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19 Jan 2009
Functional Genomics and
Microarray Analysis (2)
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Data Clustering
Lecture Overview
Introduction: What is Data Clustering
Key Terms & Concepts
–
Dimensionality
–
Centroids & Distance
–
Distance & Similarity measures
–
Data Structures Used
–
Hierarchical & non

hierarchical
Hierarchical Clustering
–
Algorithm
–
Single/complete/average linkage
–
Dendrograms
K

means Clustering
–
Algorithm
Other Related Concepts
–
Self Organising Maps (SOM)
–
Dimensionality Reduction: PCA & MDS
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Introduction
Analysis of Gene Expression Matrices
Samples
Genes
Gene
expression
levels
Gene Expression Matrix
In a gene expression matrix, rows represent
genes and columns represent
measurements from different experimental
conditions measured on individual arrays.
The values at each position in the matrix
characterise the expression level (absolute
or relative) of a particular gene under a
particular experimental condition.
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Introduction
Identifying Similar Patterns
The goal of microarray data analysis is to find relationships and
patterns in the data to achieve insights in underlying biology.
Clustering algorithms can be applied to the resulting data to find
groups of similar genes or groups of similar samples.
–
e.g. Groups of genes with “similar expression profiles (Co

expressed
Genes)

similar rows in the gene expression matrix
–
or Groups of samples (disease cell lines/tissues/toxicants) with “similar
effects” on gene expression

similar columns in the gene expression
matrix
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Introduction
What is Data Clustering
Clustering of data is a method by which large sets of data is grouped into
clusters (groups) of smaller sets of similar data.
Example: There are a total of 10 balls which are of three different colours.
We are interested in clustering the balls into three different groups.
An intuitive solution is that balls of same colour are clustered (grouped
together) by colour
Identifying similarity by colour was easy, however we want to extend this
to numerical values to be able to deal with gene expression matrices, and
also to cases when there are more features (not just colour).
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Introduction
Clustering Algorithms
A clustering algorithm attempts to find natural groups of components (or
data) based on some notion similarity over the features describing them.
Also, the clustering algorithm finds the centroid of a group of data sets.
To determine cluster membership, many algorithms evaluate the distance
between a point and the cluster centroids.
The output from a clustering algorithm is basically a statistical description
of the cluster centroids with the number of components in each cluster.
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Key Terms and Concepts
Dimensionality of gene expression matrix
Clustering algorithms work by calculating
distances (or alternatively similarity in higher

dimensional spaces), i.e. when the elements
are described by many features (e.g. colour,
size, smoothness, etc for the balls example)
A gene expression matrix of N Genes x M
Samples can be viewed as:
–
N genes, each represented in an M

dimensional
space.
–
M samples, each represented in N

dimensional
space
We will show graphical examples mainly in 2

D
spaces
–
i.e. when N= 2 or M=2
Samples
Genes
Gene
expression
levels
Gene Expression Matrix
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Key Terms and Concepts
Centroid and Distance
+
+
gene A
gene B
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
centroid
In the first example (2 genes & 25 samples) the expression values of 2
Genes are plotted for 25 samples and Centroid shown)
In the second (2 genes & 2 samples) example the distance between the
expression values of the 2 genes is shown
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Key Terms and Concepts
Centriod and Distance
Cluster centroid :
The centroid of a cluster is a point whose parameter values are
the mean of the parameter values of all the points in the clusters.
Distance:
Generally, the distance between two points is taken as a common
metric to assess the similarity among the components of a
population. The commonly used distance measure is the
Euclidean metric which defines the distance between two points
p= ( p1, p2, ....) and q = ( q1, q2, ....) is given by :
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Key Terms and Concepts
Distance/Similarity Measures
Euclidean (L
2
) distance
Manhattan (L
1
) distance
L
m
: (
x
1

x
2

m
+y
1

y
2

m
)
1/m
L
∞
: max(x
1

x
2
,y
1

y
2
)
Inner product: x
1
x
2
+y
1
y
2
Correlation coefficient
Spearman rank correlation coefficient
For simplicity we will concentrate on Euclidean and Manhattan
distances in this course
(x
1
, y
1
)
(x
2
,y
2
)
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Key Terms and Concepts
Distance Measures: Minkowski Metric
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Key Terms
Commonly Used Minkowski Metrics
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Key Terms and Concepts
Distance/Similarity Matrices
Gene Expression Matrix
–
N Genes x M Samples
Clustering is based on distances, this
leads to a new useful data structure:
Similarity/Dissimilarity matrix
–
Represents the distance between
either N Genes (NxN) or M Samples
(MxM)
–
Only need half the matrix, since it is
symmetrical
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Key Terms
Hierarchical vs. Non

hierarchical
Hierarchical clustering is the most commonly used methods for
identifying groups of closely related genes or tissues. Hierarchical
clustering is a method that successively links genes or samples
with similar profiles to form a tree structure
–
much like
phylognentic tree.
K

means clustering is a method for non

hierarchical (flat)
clustering that requires the analyst to supply the number of
clusters in advance and then allocates genes and samples to
clusters appropriately.
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Hierarchical Clustering
Algorithm
•
Given a set of N items to be clustered, and an NxN distance (or
similarity) matrix, the basic process hierarchical clustering is this:
1.
Start by assigning each item to its own cluster, so that if you have N
items, you now have N clusters, each containing just one item.
2.
Find the closest (most similar) pair of clusters and merge them into a
single cluster, so that now you have one less cluster.
3.
Compute distances (similarities) between the new cluster and each of
the old clusters.
4.
Repeat steps 2 and 3 until all items are clustered into a single cluster of
size N.
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Hierarchical Cluster Analysis
Scan matrix for
minimum
Join genes to 1 node
2
3
Update matrix
1
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Hierarchical Clustering
Distance Between Two Clusters
Min
distance
Average
distance
Max
distance
Single

Link
Method / Nearest Neighbor
Complete

Link
/ Furthest Neighbor
Their
Centroids
.
Average
of all cross

cluster pairs.
Whereas it is straightforward to
calculate distance between two
points, we do have various options
when calculating distance between
clusters.
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Key Terms
Linkage Methods for hierarchical clustering
Single

link clustering
(also called the connectedness or minimum
method) : we consider the distance between one cluster and another
cluster to be equal to the shortest distance from any member of one
cluster to any member of the other cluster. If the data consist of
similarities, we consider the similarity between one cluster and another
cluster to be equal to the greatest similarity from any member of one
cluster to any member of the other cluster.
Complete

link clustering
(also called the diameter or maximum
method): we consider the distance between one cluster and another
cluster to be equal to the longest distance from any member of one
cluster to any member of the other cluster.
Average

link clustering
we consider the distance between one cluster
and another cluster to be equal to the average distance from any member
of one cluster to any member of the other cluster.
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Single

Link Method
b
a
Distance Matrix
Euclidean Distance
(1)
(2)
(3)
a,b,c
c
c
d
a,b
d
d
a,b,c,d
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Complete

Link Method
b
a
Distance Matrix
Euclidean Distance
(1)
(2)
(3)
a,b
c
c
d
a,b
d
c,d
a,b,c,d
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Key Terms and Concepts
Dendrograms and Linkage
2
4
6
0
Single

Link
Complete

Link
The resulting tree structure is usally referred to as a dendrogram.
In a dendrogram the length of each tree branch represents the distance
between clusters it joins.
Different dendrograms may arise when different Linkage methods are used
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Two Way Hierarchical Clustering
Note we can do two way
clustering by performing
clustering on both the rows and
the columns
It is common to visualise the
data as shown using a
heatmap.
Don’t confuse the heatmap
with the colours of a
microarray image.
They are different !
Why?
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Basic Ideas : using cluster centroids (means) to represent cluster
Assigning data elements to the closet cluster (centroid).
Goal: Minimise square error (intra

class dissimilarity)
K

Means Clustering
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K

means Clustering
Algorithm
1) Select an initial partition of k clusters
2) Assign each object to the cluster with the closest centroid
3) Compute the new centeroid of the clusters:
4) Repeat step 2 and 3 until no object changes cluster
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The
K

Means
Clustering Method
Example
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Summary
Clustering algorithms used to find similarity relationships between genes,
diseases, tissue or samples
Different similarity metrics can be used
–
mainly Euclidean and Manhattan)
Hierarchical clustering
–
Similarity matrix
–
Algorithm
–
Linkage methods
K

means clustering algorithm
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Data Classification
Lecture Overview
Introduction: Diagnostic and Prognostic Tools
Data Classification
Classification vs. Classification
Examples of Simple Classification Algorithms
–
Centroid

based
–
K

NN
Decision Trees
–
Basic Concept
–
Algorithm
–
Entropy and Information Gain
–
Extracting rules from trees
Bayesian Classifiers
Evaluating Classifiers
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Introduction
Predictive Modelling
Diagnostic Tools:
One of the most exciting areas of Microarray research
is the use of Microarrays to find groups of gene that can be used
diagnostically to determine the disease that an individual is suffering.
–
Tissue Classification Tools: a simple example is given measurements from
one tissue type is to be able to ascertain whether the tissue has markers of
cancer or not, and if so which type of cancer.
Prognostic Tools:
Another exciting area
is given measurements from
an individual’s sample is to prognostically predict the success of a course
of a particular therapy
In both cases we can train a classification algorithm on previously
collected data so as to obtain a predictive modelling tool. The aim of the
algorithm is to find a small set of features and their values (e.g. set of
genes and their expression values) that can be used in future predictions
(or classification) on unseen samples
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Classification:
Obtaining a labeled training data set
Goal: Identify subset of genes that distinguish between treatments,
tissues, etc.
Method
–
Collect several samples grouped by type (e.g. Diseased vs. Healthy) or by
treatment outcome (e.g. Success vs. Failure).
–
Use genes as “features”
–
Build a classifier to distinguish treatments
ID G1
G2
G3
G4
Cancer
1 11.12
1.34
1.97
11.0
No
2 12.34
2.01
1.22
11.1
No
3 13.11
1.34
1.34
2.0
Yes
4 13.34
11.11
1.38
2.23
Yes
5 14.11
13.10
1.06
2.44
Yes
6 11.34
14.21
1.07
1.23
No
7 21.01
12.32
1.97
1.34
Yes
8 66.11
33.3
1.97
1.34
Yes
9 33.11
44.1
1.96
11.23
Yes
To Predict categorical
class labels
construct a
mode
l based on the
training set, and then use the
model in classifying new unseen
data
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Classification:
Generating a predictive model
The output of a classifier is a predictive model that can be used to
classify unseen based on the values of their gene expressions.
The model shown below is a special type of classification models,
known a
Decision Tree.
G1
>22
G3
G4
<=12
>12
No
Yes
No
Yes
<=52
>52
<=22
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Classification
Overview
Training
Data:
Inductive
Learning
System
Classifiers
(Derived Hypotheses)
Task: determine which of a fixed set of classes an example
belongs to
Inductive Learning System:
–
Input: training set of examples annotated with class values.
–
Output:induced hypotheses (model/concept description/classifiers)
Learning : Induce classifiers from training data
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Classification
Overview
Using a Classifier for Prediction
Data to be classified
Classifier
Decision on class
assignment
Using Hypothesis for Prediction: classifying any example described in the
same manner as the data used in training the system (i.e. same set of
features)
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Classification
Examples in all walks of life
The values of the features
in the table can be
categorical or numerical.
However, we only deal
with categorical variables
in this course
The Class Value has to be
Categorical.
Outlook
Sunny
Overcast
Rain
Humidity
Yes
Wind
High
Normal
No
Yes
No
Yes
true
false
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Classification vs. Clustering
Classification
Clustering
•
known number of classes
•
based on a training set
•
used to classify future observations
•
unknown number of classes
•
no prior knowledge
•
used to understand (explore) data
•
Classification is a form of
supervised learning
•
Clustering a form of unsupervised
learning
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Typical Classification Algorithms
Centroid Classifiers
kNN: k Nearest Neigbours
Bayesian Classification: Naïve Bayes and Bayesian
Networks
Decision trees
Neural Networks
Linear Discriminant Analysis
Support Vector Machines
…..
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G2
*
*
*
*
*
*
*
*
*
o
o
o
o
o
o
o
o
o
*
*
o
G2
G1
a*G1 + b*G2 > t

> o !
*
*
*
*
*
*
*
*
*
o
o
o
o
o
o
o
o
o
*
*
o
G1
Linear Classifier:
Non Linear Classifier:
Types of Classifiers
Linear vs. non linear
Linear Classifiers are easier to develop e.g Linear Discriminant Analysis (LDA)
Method, which tries to find a good regression line by minimising the squared
errors of the training data
Linear Classifiers, however, may produce models that are not perfect on the
training data.
Non

linear classifiers tend to be more accurate, may over

fit the data
By over

fitting the data, they may actually perform worse on unseen data
A linear discriminant in 2

D is a
straight line.
In N

D it is a hyperplace
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Types of Classifiers
K

Nearest Neighbour Classifiers
K

NN works by assigning a data point to the
class of its k closest neighbors (e.g. based on
Euclidean or Manhattan distance).
K

NN returns the most common class label
among the k training examples nearest to
x
.
We usually set
K
> 1 to avoid outliers
Variations:
–
Can also use a radius threshold rather than K.
–
We can also set a weight for each neighbour
that takes into account how far it is from the
query point
.
_
+
_
x
+
_
_
+
_
_
+
+
+
+
+
+
+
_
_
_
_
_
Model Training:
None.
Classification:
–
Given a data point,
Locate
K
nearest points.
–
Assign the majority
class of the
K
points
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Types of Classifiers
Decision Trees
Decision tree
–
A flow

chart

like tree structure
–
Internal node denotes a test on an attribute
–
Branch represents an outcome of the test
–
Leaf nodes represent class labels or class distribution
Decision tree generation
–
At start, all the training examples are at the root
–
Partition examples recursively based on selected attributes
Use of decision tree: Classifying an unknown sample
–
Test the attribute values of the sample against the decision tree
Outlook
Sunny
Overcast
Rain
Humidity
Yes
Wind
High
Normal
No
Yes
No
Yes
true
false
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Types of Classifiers
Decision Tree Construction
•
General idea:
•
Using the training data, choose the best feature to be used for the
logical test at the root of the tree.
•
Partition training data into sub

groups based on the values of the
logical test
•
Recursively apply the same procedure (select attribute and split) and
terminate when all the data elements in one branch are of the same
class.
•
Key to Success is how to choose the best feature at each step
•
The basic approach to select a attribute is to examine each attribute
and evaluate its likelihood for improving the overall decision
performance of the tree.
•
The most widely used node

splitting evaluation functions work by
reducing the degree of randomness or ‘impurity” in the current node.
Outlook
Sunny
Overcast
Rain
Humidity
Yes
Wind
High
Normal
No
Yes
No
Yes
true
false
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Decision Tree Construction
Algorithm
Basic algorithm (a greedy algorithm)
–
Tree is constructed in a
top

down recursive manner
–
At start, all the training examples are at the root
–
Attributes are categorical (if continuous

valued, they are discretized in
advance)
–
Examples are partitioned recursively based on selected attributes
–
Test attributes are selected on the basis of a heuristic or statistical
measure (e.g.,
information gain
)
Conditions for stopping partitioning
–
All samples for a given node belong to the same class
–
There are no remaining attributes for further partitioning
–
majority
voting
is employed for classifying the leaf
–
There are no samples left
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Decision Tree
Example
In the simple example
shown, the expression
values which are usually
numbers have been made
into discrete values.
There are more complex
methods that can deal with
numeric features, but are
beyond this course
In the example, I have chosen to use 3 discrete ranges for Gene1, two ranges
(high/low) for genes 2 and , and expressed (yes/no) for gene 3.
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Decision Trees
Using Information Gain
Select the attribute with the highest information gain
Assume there are two classes,
P
and
N
–
Let the set of examples
S
contain
p
elements of class
P
and
n
elements of class
N
–
The amount of information (
entropy) :
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Information Gain in Decision
Tree Construction
Assume that using attribute A a set
S
will be partitioned into
sets {
S
1
,
S
2
, …,
S
v
}
–
If
S
i
contains
p
i
examples of
P
and
n
i
examples of
N
, the expected
information (total entropy) in all subtrees
S
i
generated by the
partition via A is
The encoding information that would be gained by branching
on
A
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Attribute Selection by Information
Gain Computation
Class P: diseased = “yes”
Class N: diseased = “no”
I(p, n) = I(9, 5) =0.940
Compute the entropy for
G1
:
Hence
Similarly
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Extracting Classification Rules from
Trees
Decision Trees can be simplified by representing the knowledge
in the form of
IF

THEN
rules that are easier for humans to
understand
–
One rule is created for each path from the root to a leaf
–
Each attribute

value pair along a path forms a conjunction
–
The leaf node holds the class prediction
Example
IF
G1
= “<=30” AND
G3
= “
no
” THEN
diseased
= “
no
”
IF
G1
= “<=30” AND
G3
= “
yes
” THEN
diseased
= “
yes
”
IF
G1
= “31…40”
THEN
diseased
= “
yes
”
IF
G1
= “>40” AND G4 = “
high
” THEN
diseased
= “
yes
”
IF
G1
= “>40” AND
G4
= “
low
” THEN
diseased
= “
no
”
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Further Notes
We have mainly used examples with two classes in our examples,
however most classification algorithms can work on many class values so
long as they are discrete.
We have also mainly concentrated on examples that work on discrete
feature values
Note that in many cases, the data may be of very high dimensionality,
and this may cause problems for the algorithms, and might need to use
dimensionality reduction methods.
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Summary
Classification algorithms can be used to develop diagnostic and
prognostic tools based on collected data by generating predictive models
that can label unseen data into existing classes.
Simple classification methods: LDA, Centroid

based classifiers and k

NN
Decision Trees:
–
Decision Tree Induction works by choosing the best logical test for each tree
node one at a time, and recursively splitting the data and applying same
procedure
–
Entropy and Information Gain are the key concepts to apply
Not all classifiers generate 100% accuracy, confusion matrices can be
used to evaluate their accuracy.
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