Cluster 3.0 Manual
for Windows, Mac OS X, Linux, Unix
Michael Eisen; updated by Michiel de Hoon
Software copyright c⃝ Stanford University 1998-99 This manual was originally written by
Michael Eisen. It is only partially complete and is a work in progress. The manual was
updated in 2002 by Michiel de Hoon, University of Tokyo, Human Genome Center.
1
This is the manual for Cluster 3.0. Cluster was originally written by Michael Eisen while
at Stanford University. We have modified the k-means clustering algorithm in Cluster, and
extended the algorithm for Self-Organizing Maps to include two-dimensional rectangular
grids. The Euclidean distance and the city-block distance were added as new distance
measures between gene expression data. The proprietary Numerical Recipes routines, which
were used in the original version of Cluster/TreeView, have been replaced by open source
software.
Cluster 3.0 is available for Windows, Mac OS X, Linux, and Unix.
November 5, 2002.
Michiel de Hoon
Human Genome Center, University of Tokyo.
i
Table of Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Loading, filtering, and adjusting data . . . . . . . . . 3
2.1 Loading Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Filtering Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Adjusting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Log transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Mean/Median Centering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Distance/Similarity measures . . . . . . . . . . . . . . . . 10
3.1 Distance measures based on the Pearson correlation . . . . . . . . . . . 10
3.2 Non-parametric distance measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Distance measures related to the Euclidean distance . . . . . . . . . . . 12
3.3.1 Euclidean distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 City-block distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Missing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 Calculating the distance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Clustering techniques . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1.1 Centroid Linkage Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1.2 Single Linkage Clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.3 Complete Linkage Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.4 Average Linkage Clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.5 Weighting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.6 Ordering of Output File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.7 Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 The \math k-means Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . 20
4.3 Self-Organizing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Running Cluster 3.0 as a command line
program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 TreeView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7 Code Development Information . . . . . . . . . . . . . . 29
8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Chapter 1: Introduction 2
1 Introduction
Cluster and TreeView are programs that provide a computational and graphical environment
for analyzing data from DNA microarray experiments, or other genomic datasets. The
program Cluster can organize and analyze the data in a number of different ways. TreeView
allows the organized data to be visualized and browsed.
This manual is intended as a reference for using the software, and not as a comprehensive
introduction to the methods employed. Many of the methods are drawn from standard
statistical cluster analysis. There are excellent textbooks available on cluster analysis which
are listed in the bibliography at the end. The bibliography also contains citations for recent
publications in the biological sciences, especially genomics, that employ methods similar to
those used here.
Chapter 2: Loading, filtering, and adjusting data 3
2 Loading, filtering, and adjusting data
Data can be loaded into Cluster by choosing Load data file under the File menu. A number
of options are provided for adjusting and filtering the data you have loaded. These functions
are accessed via the Filter Data and Adjust Data tabs.
Chapter 2: Loading, filtering, and adjusting data 4
2.1 Loading Data
The first step in using Cluster is to import data. Currently, Cluster only reads tab-delimited
text files in a particular format, described below. Such tab-delimited text files can be created
and exported in any standard spreadsheet program, such as Microsoft Excel. An example
datafile can be found under the File format help item in the Help menu. This contains all
the information you need for making a Cluster input file.
By convention, in Cluster input tables rows represent genes and columns represent samples
or observations (e.g. a single microarray hybridization). For a simple timecourse, a
minimal Cluster input file would look like this:
Each row (gene) has an identifier (in green) that always goes in the first column. Here we
are using yeast open reading frame codes. Each column (sample) has a label (in blue) that
is always in the first row; here the labels describe the time at which a sample was taken.
The first column of the first row contains a special field (in red) that tells the program what
kind of objects are in each row. In this case, YORF stands for yeast open reading frame.
This field can be any alpha-numeric value. It is used in TreeView to specify how rows are
linked to external websites.
The remaining cells in the table contain data for the appropriate gene and sample. The
5.8 in row 2 column 4 means that the observed data value for gene YAL001C at 2 hours
was 5.8. Missing values are acceptable and are designated by empty cells (e.g. YAL005C
at 2 hours).
It is possible to have additional information in the input file. A maximal Cluster input
file would look like this:
The yellow columns and rows are optional. By default, TreeView uses the ID in column 1
as a label for each gene. The NAME column allows you to specify a label for each gene that
is distinct from the ID in column 1. The other rows and columns will be described later in
this text.
Chapter 2: Loading, filtering, and adjusting data 5
When Cluster 3.0 opens the data file, the number of columns in each row is checked. If
a given row contains less or more columns than needed, an error message is displayed.
Demo data
A demo datafile, which will be used in all of the examples here, is available
at http://rana.lbl.gov/downloads/data/demo.txt and is mirrored at
http://bonsai.ims.u-tokyo.ac.jp/~mdehoon/software/cluster/demo.txt.
The datafile contains yeast gene expression data described in Eisen et al. (1998) [see
references at end]. Download this data and load it into Cluster. Cluster will give you
information about the loaded datafile.
Chapter 2: Loading, filtering, and adjusting data 6
2.2 Filtering Data
The Filter Data tab allows you to remove genes that do not have certain desired properties
from your dataset. The currently available properties that can be used to filter data are
• % Present >= X. This removes all genes that have missing values in greater than
(100 − X) percent of the columns.
• SD (Gene Vector) >= X. This removes all genes that have standard deviations of
observed values less than X.
• At least X Observations with abs(Val) >= Y. This removes all genes that do not have
at least X observations with absolute values greater than Y .
• MaxVal-MinVal >= X. This removes all genes whose maximum minus minimum values
are less than X.
These are fairly self-explanatory. When you press filter, the filters are not immediately
applied to the dataset. You are first told how many genes would have passed the filter. If
Chapter 2: Loading, filtering, and adjusting data 7
you want to accept the filter, you press Accept, otherwise no changes are made.
2.3 Adjusting Data
From the Adjust Data tab, you can perform a number of operations that alter the underlying
data in the imported table. These operations are
• Log Transform Data: replace all data values x by log2 (x).
• Center genes [mean or median]: Subtract the row-wise mean or median from the values
in each row of data, so that the mean or median value of each row is 0.
• Center arrays [mean or median]: Subtract the column-wise mean or median from the
values in each column of data, so that the mean or median value of each column is 0.
• Normalize genes: Multiply all values in each row of data by a scale factor S so that
the sum of the squares of the values in each row is 1.0 (a separate S is computed for
each row).
Chapter 2: Loading, filtering, and adjusting data 8
• Normalize arrays: Multiply all values in each column of data by a scale factor S so that
the sum of the squares of the values in each column is 1.0 (a separate S is computed
for each column).
These operations are not associative, so the order in which these operations is applied
is very important, and you should consider it carefully before you apply these operations.
The order of operations is (only checked operations are performed):
• Log transform all values.
• Center rows by subtracting the mean or median.
• Normalize rows.
• Center columns by subtracting the mean or median.
• Normalize columns.
2.3.1 Log transformation
The results of many DNA microarray experiments are fluorescent ratios. Ratio measurements
are most naturally processed in log space. Consider an experiment where you are
looking at gene expression over time, and the results are relative expression levels compared
to time 0. Assume at timepoint 1, a gene is unchanged, at timepoint 2 it is up 2-fold and
at timepoint three is down 2-fold relative to time 0. The raw ratio values are 1.0, 2.0 and
0.5. In most applications, you want to think of 2-fold up and 2-fold down as being the
same magnitude of change, but in an opposite direction. In raw ratio space, however, the
difference between timepoint 1 and 2 is +1.0, while between timepoint 1 and 3 is -0.5. Thus
mathematical operations that use the difference between values would think that the 2-fold
up change was twice as significant as the 2-fold down change. Usually, you do not want
this. In log space (we use log base 2 for simplicity) the data points become 0,1.0,-1.0.With
these values, 2-fold up and 2-fold down are symmetric about 0. For most applications, we
recommend you work in log space.
2.3.2 Mean/Median Centering
Consider a now common experimental design where you are looking at a large number of
tumor samples all compared to a common reference sample made from a collection of celllines.
For each gene, you have a series of ratio values that are relative to the expression
level of that gene in the reference sample. Since the reference sample really has nothing
to do with your experiment, you want your analysis to be independent of the amount of
a gene present in the reference sample. This is achieved by adjusting the values of each
gene to reflect their variation from some property of the series of observed values such as
the mean or median. This is what mean and/or median centering of genes does. Centering
makes less sense in experiments where the reference sample is part of the experiment, as
it is many timecourses. Centering the data for columns/arrays can also be used to remove
certain types of biases. The results of many two-color fluorescent hybridization experiments
are not corrected for systematic biases in ratios that are the result of differences in RNA
amounts, labeling efficiency and image acquisition parameters. Such biases have the effect
of multiplying ratios for all genes by a fixed scalar. Mean or median centering the data
in log-space has the effect of correcting this bias, although it should be noted that an
assumption is being made in correcting this bias, which is that the average gene in a given
experiment is expected to have a ratio of 1.0 (or log-ratio of 0).
Chapter 2: Loading, filtering, and adjusting data 9
In general, I recommend the use of median rather than mean centering, as it is more
robust against outliers.
2.3.3 Normalization
Normalization sets the magnitude (sum of the squares of the values) of a row/column vector
to 1.0. Most of the distance metrics used by Cluster work with internally normalized data
vectors, but the data are output as they were originally entered. If you want to output
normalized vectors, you should select this option. A sample series of operations for raw
data would be:
• Adjust Cycle 1) log transform
• Adjust Cycle 2) median center genes and arrays
• repeat (2) five to ten times
• Adjust Cycle 3) normalize genes and arrays
• repeat (3) five to ten times
This results in a log-transformed, median polished (i.e. all row-wise and column-wise
median values are close to zero) and normal (i.e. all row and column magnitudes are close
to 1.0) dataset. After performing these operations you should save the dataset.
Chapter 3: Distance/Similarity measures 10
3 Distance/Similarity measures
The first choice that must be made is how similarity (or alternatively, distance) between
gene expression data is to be defined. There are many ways to compute how similar two
series of numbers are. Cluster provides eight options.
3.1 Distance measures based on the Pearson correlation
The most commonly used similarity metrics are based on Pearson correlation. The Pearson
correlation coefficient between any two series of numbers x = {x1, x2, . . . , xn} and y =
{y1, y2, . . . , yn} is defined as
r =
1
n
n∑
i=1
(
xi − x
σx
) (
yi − y
σy
)
,
where x is the average of values in x, and σx is the standard deviation of these values.
There are many ways of conceptualizing the correlation coefficient. If you were to make
a scatterplot of the values of x against y (pairing x1 with y1, x2 with y2, etc), then r reports
how well you can fit a line to the values.
The simplest way to think about the correlation coefficient is to plot x and y as curves,
with r telling you how similar the shapes of the two curves are. The Pearson correlation
coefficient is always between -1 and 1, with 1 meaning that the two series are identical, 0
meaning they are completely uncorrelated, and -1 meaning they are perfect opposites. The
correlation coefficient is invariant under linear transformation of the data. That is, if you
multiply all the values in y by 2, or add 7 to all the values in y, the correlation between x and
y will be unchanged. Thus, two curves that have identical shape, but different magnitude,
will still have a correlation of 1.
Cluster actually uses four different flavors of the Pearson correlation. The textbook
Pearson correlation coefficient, given by the formula above, is used if you select Correlation
(centered) in the Similarity Metric dialog box. Correlation (uncentered) uses the following
modified equations:
r =
1
n
n∑
i=1
(
xi
σ
(0)
x
) (
yi
σ
(0)
y
)
,
in which
σ(0)
x =
1
n
n∑
i=1
(xi)
2
;
σ(0)
y =
1
n
n∑
i=1
(yi)
2
.
This is basically the same function, except that it assumes the mean is 0, even when it
is not. The difference is that, if you have two vectors x and y with identical shape, but
which are offset relative to each other by a fixed value, they will have a standard Pearson
Chapter 3: Distance/Similarity measures 11
correlation (centered correlation) of 1 but will not have an uncentered correlation of 1. The
uncentered correlation is equal to the cosine of the angle of two n-dimensional vectors x
and y, each representing a vector in n-dimensional space that passes through the origin.
Cluster provides two similarity metrics that are the absolute value of these two correlation
functions, which consider two items to be similar if they have opposite expression patterns;
the standard correlation coefficients consider opposite genes to be very distant.
3.2 Non-parametric distance measures
The Spearman rank correlation and Kendall’s τ are two additional metrics, which are nonparametric
versions of the Pearson correlation coefficient. These methods are more robust
against outliers.
The Spearman rank correlation calculates the correlation between the ranks of the data
values in the two vectors. For example, if we have two data vectors
x = {2.3, 6.7, 4.5, 20.8} ;
y = {2.1, 5.9, 4.4, 4.2} ,
then we first replace them by their ranks:
x = {1, 3, 2, 4} ;
y = {1, 4, 3, 2} .
Now we calculate the correlation coefficient in their usual manner from these data vectors,
resulting in
rSpearman = 0.4.
In comparison, the regular Pearson correlation between these data is r = 0.2344. By
replacing the data values by their ranks, we reduced the effect of the outlier 20.8 on the
value of the correlation coefficient. The Spearman rank correlation can be used as a test
statistic for independence between x and y. For more information, see Conover (1980).
Kendall’s τ goes a step further by using only the relative ordering of x and y to calculate
the correlation (Snedecor & Cochran). To calculate Kendall’s τ, consider all pairs of data
points (xi, yi) and (xj, yj). We call a pair concordant if
• xi < xj and yi < yj; or
• xi > xj and yi > yj,
and discordant if
• xi < xj and yi > yj; or
• xi > xj and yi < yj.
We can represent this by a table:
− (2.3, 2.1) (6.7, 5.9) (4.5, 4.4) (20.8, 4.2)
(2.3, 2.1) − << << <<
(6.7, 5.9) >> − >> <>
(4.5, 4.4) >> << − <>
(20.8, 4.2) >> >< >< −
Chapter 3: Distance/Similarity measures 12
From this table, we find that there are four concordant pairs and two discordant pairs:
nc = 4;
nd = 2;
Kendall’s τ is calculated as
τ = Nc − Nd
N (N − 1) /2,
which in this case evaluates as 0.33. In the C Clustering Library, the calculation of Kendall’s
τ is corrected for the possibility that two ranks are equal. As in case of the Spearman rank
correlation, we may use Kendall’s τ to test for independence between x and y.
3.3 Distance measures related to the Euclidean distance
3.3.1 Euclidean distance
A newly added distance function is the Euclidean distance, which is defined as
d
(
x, y
)
=
1
n
n∑
i=1
(xi − yi)
2
.
The Euclidean distance takes the difference between two gene expression levels directly. It
should therefore only be used for expression data that are suitably normalized, for example
by converting the measured gene expression levels to log-ratios. In the sum, we only include
terms for which both xi and yi are present, and divide by n accordingly.
Unlike the correlation-based distance measures, the Euclidean distance takes the magnitude
of changes in the gene expression levels into account. An example of the Euclidean
distance applied to k-means clustering can be found in De Hoon, Imoto, and Miyano (2002).
3.3.2 City-block distance
The city-block distance, alternatively known as the Manhattan distance, is related to the
Euclidean distance. Whereas the Euclidean distance corresponds to the length of the shortest
path between two points, the city-block distance is the sum of distances along each
dimension:
d =
n∑
i=1
|xi − yi| .
This is equal to the distance you would have to walk between two points in a city, where
you have to walk along city blocks. The city-block distance is a metric, as it satisfies the
triangle inequality. Again we only include terms for which both xi and yi are present, and
divide by n accordingly.
As for the Euclidean distance, the expression data are subtracted directly from each
other, and we should therefore make sure that they are properly normalized.
3.4 Missing values
When either x or y has missing values, only observations present for both x and y are used
in computing similarities.
Chapter 3: Distance/Similarity measures 13
3.5 Calculating the distance matrix
With any specified metric, the first step in the hierarchical clustering routines described
below is to compute the distance (the opposite of similarity; for all correlation metrics
distance = 1.0 - correlation) between all pairs of items to be clustered (e.g. the set of
genes in the current dataset). This can often be time consuming, and, except for pairwise
single-linkage clustering, memory intensive (the maximum amount of memory required is
4 × N × N bytes, where N is the number of items being clustered). The algorithm for
pairwise single-linkage hierarchical clustering is less memory-intensive (linear in N).
Chapter 4: Clustering techniques 14
4 Clustering techniques
The Cluster program provides several clustering algorithms. Hierarchical clustering methods
organizes genes in a tree structure, based on their similarity. Four variants of hierarchical
clustering are available in Cluster. In k-means clustering, genes are organized into
k clusters, where the number of clusters k needs to be chosen in advance. Self-Organizing
Maps create clusters of genes on a two-dimensional rectangular grid, where neighboring
clusters are similar. For each of these methods, one of the eight different distance meaures
can be used. Finally, in Principal Component Analysis, clusters are organized based on the
principal component axes of the distance matrix.
4.1 Hierarchical Clustering
The Hierarchical Clustering tab allows you to perform hierarchical clustering on your data.
This is a powerful and useful method for analyzing all sorts of large genomic datasets. Many
published applications of this analysis are given in the references section at the end.
Cluster currently performs four types of binary, agglomerative, hierarchical clustering.
The basic idea is to assemble a set of items (genes or arrays) into a tree, where items are
joined by very short branches if they are very similar to each other, and by increasingly
longer branches as their similarity decreases.
The first step in hierarchical clustering is to calculate the distance matrix between the
gene expression data. Once this matrix of distances is computed, the clustering begins.
Agglomerative hierarchical processing consists of repeated cycles where the two closest
Chapter 4: Clustering techniques 15
remaining items (those with the smallest distance) are joined by a node/branch of a tree,
with the length of the branch set to the distance between the joined items. The two joined
items are removed from list of items being processed and replaced by a item that represents
the new branch. The distances between this new item and all other remaining items are
computed, and the process is repeated until only one item remains.
Note that once clustering commences, we are working with items that are true items (e.g.
a single gene) and items that are pseudo-items that contain a number of true items. There
are a variety of ways to compute distances when we are dealing with pseudo-items, and
Cluster currently provides four choices, which are called centroid linkage, single linkage,
complete linkage, and average linkage. Note that in older versions of Cluster, centroid
linkage was referred to as average linkage.
4.1.1 Centroid Linkage Clustering
If you click Centroid Linkage Clustering, a vector is assigned to each pseudo-item, and this
vector is used to compute the distances between this pseudo-item and all remaining items or
pseudo-items using the same similarity metric as was used to calculate the initial similarity
matrix. The vector is the average of the vectors of all actual items (e.g. genes) contained
within the pseudo-item. Thus, when a new branch of the tree is formed joining together a
branch with 5 items and an actual item, the new pseudo-item is assigned a vector that is
the average of the 6 vectors it contains, and not the average of the two joined items (note
that missing values are not used in the average, and a pseudo-item can have a missing value
if all of the items it contains are missing values in the corresponding row/column).
Note that from a theoretical perspective, Centroid Linkage Clustering is peculiar if it
is used in combination with one of the distance measures that are based on the Pearson
correlation. For these distance measures, the data vectors are implicitly normalized when
calculating the distance (for example, by subtracting the mean and dividing by the standard
deviation when calculating the Pearson correlation. However, when two genes are joined
and their centroid is calculated by averaging their data vectors, no normalization is applied.
This may lead to the surprising result that distances may decrease when we go up in the
tree representing the hierarchical clustering result. For example, consider this data set:
Exp1 Exp2 Exp3 Exp4
Gene1 0.96 0.07 0.97 0.98
Gene2 0.50 0.28 0.29 0.77
Gene3 0.08 0.96 0.51 0.51
Gene4 0.14 0.19 0.41 0.51
Performing pairwise centroid-linkage hierarchical clustering on this data set, using the Pearson
distance as the distance measure, produces the clustering result
• Gene 1 joins Gene 2 at distance 0.47
• (Gene 1, Gene 2) joins Gene 4 at distance 0.46
• (Gene 1, Gene 2, Gene 4) joins Gene 3 at distance 1.62
This may result in ill-formed dendrograms. For an example, see the Java TreeView manual.
A solution is to use the Euclidean or the city-block distance, or to use one of the other
hierarchical clustering routines, which don’t suffer from this issue regardless of the distance
measure being used.
Chapter 4: Clustering techniques 16
4.1.2 Single Linkage Clustering
In Single Linkage Clustering the distance between two items x and y is the minimum of all
pairwise distances between items contained in x and y. Unlike centroid linkage clustering,
in single linkage clustering no further distances need to be calculated once the distance
matrix is known.
In Cluster 3.0, as of version 1.29 the implementation of single linkage clustering is based
on the SLINK algorithm (see Sibson, 1973). Whereas this algorithm yields the exact same
clustering result as conventional single-linkage hierarchical clustering, it is much faster and
more memory-efficient (being linear in the memory requirements, compared to quadratic
for the conventional algorithm). Hence, single-linkage hierarchical clustering can be used to
cluster large gene expression data sets, for which centroid-, complete-, and average-linkage
fail due to lack of memory.
4.1.3 Complete Linkage Clustering
In Complete Linkage Clustering the distance between two items x and y is the maximum of
all pairwise distances between items contained in x and y. As in single linkage clustering,
no other distances need to be calculated once the distance matrix is known.
4.1.4 Average Linkage Clustering
In average linkage clustering, the distance between two items x and y is the mean of all
pairwise distances between items contained in x and y.
4.1.5 Weighting
Weighting: By default, all of the observations for a given item are treated equally. In some
cases you may want to give some observations more weight than others. For example, if
you have duplicate copies of a gene on your array, you might want to downweight each
individual copy when computing distances between arrays. You can specify weights using
the ‘GWEIGHT’ (gene weight) and ‘EWEIGHT’ (experiment weight) parameters in the input
file. By default all weights are set to 1.0. Thus, the actual formula, with weights included,
for the Pearson correlation of x = {x1, x2, . . . , xn} and y = {y1, y2, . . . , yn} with observation
weights of w = {w1, w2, . . . , wn} is
r =
1
∑N
i=1 wi
N∑
i=1
wi
(
Xi − X
σX
) (
Yi − Y
σY
)
Note that when you are clustering rows (genes), you are using column (array) weights. It
is possible to compute weights as well based on a not entirely well understood function. If
you want to compute weights for clustering genes, select the check box in the Genes panel
Chapter 4: Clustering techniques 17
of the Hierarchical Clustering tab.
This will expose a Weight Options dialog box in the Arrays panel (I realize this placement
is a bit counterintuitive, but it makes sense as you will see below). The idea behind the
Calculate Weights option is to weight each row (the same idea applies to columns as well)
based on the local density of row vectors in its vicinity, with a high density vicinity resulting
in a low weight and a low density vicinity resulting in a higher weight. This is implemented
by assigning a local density score L (i) to each row i.
L (i) =
∑
j with d(i,j)<k
(
k − d (i, j)
k
)n
,
where the cutoff k and the exponent n are user supplied parameters. The weight for each
row is 1/L. Note that L (i) is always at least 1, since d (i, i) = 0. Each other row that is
within the distance k of row i increases L (i) and decreases the weight. The larger d (i, j),
the less L (i) is increased. Values of n greater than 1 mean that the contribution to L (i)
drops off rapidly as d (i, j) increases.
4.1.6 Ordering of Output File
The result of a clustering run is a tree or pair of trees (one for genes one for arrays).
However, to visualize the results in TreeView, it is necessary to use this tree to reorder the
rows and/or columns in the initial datatable. Note that if you simply draw all of the node
Chapter 4: Clustering techniques 18
in the tree in the following manner, a natural ordering of items emerges:
Thus, any tree can be used to generate an ordering. However, the ordering for any given
tree is not unique. There is a family of 2N−1
ordering consistent with any tree of N items;
you can flip any node on the tree (exchange the bottom and top branches) and you will
get a new ordering that is equally consistent with the tree. By default, when Cluster joins
two items, it randomly places one item on the top branch and the other on the bottom
branch. It is possible to guide this process to generate the best ordering consistent with a
given tree. This is done by using the ‘GORDER’ (gene order) and ‘EORDER’ (experiment order)
parameters in the input file, or by running a self-organizing map (see section below) prior
to clustering. By default, Cluster sets the order parameter for each row/column to 1. When
a new node is created, Cluster compares the order parameters of the two joined items, and
places the item with the smaller order value on the top branch. The order parameter for
a node is the average of the order parameters of its members. Thus, if you want the gene
order produced by Cluster to be as close as possible (without violating the structure of the
tree) to the order in your input file, you use the ‘GORDER’ column, and assign a value that
increases for each row. Note that order parameters do not have to be unique.
4.1.7 Output Files
Cluster writes up to three output files for each hierarchical clustering run. The root filename
of each file is whatever text you enter into the Job Name dialog box. When you load a
file, Job Name is set to the root filename of the input file. The three output files are
‘JobName.cdt’, ‘JobName.gtr’, ‘JobName.atr’ The ‘.cdt’ (for clustered data table) file
contains the original data with the rows and columns reordered based on the clustering
result. It is the same format as the input files, except that an additional column and/or
row is added if clustering is performed on genes and/or arrays. This additional column/row
contains a unique identifier for each row/column that is linked to the description of the
tree structure in the ‘.gtr’ and ‘.atr’ files. The ‘.gtr’ (gene tree) and ‘.atr’ (array tree)
files are tab-delimited text files that report on the history of node joining in the gene or
array clustering (note that these files are produced only when clustering is performed on the
corresponding axis). When clustering begins each item to be clustered is assigned a unique
identifier (e.g. ‘GENE1X’ or ‘ARRY42X’ — the ‘X’ is a relic from the days when this was
written in Perl and substring searches were used). These identifiers are added to the .cdt
Chapter 4: Clustering techniques 19
file. As each node is generated, it receives a unique identifier as well, starting is ‘NODE1X’,
‘NODE2X’, etc. Each joining event is stored in the ‘.gtr’ or ‘.atr’ file as a row with the
node identifier, the identifiers of the two joined elements, and the similarity score for the
two joined elements. These files look like:
NODE1X GENE1X GENE4X 0.98
NODE2X GENE5X GENE2X 0.80
NODE3X NODE1X GENE3X 0.72
NODE4X NODE2X NODE3X 0.60
The ‘.gtr’ and/or ‘.atr’ files are automatically read in TreeView when you open the
corresponding ‘.cdt’ file.
Chapter 4: Clustering techniques 20
4.2 The k-means Clustering Algorithm
The k-means clustering algorithm is a simple, but popular, form of cluster analysis. The
basic idea is that you start with a collection of items (e.g. genes) and some chosen number
of clusters (k) you want to find. The items are initially randomly assigned to a cluster. The
k-means clustering proceeds by repeated application of a two-step process where
1. the mean vector for all items in each cluster is computed;
2. items are reassigned to the cluster whose center is closest to the item.
Since the initial cluster assignment is random, different runs of the k-means clustering
algorithm may not give the same final clustering solution. To deal with this, the k-means
clustering algorithms is repeated many times, each time starting from a different initial
clustering. The sum of distances within the clusters is used to compare different clustering
solutions. The clustering solution with the smallest sum of within-cluster distances is saved.
The number of runs that should be done depends on how difficult it is to find the optimal
solution, which in turn depends on the number of genes involved. Cluster therefore shows
in the status bar how many times the optimal solution has been found. If this number is
one, there may be a clustering solution with an even lower sum of within-cluster distances.
The k-means clustering algorithm should then be repeated with more trials. If the optimal
solution is found many times, the solution that has been found is likely to have the lowest
possible within-cluster sum of distances. We can then assume that the k-means clustering
procedure has then found the overall optimal clustering solution.
Chapter 4: Clustering techniques 21
It should be noted that generally, the k-means clustering algorithm finds a clustering
solution with a smaller within-cluster sum of distances than the hierarchical clustering
techniques.
The parameters that control k-means clustering are
• the number of clusters (k);
• the number of trials.
The output is simply an assignment of items to a cluster. The implementation here
simply rearranges the rows and/or columns based on which cluster they were assigned to.
The output data file is ‘JobName_K_GKg_AKa.cdt’, where ‘_GKg’ is included if genes were
organized, and ‘_AKa’ is included if arrays were organized. Here, ‘Kg’ and ‘Ka’ represent the
number of clusters for gene clustering and array clustering, respectively. This file contains
the gene expression data, organized by cluster by rearranging the rows and columns. In
addition, the files ‘JobName_K_GKg.kgg’ and ‘JobName_K_AKa.kag’ are created, containing
a list of genes/arrays and the cluster they were assigned to.
Whereas k-means clustering as implemented in Cluster 3.0 allows any of the eight distance
measures to be used, we recommend using the Euclidean distance or city-block distance
instead of the distance measures based on the Pearson correlation, for the same
reason as in case of pairwise centroid-linkage hierarchical clustering. The distance measures
based on the Pearson correlation effectively normalize the data vectors when calculating the
distance, whereas no normalization is used when calculating the cluster centroid. To use
k-means clustering with a distance measure based on the Pearson correlation, it is better
to first normalize the data appropriately (using the "Adjust Data" tab) before running the
k-means algorithm.
Cluster also implements a slight variation on k-means clustering, known as k-medians
clustering, in which the median instead of the mean of items in a node are used. In a
theoretical sense, it is best to use k-means with the Euclidean distance, and k-medians with
the city-block distance.
Chapter 4: Clustering techniques 22
4.3 Self-Organizing Maps
Self-Organizing Maps (SOMs) is a method of cluster analysis that are somewhat related
to k-means clustering. SOMs were invented in by Teuvo Kohonen in the early 1980s, and
have recently been used in genomic analysis (see Chu 1998, Tamayo 1999 and Golub 1999
in references). The Tamayo paper contains a simple explanation of the methods. A more
detailed description is available in the book by Kohonen, Self-Organizing Maps, 1997.
The current implementation varies slightly from that of Tamayo et al., in that it restricts
the analysis one-dimensional SOMs along each axis, as opposed to a two-dimensional
network. The one-dimensional SOM is used to reorder the elements on whichever axes are
selected. The result is similar to the result of k-means clustering, except that, unlike in
k-means clustering, the nodes in a SOM are ordered. This tends to result in a relatively
smooth transition between groups.
The options for SOMs are
• whether or not you will organize each axis;
• the number of nodes for each axis (the default is n1/4
, where n is the number of items;
the total number of clusters is then equal to the square root of the number of items);
• the number of iterations to be run.
The output file is of the form ‘JobName_SOM_GXg-Yg_AXa-Ya.txt’, where ‘GXg-Yg’ is
included if genes were organized, and ‘AXg-Yg’ is included if arrays were organized. ‘X’ and
‘Y’ represent the dimensions of the corresponding SOM. Up to two additional files (‘.gnf’
and ‘.anf’) are written containing the vectors for the SOM nodes.
Chapter 4: Clustering techniques 23
In previous versions of Cluster, only one-dimensional SOMs were supported. The current
version of the Cluster introduces two-dimensional SOMs.
SOMs and hierarchical clustering: Our original use of SOMs (see Chu et al., 1998) was
motivated by the desire to take advantage of the properties of both SOMs and hierarchical
clustering. This was accomplished by first computing a one dimensional SOM, and using
the ordering from the SOM to guide the flipping of nodes in the hierarchical tree. In
Cluster, after a SOM is run on a dataset, the GORDER and/or EORDER fields are set to
the ordering from the SOM so that, for subsequent hierarchical clustering runs, the output
ordering will come as close as possible to the ordering in the SOM without violating the
structure of the tree.
4.4 Principal Component Analysis
Principal Component Analysis (PCA) is a widely used technique for analyzing multivariate
data. A practical example of applying Principal Component Analysis to gene expression
data is presented by Yeung and Ruzzo (2001).
In essence, PCA is a coordinate transformation in which each row in the data matrix is
written as a linear sum over basis vectors called principal components, which are ordered
and chosen such that each maximally explains the remaining variance in the data vectors.
For example, an n × 3 data matrix can be represented as an ellipsoidal cloud of n points in
three dimensional space. The first principal component is the longest axis of the ellipsoid,
the second principal component the second longest axis of the ellipsoid, and the third
principal component is the shortest axis. Each row in the data matrix can be reconstructed
Chapter 4: Clustering techniques 24
as a suitable linear combination of the principal components. However, in order to reduce
the dimensionality of the data, usually only the most important principal components are
retained. The remaining variance present in the data is then regarded as unexplained
variance.
The principal components can be found by calculating the eigenvectors of the covariance
matrix of the data. The corresponding eigenvalues determine how much of the variance
present in the data is explained by each principal component.
Before applying PCA, typically the mean is subtracted from each column in the data
matrix. In the example above, this effectively centers the ellipsoidal cloud around its centroid
in 3D space, with the principal components describing the variation of poins in the
ellipsoidal cloud with respect to their centroid.
In Cluster, you can apply PCA to the rows (genes) of the data matrix, or to the columns
(microarrays) of the data matrix. In each case, the output consists of two files. When
applying PCA to genes, the names of the output files are ‘JobName_pca_gene.pc.txt’
and ‘JobName_pca_gene.coords.txt’, where the former contains contains the principal
components, and the latter contains the coordinates of each row in the data matrix
with respect to the principal components. When applying PCA to the columns in
the data matrix, the respective file names are ‘JobName_pca_array.pc.txt’ and
‘JobName_pca_array.coords.txt’. The original data matrix can be recovered from the
principal components and the coordinates.
As an example, consider this input file:
UNIQID EXP1 EXP2 EXP3
GENE1 3 4 -2
GENE2 4 1 -3
GENE3 1 -8 7
GENE4 -6 6 4
GENE5 0 -3 8
Applying PCA to the rows (genes) of the data in this input file generates a coordinate file
containing
UNIQID NAME GWEIGHT 13.513398 10.162987 2.025283
GENE1 GENE1 1.000000 6.280326 -2.404095 -0.760157
GENE2 GENE2 1.000000 4.720801 -4.995230 0.601424
GENE3 GENE3 1.000000 -8.755665 -2.117608 0.924161
GENE4 GENE4 1.000000 3.443490 8.133673 0.621082
GENE5 GENE5 1.000000 -5.688953 1.383261 -1.386509
where the first line shows the eigenvalues of the principal components, and a prinpical
component file containing
EIGVALUE EXP1 EXP2 EXP3
MEAN 0.400000 0.000000 2.800000
13.513398 0.045493 0.753594 -0.655764
10.162987 -0.756275 0.454867 0.470260
2.025283 -0.652670 -0.474545 -0.590617
Chapter 4: Clustering techniques 25
with the eigenvalues of the principal components shown in the first column. From this
principal component decomposition, we can regenerate the original data matrix as follows:






6.280326 −2.404095 −0.760157
4.720801 −4.995230 0.601424
−8.755665 −2.117608 0.924161
3.443490 8.133673 0.621082
−5.688953 1.383261 −1.386509






·


0.045493 0.753594 −0.655764
−0.756275 0.454867 0.470260
−0.652670 −0.474545 −0.590617


+






0.400000 0.000000 2.800000
0.400000 0.000000 2.800000
0.400000 0.000000 2.800000
0.400000 0.000000 2.800000
0.400000 0.000000 2.800000






=






3 4 −2
4 1 −3
1 −8 7
−6 6 4
0 −3 8






Note that the coordinate file ‘JobName_pca_gene.coords.txt’ is a valid input file to Cluster
3.0. Hence, it can be loaded into Cluster 3.0 for further analysis, possibly after removing
columns with low eigenvalues.
Chapter 5: Running Cluster 3.0 as a command line program 26
5 Running Cluster 3.0 as a command line
program
Cluster 3.0 can also be run as a command line program. This may be useful if you want
to run Cluster 3.0 on a remote server, and also allows automatic processing a large number
of data files by running a batch script. Note, however, that the Python and Perl interfaces
to the C Clustering Library may be better suited for this task, as they are more
powerful than the command line program (see the manual for the C Clustering Library at
http://bonsai.ims.u-tokyo.ac.jp/~mdehoon/software/cluster/cluster.pdf).
The GUI version of Cluster 3.0 can be used as a command line program by applying
the appropriate command line parameters. You can also compile Cluster 3.0 without GUI
support (if you will be using it from the command line only) by downloading the source code
from http://bonsai.ims.u-tokyo.ac.jp/~mdehoon/software/cluster, and running
configure --without-x
make
make install
The executable is called cluster. To run this program, execute
cluster [options]
in which the options consist of the following command line parameters:
-f filename
File loading
-l Specifies to log-transform the data before clustering (default is no
log-transform)
-cg a|m Specifies whether to center each row (gene) in the data set:
a: Subtract the mean of each row
m: Subtract the median of each row
(default is no centering)
-ng Specifies to normalize each row (gene) in the data set (default is no normaliza-
tion)
-ca a|m Specifies whether to center each column (microarray) in the data set:
a: Subtract the mean of each column
m: Subtract the median of each column
(default is no centering)
-na Specifies to normalize each column (microarray) in the data set (default is no
normalization)
-u jobname
Allows you to specify a different name for the output files (default is derived
from the input file name)
-g [0..9] Specifies the distance measure for gene clustering. 0 means no gene clustering;
for the values 1 through 9, see below (default: 0)
-e [0..9] Specifies the distance measure for microarray clustering. 0 means no microarray
clustering; for the values 1 through 9, see below (default: 0)
Chapter 5: Running Cluster 3.0 as a command line program 27
-m [msca] Specifies which hierarchical clustering method to use:
m: Pairwise complete- (maximum-) linkage (default)
s: Pairwise single-linkage
c: Pairwise centroid-linkage
a: Pairwise average-linkage
-k number
Specifies whether to run k-means clustering instead of hierarchical clustering,
and the number of clusters k to use (default: 0, no k-means clustering)
-pg Specifies to apply Principal Component Analysis to genes instead of clustering
-pa Specifies to apply Principal Component Analysis to arrays instead of clustering
-s Specifies to calculate an SOM instead of hierarchical clustering
-x number
Specifies the horizontal dimension of the SOM grid (default: 2)
-y number
Specifies the vertical dimension of the SOM grid (default: 1)
-v, --version
Display version information
-h, --help
Display help information
For the command line options ‘-g’, ‘-e’, the following integers can be used to specify
the distance measure:
0 No clustering
1 Uncentered correlation
2 Pearson correlation
3 Uncentered correlation, absolute value
4 Pearson correlation, absolute value
5 Spearman’s rank correlation
6 Kendall’s tau
7 Euclidean distance
8 City-block distance
By default, no clustering is done, allowing you to use cluster for normalizing a data
set only.
Chapter 6: TreeView 28
6 TreeView
TreeView is a program that allows interactive graphical analysis of the results from Cluster.
TreeView reads in matching ‘*.cdt’ and ‘*.gtr’, ‘*.atr’, ‘*.kgg’, or ‘*.kag’ files produced
by Cluster. We recommend using the Java program Java TreeView, which is based on the
original TreeView. Java TreeView was written by Alok Saldanha at Stanford University;
it can be downloaded from http://jtreeview.sourceforge.net/. Java TreeView runs
on Windows, Macintosh, Linux, and Unix computers, and can show both hierarchical and
k-means results.
Chapter 7: Code Development Information 29
7 Code Development Information
In previous versions of Cluster, the proprietary Numerical Recipes routines were heavily
used. We have replaced these routines by the C clustering library, which was released under
the Python License. Accordingly, the complete source code of Cluster is now open. It can
be downloaded from http://bonsai.ims.u-tokyo.ac.jp/~mdehoon/software/cluster.
We used the GNU C compiler in order to enable anybody to compile the code.
No commercial compiler is required. The GNU C compiler is available at
http://www.gnu.org. There you can also find texinfo, which was used to generate
the printed and the HTML documentation. To convert the picture files to EPS files
for the printed documentation, we used pngtopnm and pnmtops of Netpbm, which can
be found at http://netpbm.sourceforge.net. The HTML Help file was generated
using the HTML Help Workshop, which is freely available at the Microsoft site
(http://msdn.microsoft.com). The Windows Installer was created with the Inno Setup
Compiler, which is available at http://www.innosetup.com.
For Mac OS X, we used the Project Builder and the Interface Builder, which are part of
the Mac OS X Development Tools. The prebuilt package was created with PackageMaker,
which is also part of Mac OS X. The project files needed to recompile Cluster 3.0 are
included in the source code. From the command prompt, Cluster 3.0 can be recompiled by
running make from the mac subdirectory; this produces a universal binary for PowerPC and
Intel processors.
For Cluster 3.0 on Linux/Unix, we used the Motif libraries that are installed on most
Linux/Unix computers. The include files are typically located in /usr/X11R6/include/Xm.
You will need a version of Motif that is compliant with Motif 2.1, such as Open Motif
(http://www.opengroup.org), which is available at http://www.motifzone.net.
To improve the portability of the code, we made use of GNU’s automake and autoconf.
The corresponding Makefile.am and configure.ac files are included in the source code
distribution.
Chapter 8: Bibliography 30
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