DNA Chips and Their Analysis Comp. Genomics: Lecture 13.

DNA Chips and Their Analysis

Comp. Genomics: Lecture 13

What is a DNA Microarray?

• An experiment on the order of 10k elements

• A way to explore the function of a gene

• A snapshot of the expression level of an entire phenotype under given test conditions

Some Microarray Terminology

• Probe: ssDNA printed on the solid substrate (nylon or glass) These are the genes we are going to be testing

• Target: cDNA which has been labeled and is to be washed over the probe

Microarray Fabrication

• Deposition of DNA fragments– Deposition of PCR-amplified cDNA clones– Printing of already synthesized

oligonucleotieds

• In Situ synthesis– Photolithography– Ink Jet Printing– Electrochemical Synthesis

From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

cDNA Microarrays and Oligonucleotide Probes


cDNA Arrays Oligonucleotide Arrays

•Long Sequences•Spot Unknown Sequences•More variability

•Short Sequences•Spot Known Sequences•More reliable data

In Situ Synthesis

• Photochemically synthesized on the chip• Reduces noise caused by PCR, cloning,

and Spotting• As previously mentioned, three kinds of In

Situ Synthesis– Photolithography– Ink Jet Printing– Electrochemical Synthesis


Photolithography

• Similar to process used to build VLSI circuits

• Photolithographic masks are used to add each base

• If base is present, there will be a hole in the corresponding mask

• Can create high density arrays, but sequence length is limited


Photodeprotection

mask

C

Ink Jet Printing

• Four cartridges are loaded with the four nucleotides: A, G, C,T

• As the printer head moves across the array, the nucleotides are deposited where they are needed


Electrochemical Synthesis

• Electrodes are embedded in the substrate to manage individual reaction sites

• Electrodes are activated in necessary positions in a predetermined sequence that allows the sequences to be constructed base by base

• Solutions containing specific bases are washed over the substrate while the electrodes are activated


Color Coding

• Tables are difficult to read• Data is presented with a color scale• Coding scheme:

– Green = repressed (less mRNA) gene in experiment

– Red = induced (more mRNA) gene in experiment

– Black = no change (1:1 ratio)

• Or– Green = control condition (e.g. aerobic)

– Red = experimental condition (e.g. anaerobic)

• We only use ratio

Campbell & Heyer, 2003

http://www.bio.davidson.edu/courses/genomics/chip/chip.html

Application of Microarrays

• We only know the function of about 20% of the 30,000 genes in the Human Genome– Gene exploration– Faster and better

• Can be used for DNA computing

http://www.gene-chips.com/sample1.html


A Data Mining Problem

• On a given Microarray we test on the order of 10k elements at a time

• Data is obtained faster than it can be processed

• We need some ways to work through this large data set and make sense of the data

Example data: fold change (ratios)

Name 0 hours 2 hours 4 hours 6 hours 8 hours 10 hours

Gene C 1 8 12 16 12 8

Gene D 1 3 4 4 3 2

Gene E 1 4 8 8 8 8

Gene F 1 1 1 0.25 0.25 0.1

Gene G 1 2 3 4 3 2

Gene H 1 0.5 0.33 0.25 0.33 0.5

Gene I 1 4 8 4 1 0.5

Gene J 1 2 1 2 1 2

Gene K 1 1 1 1 3 3

Gene L 1 2 3 4 3 2

Gene M 1 0.33 0.25 0.25 0.33 0.5

Gene N 1 0.125 0.0833 0.0625 0.0833 0.125


Example data: log2 transformation


Gene C 0 3 3.58 4 3.58 3

Gene D 0 1.58 2 2 1.58 1

Gene E 0 2 3 3 3 3

Gene F 0 0 0 -2 -2 -3.32

Gene G 0 1 1.58 2 1.58 1

Gene H 0 -1 -1.60 -2 -1.60 -1

Gene I 0 2 3 2 0 -1

Gene J 0 1 0 1 0 1

Gene K 0 0 0 0 1.58 1.58

Gene L 0 1 1.58 2 1.58 1

Gene M 0 -1.60 -2 -2 -1.60 -1

Gene N 0 -3 -3.59 -4 -3.59 -3


Pearson Correlation Coefficient, r values in [-1,1] interval

• Gene expression over time is a vector, e.g. for gene C: (0, 3, 3.58, 4, 3.58, 3)

• Given two vectors X and Y that contain N elements, we calculate r as follows:

Cho & Won, 2003

Pearson Correlation Coefficient, r (cont.)

• X = Gene C = (0, 3.00, 3.58, 4, 3.58, 3)Y = Gene D = (0, 1.58, 2.00, 2, 1.58, 1)

• ∑XY = (0)(0)+(3)(1.58)+(3.58)(2)+(4)(2)+(3.58)(1.58)+(3)(1) = 28.5564

• ∑X = 3+3.58+4+3.58+3 = 17.16• ∑X2 = 32+3.582+42+3.582+32 = 59.6328• ∑Y = 1.58+2+2+1.58+1 = 8.16• ∑Y2 = 1.582+22+22+1.582+12 = 13.9928• N = 6• ∑XY – ∑X∑Y/N = 28.5564 – (17.16)(8.16)/6 = 5.2188 • ∑X2 – (∑X)2/N = 59.6328 – (17.16)2/6 = 10.5552• ∑Y2 – (∑Y)2/N = 13.9928 – (8.16)2/6 = 2.8952• r = 5.2188 / sqrt((10.5552)(2.8952)) = 0.944

Example data: Pearson correlation coefficient

Gene C Gene D Gene E Gene F Gene G Gene H Gene I Gene J Gene K Gene L Gene M Gene N

Gene C 1 0.94 0.96 -0.40 0.95 -0.95 0.41 0.36 0.23 0.95 -0.94 -1

Gene D 0.94 1 0.84 -0.10 0.94 -0.94 0.68 0.24 -0.07 0.94 -1 -0.94

Gene E 0.96 0.84 1 -0.57 0.89 -0.89 0.21 0.30 0.43 0.89 -0.84 -0.96

Gene F -0.40 -0.10 -0.57 1 -0.35 0.35 0.60 -0.43 -0.79 -0.35 0.10 0.40

Gene G 0.95 0.94 0.89 -0.35 1 -1 0.48 0.22 0.11 1 -0.94 -0.95

Gene H -0.95 -0.94 -0.89 0.35 -1 1 -0.48 -0.21 -0.11 -1 0.94 0.95

Gene I 0.41 0.68 0.21 0.60 0.48 -0.48 1 0 -0.75 0.48 -0.68 -0.41

Gene J 0.36 0.24 0.30 -0.43 0.22 -0.21 0 1 0 0.22 -0.24 -0.36

Gene K 0.23 -0.07 0.43 -0.79 0.11 -0.11 -0.75 0 1 0.11 0.07 -0.23

Gene L 0.95 0.94 0.89 -0.35 1 -1 0.48 0.22 0.11 1 -0.94 -0.95

Gene M -0.94 -1 -0.84 0.10 -0.94 0.94 -0.68 -0.24 0.07 -0.94 1 0.94

Gene N -1 -0.94 -0.96 0.40 -0.95 0.95 -0.41 -0.36 -0.23 -0.95 0.94 1Campbell & Heyer, 2003

Example: Reorganization of data



Gene M 1 0.33 0.25 0.25 0.33 0.5

Gene N 1 0.125 0.0833 0.0625 0.0833 0.125

Gene H 1 0.5 0.33 0.25 0.33 0.5

Gene K 1 1 1 1 3 3

Gene J 1 2 1 2 1 2

Gene E 1 4 8 8 8 8

Gene C 1 8 12 16 12 8

Gene L 1 2 3 4 3 2

Gene G 1 2 3 4 3 2

Gene D 1 3 4 4 3 2

Gene I 1 4 8 4 1 0.5

Gene F 1 1 1 0.25 0.25 0.1

• Replace each entry xi by its rank in vector x.

• Then compute Pearson correlation coefficients of rank vectors.

• Example: X = Gene C = (0, 3.00, 3.41, 4, 3.58, 3.01) Y = Gene D = (0, 1.51, 2.00, 2.32, 1.58, 1)

• Ranks(X)= (1,2,4,6,5,3)• Ranks(Y)= (1,3,5,6,4,2)• Ties should be taken care of: (1) rare

(2) randomize (small effect)

Pearson Rank Correlation Coefficient

Grouping and Reduction

• Grouping: discovers patterns in the data from a microarray

• Reduction: reduces the complexity of data by removing redundant probes (genes) that will be used in subsequent assays

Unsupervised Grouping: Clustering

• Pattern discovery via grouping

similarly expressed genes together

• Techniques most often used k-Means Clustering Hierarchical Clustering Biclustering Additional Methods: Self Organizing Maps (SOMS),

plaid models, singular value decomposition (SVD),

order preserving submatrices (OPSM),……

Clustering Overview

• Different similarity measures– Pearson Correlation Coefficient– Cosine Coefficient– Euclidean Distance– Information Gain– Mutual Information– Signal to noise ratio– Simple Matching for Nominals

Clustering Overview (cont.)

• Different Clustering Methods– Unsupervised

• k-means Clustering (k nearest neighbors)• Hierarchical Clustering• Self-organizing map

– Supervised• Support vector machine• Ensemble classifier

Data Mining

Clustering Limitations

• Any data can be clustered, therefore we must be careful what conclusions we draw from our results

• Clustering is often randomized and can and will produce different results for different runs on same data

K-means Clustering

• Given a set of m data points in

n-dimensional space and an integer k

• We want to find the set of k points in

n-dimensional space that minimizes the Euclidean (mean squared) distance from each data point to its nearest center

• No exact polynomial-time algorithms are

known for this problem (no wonder, NP-hard!)“A Local Search Approximation Algorithm for k-Means Clustering” by Kanungo

et. al

K-means Heuristic (Lloyd’s Algorithm)

• Has been shown to converge to a locally optimal solution

• But can converge to a solution arbitrarily bad compared to the optimal solution

•“K-means-type algorithms: A generalized convergence theorem and characterization of local optimality” by Selim and Ismail

•“A Local Search Approximation Algorithm for k-Means Clustering” by Kanungo et al.

K=3

Data Points

Optimal Centers

Heuristic Centers

Euclidean Distance

n

iiiE yxyxd

1

2)(),(

543),( 22 AOd E

Now to find the distance between two points, say the origin and the point (3,4):

Simple and Fast! Remember this when we consider the complexity!

Finding a Centroid

We use the following equation to find the n dimensional centroid point (center of mass) amid k (n dimensional) points:

),...,2

,1

(),...,,( 11121 k

xnth

k

ndx

k

stxxxxCP

k

ii

k

ii

k

ii

k

Example: Let’s find the midpoint between three 2D points, say: (2,4) (5,2) (8,9)

2 5 8 4 2 9( , ) (5,5)

3 3CP

K-means Iterative Heuristic

• Choose k initial center points “randomly”• Cluster data using Euclidean distance (or other distance

metric)• Calculate new center points for each cluster, using only points within the cluster• Re-Cluster all data using the new center points

(this step could cause some data points to be placed in a different cluster)

• Repeat steps 3 & 4 until no data points are moved from one cluster to another (stabilization), or till some other convergence criteria is met


An example with 2 clusters

1. We Pick 2 centers at random

2. We cluster our data around these center points

Figure Reproduced From “Data Analysis Tools for DNA Microarrays” by Sorin Draghici

K-means example with k=2

3. We recalculate centers based on our current clusters



4. We re-cluster our data around our new center points



5. We repeat the last two steps until no more data points are moved into a different cluster


Choosing k

• Run algorithm on data with several different values of k

• Use advance knowledge about the characteristics of your test(e.g. Cancerous vs Non-Cancerous Tissues,

in case the experiments are being clustered)

Cluster Quality

• Since any data can be clustered, how do we know our clusters are meaningful?– The size (diameter) of the cluster

vs. the inter-cluster distance– Distance between the members of a cluster and the

cluster’s center– Diameter of the smallest sphere containing the cluster


Cluster Quality Continued

diameter=5

diameter=5distance=2

0

distance=5

Quality of cluster assessed by ratio of distance to nearest cluster and cluster diameter


Cluster Quality Continued

Quality can be assessed simply by looking at the diameter of a cluster (alone????)

A cluster can be formed by the heuristic even when there is no similarity between clustered patterns. This occurs because the algorithm forces k clusters to be created.From “Data Analysis Tools for DNA Microarrays” by

Sorin Draghici

Characteristics of k-means Clustering

• The random selection of initial center points creates the following properties– Non-Determinism– May produce clusters without patterns

• One solution is to choose the centers randomly from existing patterns


Heuristic’s Complexity

• Linear in the number of data points, N

• Can be shown to have run time cN, where c does not depend on N, but rather the number of clusters, k

• (not sure about dependence on dimension, n?)

heuristic is efficient


Hierarchical Clustering

- a different clustering paradigm


Hierarchical Clustering (cont.)

Gene C Gene D Gene E Gene F Gene G Gene H Gene I Gene J Gene K Gene L Gene M Gene N

Gene C 0.94 0.96 -0.40 0.95 -0.95 0.41 0.36 0.23 0.95 -0.94 -1

Gene D 0.84 -0.10 0.94 -0.94 0.68 0.24 -0.07 0.94 -1 -0.94

Gene E -0.57 0.89 -0.89 0.21 0.30 0.43 0.89 -0.84 -0.96

Gene F -0.35 0.35 0.60 -0.43 -0.79 -0.35 0.10 0.40

Gene G -1 0.48 0.22 0.11 1 -0.94 -0.95

Gene H -0.48 -0.21 -0.11 -1 0.94 0.95

Gene I 0 -0.75 0.48 -0.68 -0.41

Gene J 0 0.22 -0.24 -0.36

Gene K 0.11 0.07 -0.23

Gene L -0.94 -0.95

Gene M 0.94

Gene N



F

C

G

D

E

Gene C Gene D Gene E Gene F Gene G

Gene C 0.94 0.96 -0.40 0.95

Gene D 0.84 -0.10 0.94

Gene E -0.57 0.89

Gene F -0.35

Gene G

C E

1

1 Gene D Gene F Gene G

1 0.89 -0.485 0.92

Gene D -0.10 0.94

Gene F -0.35

Gene G

Average “similarity” to

Gene D: (0.94+0.84)/2 = 0.89

•Gene F: (-0.40+(-0.57))/2 = -0.485

•Gene G: (0.95+0.89)/2 = 0.92

1


F

G

D

C E

1

1 Gene D Gene F Gene G

1 0.89 -0.485 0.92

Gene D -0.10 0.94

Gene F -0.35

Gene G

G D

2


F

C E

1

G D

2

1 2 Gene F

1 0.905 -0.485

2 -0.225

Gene F 3


F C E

1

G D

2

3

3 Gene F

3 -0.355

Gene F

4

F


F C E

1

G D

2

3

4

algorithm looks familiar?

Clustering of entire yeast genome


Hierarchical Clustering:Yeast Gene Expression Data

Eisen et al., 1998

A SOFM Example With Yeast

“Interpresting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation” by Tamayo et al.

SOM Description

•Each unit of the SOM has a weighted

connection to all inputs•As the algorithm

progresses, neighboring units are grouped by similarity

Input Layer

Output Layer


An Example Using Color

Each color in the map is associated with a weight

From http://davis.wpi.edu/~matt/courses/soms/

Cluster Analysis of Microarray Expression Data Matrices

Application of cluster analysis techniques in the elucidation gene

expression data

The features of a living organism are governed principally by its genes.

If we want to fully understand living systems we must know the function of each gene.

Once we know a gene’s sequence we can design experiments to find its function:

However this approach is too slow to handle all the gene sequence information we have today (HGSP).

Function of Genes

Delete Gene X

Gene X

The Classical Approach of Assigning a function to a Gene

Conclusion: Gene X = left eye gene.

("זבוב בלי רגליים

– חרש")

Microarray AnalysisMicroarray analysis allows the monitoring of the activities of many genes over many different conditions.Experiments are carried out on a Physical Matrix like the one below:

To facilitate computational analysis the physical matrix which may contain 1000’s of gene’s is converted into a numerical matrix using image analysis equipment.

G1G2G3G4G5G6G7G6G7

C1 C2 C3 C4 C5 C6 C7LowZeroHigh

1.55 1.05 0.5 2.5 1.75 0.25 0.1

1.7 0.3 2.4 2.9 1.5 0.5 1.0

1.55 1.05 0.5 2.5 1.75 0.25 0.1

1.7 0.3 2.4 1.5 0.5 1.0

1.55 0.5 2.5 1.75 0.25 0.1

0.3 2.4 2.9 1.5 0.5 1.0

1.55 1.05 0.5 2.5 1.75 0.25 0.1

Conditions

Genes

Possible inference:

If Gene X’s activity (expression) is affected by Condition Y (Extreme Heat), then Gene X may be involved in protecting the cellular components from extreme heat.

Each Gene has its corresponding Expression Profile for a set of conditions.

This Expression Profile may be thought of as a feature profile for that gene for that set of conditions (A condition feature profile).

Cluster Analysis• Cluster Analysis is an unsupervised procedure which involves grouping of objects

based on their similarity in feature space.

• In the Gene Expression context Genes are grouped based on the similarity of their Condition feature profile.

• Cluster analysis was first applied to Gene Expression data from Brewer’s Yeast (Saccharomyces cerevisiae) by Eisen et al. (1998).

Two general conclusions can be drawn from these clusters:

• Genes clustered together may be related within a biological module/system.

• If there are genes of known function within a cluster these may help to class this biological/module system.

X

Y

A

B

C

Z

Clusters A,B and C represent groups of related genes.

Clustering

1.55 1.05 0.5 2.5 1.75 0.25 0.1

1.7 0.3 2.4 2.9 1.5 0.5 1.0

1.55 1.05 0.5 2.5 1.75 0.25 0.1

1.7 0.3 2.4 1.5 0.5 1.0

1.55 0.5 2.5 1.75 0.25 0.1

0.3 2.4 2.9 1.5 0.5 1.0

1.55 1.05 0.5 2.5 1.75 0.25 0.1

Conditions

Genes

From Data to Biological Hypothesis

System C

Cluster C with four Genes may represent System C

Relating these genes aids in elucidation of this System C

Gene Expression Microarray Cluster SetConditions (A-Z)

Gene 1Gene 2Gene 3Gene 4 Gene 5Gene 6Gene 7 X

Y

A

B

C

External Stimulus( Condition X)

Regulator Protein

Toxin

DNA Gene a Gene b Gene c Gene d

Gene Expression

Toxin Pump

Cell Membrane

Some Drawbacks of Clustering Biological Data1. Clustering works well over small numbers of conditions but a typical Microarray

may have hundreds of experimental conditions. A global clustering may not offer sufficient resolution with so many features.

2. As with other clustering applications, it may be difficult to cluster noisy expression data.

3. Biological Systems tend to be inter-related and may share numerous factors (Genes) – Clustering enforces partitions which may not accurately represent these intimacies.

4. Clustering Genes over all Conditions only finds the strongest signals in the dataset as a whole. More ‘local’ signals within the data matrix may be missed.

X

Y

A

B

C

Z

Inter-related(3)

Local Signals(4)

May represent more complex system such as:

How do we better model more complex systems?

• One technique that allows detection of all signals in the data is biclustering.

• Instead of clustering genes over all conditions biclustering clusters genes with respect to subsets of conditions.

-interrelated clusters (genes may belong more than one bicluster).

-local signals (genes correlated over only a few conditions).

-noisy data (allows erratic genes to belong to no cluster).

This enables better representation of:

Biclustering

• Technique first described by J.A. Hartigan in 1972 and termed ‘Direct Clustering’.

• First Introduced to Microarray expression data by Cheng and Church(2000)

Gene 1Gene 2Gene 3Gene 4Gene 5Gene 6Gene 7Gene 8Gene 9

A B C D E F G H

Gene 1

Gene 4

Gene 6

Gene 7

Gene 9

B E F

BiclusteringA B D E F G H

Gene 1

Gene 4

Gene 9

Clustering misses local signal {(B,E,F),(1,4,6,7,9)} present over subset of conditions.

Gene 1

Gene 4

Gene 9

A B C D E F G H

Clustering

Biclustering discovers local coherences over a subset of conditions

Conditions

Approaches to Biclustering Microarray Gene Expression

• First applied to Gene Expression Data by Cheng and Church(2000).– Used a sub-matrix scoring technique to locate biclusters.

• Tanay et al.(2000)– Modelled the expression data on Bipartite graphs and

used graph techniques to find ‘complete graphs’ or biclusters.

• Lazzeroni and Owen– Used matrix reordering to represent different ‘layers’ of

signals (biclusters) ‘Plaid Models’ to represent multiple signals within data.

• Ben-Dor et al. (2002) – “Biclusters” depending on order relations (OPSM).

Bipartite Graph Modelling•First proposed in: “Discovering statically significant biclusters in

gene expressing data” Tanay et al. Bioinformatics 2000

Within the graph modelling paradigm biclusters are equivalent to complete bipartite sub-graphs.

Tanay and colleagues used probabilistic models to determine the least probable sub-graphs (those showing most order and consequently most surprising) to identify biclusters.

1234567

1234567

A B C D E FA

B

C

D

E

F

146

AD146

AD

Graph G

Sub-graph H(Bicluster)

Data Matrix M

Sub-Matrix (Bicluster)

Genes

Genes

Conditions

Conditions

The Cheng and Church Approach

ija

The core element in this approach is the development of a scoring to prioritise sub-matrices.

This scoring is based on the concept of the residue of an entry in a matrix.

In the Matrix (I,J) the residue score of element is given by:

IJIjiJijij aaaaaR )(

ai

jIJ

In words, the residue of an entry is the value of the entry minus the row average, minus the column average, plus the average value in the matrix.

This score gives an idea of how the value fits into the data in the surrounding matrix.

ija

The mean squared residue score (H) for a matrix (I,J) is then calculated :

JjIi

IJIjiJij aaaaJI

JIH,

2)(||||

1),(

This Global H score gives an indication of how the data fits together within that matrix- whether it has some coherence or is random.

The Cheng and Church Approach(2)

A low H score means that there is a correlation in the matrix

- a score of H(I,J)= 0 would mean that the data in the matrix fluctuates in unison i.e. the sub-matrix is a bicluster

A high H value signifies that the data is uncorrelated.

- a matrix of equally spread random values over the range [a,b], has an expected H score of (b-a)/12. range = [0,800] then H(I,J) = 53,333

Worked example of H score:

IJIjiJijij aaaaaR )(

R(1) = 1- 2 - 5.4 + 6.5 = 0.1

R(2) = 2 - 2 - 6.4 + 6.5 = 0.1: :: :

R(12) = 12 - 11 -7.4 + 6.5 = 0.1

Col Avg. 5.4 6.4 7.4

1 2 34 5 67 8 9

10 11 12

Row Avg.

25811

Matrix (M) Avg. = 6.5

H (M) = (0.01x12)/12 = 0.01

If 5 was replaced with 3 then the score would changed to:

H(M2) = 2.06

If the matrix was reshuffled randomly the score would be around:

H(M3) = sqr(12-1)/12 = 10.08

In order to find all possible biclusters in an Expression Matrix all sub-matrices must be tested using the H score.

The Cheng and Church Approach: Node Deletion Biclustering Algorithm

In a node deletion algorithm all columns and rows are tested for deletion. If removing a row or column decreases the H score of the Matrix than it is removed.

This continues until it is not possible to decrease the H score further. This low H score coherent sub-matrix (bicluster) is then returned.

The process then masks this located bicluster by inserting random numbers in place of it.

And reiterates the process.

R

R

R

R

Node Deletion

Node Deletion

The Cheng and Church Approach:

No. of genes, no. of conditions

4, 96 10, 29 11, 25

103, 25 127, 13 13, 21

10, 57 2, 96 25, 12

9, 51 3, 96 2, 96

Some results on lymphoma data (402696):

Conclusions:

• High throughput Functional Genomics (Microarrays) requires Data Mining Applications.

• Biclustering resolves Expression Data more effectively than single dimensional Cluster Analysis.

• Cheng and Church Approach offers good base for future work.

Future Research/Question’s:

• Implement a simple H score program to facilitate study if H score concept.

• Are there other alternative scorings which would better apply to gene expression data?

• Have unbiclustered genes any significance? Horizontally transferred genes?

• Implement full scale biclustering program and look at better adaptation to expression data sets and the biological context.

Support Vector Machines (cont.)

support vectors

•Convex hull of points is the tightest

enclosing polygon•Maximum margin

hyperplane•Instances closest to

hyperplane are called support

vectors•Support vectors

define maximum margin hyperplane

uniquelyWitten & Frank, 2000

References

•Basic microarray analysis: grouping and feature reduction by Soumya Raychaudhuri, Patrick D. Sutphin, Jeffery T. Chang and

Russ B. Altman; Trends in Biotechnology Vol. 19 No. 5 May 2001•Self Organizing Maps, Tom Germano ,

http://davis.wpi.edu/~matt/courses/soms•“Data Analysis Tools for DNA Microarrays” by Sorin Draghici;

Chapman & Hall/CRC 2003•Self-Organizing-Feature-Maps versus Statistical Clustering

Methods: A Benchmark by A. Ultsh, C. Vetter; FG Neuroinformatik & Kunstliche Intelligenz Research Report 0994

http://davis.wpi.edu/~matt/courses/soms

References

•Interpreting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al.

•A Local Search Approximation Algorithm for k-Means Clustering by Kanungo et al.

•K-means-type algorithms: A generalized convergence theorem and characterization of local optimality by Selim

and Ismail

DNA Chips and Their Analysis Comp. Genomics: Lecture 13.

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