# Introduction to Bioinformatics 1. Course Overview

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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

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
-

Method / Nearest Neighbor

Complete
-

/ 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

Single
-

(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
-

(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
-

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
-

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
-

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

2

4

6

0

Single
-

Complete
-

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

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

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
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.