierarchical clustering (1)
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Hierarchical clustering (2)
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Hierarchical Clustering Results
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Outline

• Gene expression
• Similarity between gene expression profiles
• Concept of clustering
• K-Means clustering
• Hierarchical clustering
• Minimum spanning tree-based clustering

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Graph Representation

• Represent a set of n-dimensional points as a
  graph
• each data point (gene) represented as a node
• each pair of genes represented as an edge with a
  weight defined by the dissimilarity between the
  two genes

n-D data points
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Minimum Spanning Tree

• Spanning tree a sub-graph that has all nodes
  connected and has no cycles
• Minimum spanning tree (MST) a spanning tree with
  the minimum total distance

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How to ConstructMinimum Spanning Tree

• Prims algorithm and Kruskals algorithm
• Kruskals algorithm
• step 1 select an edge with the smallest distance
  from graph
• step 2 add to tree as along as no cycle is
  formed
• step 3 remove the edge from graph
• step 4 repeat steps 1-3 till all nodes are
  connected in tree.

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8
7
10
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3
6
(a)
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Foundation of MST Approach

• Significantly simplifies the data clustering
  problem, while losing very little essential
  information for clustering.
• We have mathematically proved

A multi-dimensional clustering problem is
equivalent to a tree-partitioning problem!
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Clustering by Cutting Long Edge
Hierarchical cutting 1st cut longest edge 2nd
cut second longest edge Work well for easy
cases. Produce many clusters with single element
for some difficult cases.
1
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Tree-Based Clustering

• For each edge, calculate the assessment value
• Find the edge that give the minimum assessment
  value as the place to cut

• Clustering using iterative method
• guarantee to find the global optimality
• using tree-based dynamic programming

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Clustering through Removing Long MST-Edges

• Objective partition an MST into K subtrees so
  that the total edge-distance of all the K
  subtrees in minimized
• Finding K-1 longest MST-edges and cutting them gt
  we get K clusters
• This works as long as the inter-cluster
  edge-distances are clearly larger than the
  intra-cluster edge-distances

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An Iterative Clustering Algorithm

• Find K subtrees Ti of an MST such that to
  minimize
• Informally, the total distance between the center
  of each cluster and its data points is minimized
• The center c of a cluster C is defined as
• the sum of the distances between c and all the
  data points in C is minimized
• Does not work well if the cluster boundary is not
  convex

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A Globally Optimal Clustering Algorithm

• Given an MST T, partition T into K subtrees Ti
  and find a set of data points di, i 1k, di in
  D such that to minimize
• Informally, group data points around the best
  representatives rather than around the center
• Using Dynamic Programming for this algorithm

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Automated Selection of Number of Clusters

• Select transition point in the assessment value
• as thecorrect number of clusters.

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Transition Profiles

• indicatorn (An-1 An) / (An An1)
• Ak is the assessment value for partition with k
  clusters

Our clustering of yeast data
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Reference

• 1 Ying Xu, Victor Olman, and Dong Xu.
  Clustering Gene Expression Data Using a
  Graph-Theoretic Approach An Application of
  Minimum Spanning Trees. Bioinformatics.
  18526-535, 2002.
• 2 Dong Xu, Victor Olman, Li Wang, and Ying Xu.
  EXCAVATOR a computer program for gene expression
  data analysis. Nucleic Acid Research. 31
  5582-5589. 2003.
• Using slides from
• Michael Hongbo Xie, Temple University (in
  2006)
• Vipin Kumar, University of Minnesota
• Dong Xu, University of Missouri

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Acknowledgement