Source of the materials: Biopython cookbook (adapted) Status: Draft
Cluster analysis is the grouping of items into clusters based on the similarity of the items to each other. In bioinformatics, clustering is widely used in gene expression data analysis to find groups of genes with similar gene expression profiles. This may identify functionally related genes, as well as suggest the function of presently unknown genes.
The Biopython module Bio.Cluster provides commonly used clustering
algorithms and was designed with the application to gene expression data
in mind. However, this module can also be used for cluster analysis of
other types of data. Bio.Cluster and the underlying C Clustering
Library is described by De Hoon et al. @dehoon2004.
The following four clustering approaches are implemented in
Bio.Cluster:
Hierarchical clustering (pairwise centroid-, single-, complete-, and average-linkage);
$k$-means, $k$-medians, and $k$-medoids clustering;
Self-Organizing Maps;
Principal Component Analysis.
The data to be clustered are represented by a $n \times m$ Numerical
Python array data. Within the context of gene expression data
clustering, typically the rows correspond to different genes whereas the
columns correspond to different experimental conditions. The clustering
algorithms in Bio.Cluster can be applied both to rows (genes) and to
columns (experiments).
Often in microarray experiments, some of the data values are missing,
which is indicated by an additional $n \times m$ Numerical Python
integer array mask. If mask[i,j]==0, then data[i,j] is missing and
is ignored in the analysis.
The $k$-means/medians/medoids clustering algorithms and Self-Organizing
Maps (SOMs) include the use of a random number generator. The uniform
random number generator in Bio.Cluster is based on the algorithm by
L’Ecuyer @lecuyer1988, while random numbers following the binomial
distribution are generated using the BTPE algorithm by Kachitvichyanukul
and Schmeiser @kachitvichyanukul1988. The random number generator is
initialized automatically during its first call. As this random number
generator uses a combination of two multiplicative linear congruential
generators, two (integer) seeds are needed for initialization, for which
we use the system-supplied random number generator rand (in the C
standard library). We initialize this generator by calling srand with
the epoch time in seconds, and use the first two random numbers
generated by rand as seeds for the uniform random number generator in
Bio.Cluster.
In order to cluster items into groups based on their similarity, we
should first define what exactly we mean by similar. Bio.Cluster
provides eight distance functions, indicated by a single character, to
measure similarity, or conversely, distance:
'e': Euclidean distance;
'b': City-block distance.
'c': Pearson correlation coefficient;
'a': Absolute value of the Pearson correlation coefficient;
'u': Uncentered Pearson correlation (equivalent to the cosine of
the angle between two data vectors);
'x': Absolute uncentered Pearson correlation;
's': Spearman’s rank correlation;
'k': Kendall’s $\tau$.
The first two are true distance functions that satisfy the triangle inequality: $$d\left(\underline{u},\underline{v}\right) \leq d\left(\underline{u},\underline{w}\right) + d\left(\underline{w},\underline{v}\right) \textrm{ for all } \underline{u}, \underline{v}, \underline{w},$$ and are therefore refered to as metrics. In everyday language, this means that the shortest distance between two points is a straight line.
The remaining six distance measures are related to the correlation coefficient, where the distance $d$ is defined in terms of the correlation $r$ by $d=1-r$. Note that these distance functions are semi-metrics that do not satisfy the triangle inequality. For example, for $$\underline{u}=\left(1,0,-1\right);$$ $$\underline{v}=\left(1,1,0\right);$$ $$\underline{w}=\left(0,1,1\right);$$ we find a Pearson distance $d\left(\underline{u},\underline{w}\right) = 1.8660$, while $d\left(\underline{u},\underline{v}\right)+d\left(\underline{v},\underline{w}\right) = 1.6340$.
In Bio.Cluster, we define the Euclidean distance as
$$d = {1 \over n} \sum_{i=1}^{n} \left(x_i-y_i\right)^{2}.$$ Only those
terms are included in the summation for which both $x_i$ and $y_i$ are
present, and the denominator $n$ is chosen accordingly. As the
expression data $x_i$ and $y_i$ are subtracted directly from each other,
we should make sure that the expression data are properly normalized
when using the Euclidean 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. As
gene expression data tend to have missing values, in Bio.Cluster we
define the city-block distance as the sum of distances divided by the
number of dimensions:
$$d = {1 \over n} \sum_{i=1}^n \left|x_i-y_i\right|.$$ 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. 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.
The Pearson correlation coefficient is defined as $$r = \frac{1}{n} \sum_{i=1}^n \left( \frac{x_i -\bar{x}}{\sigma_x} \right) \left(\frac{y_i -\bar{y}}{\sigma_y} \right),$$ in which $\bar{x}, \bar{y}$ are the sample mean of $x$ and $y$ respectively, and $\sigma_x, \sigma_y$ are the sample standard deviation of $x$ and $y$. The Pearson correlation coefficient is a measure for how well a straight line can be fitted to a scatterplot of $x$ and $y$. If all the points in the scatterplot lie on a straight line, the Pearson correlation coefficient is either +1 or -1, depending on whether the slope of line is positive or negative. If the Pearson correlation coefficient is equal to zero, there is no correlation between $x$ and $y$.
The Pearson distance is then defined as $$d_{\textrm{P}} \equiv 1 - r.$$ As the Pearson correlation coefficient lies between -1 and 1, the Pearson distance lies between 0 and 2.
By taking the absolute value of the Pearson correlation, we find a number between 0 and 1. If the absolute value is 1, all the points in the scatter plot lie on a straight line with either a positive or a negative slope. If the absolute value is equal to zero, there is no correlation between $x$ and $y$.
The corresponding distance is defined as $$d_{\textrm A} \equiv 1 - \left|r\right|,$$ where $r$ is the Pearson correlation coefficient. As the absolute value of the Pearson correlation coefficient lies between 0 and 1, the corresponding distance lies between 0 and 1 as well.
In the context of gene expression experiments, the absolute correlation is equal to 1 if the gene expression profiles of two genes are either exactly the same or exactly opposite. The absolute correlation coefficient should therefore be used with care.
In some cases, it may be preferable to use the uncentered correlation instead of the regular Pearson correlation coefficient. The uncentered correlation is defined as $$r_{\textrm U} = \frac{1}{n} \sum_{i=1}^{n} \left(\frac{x_i}{\sigma_x^{(0)}} \right) \left(\frac{y_i}{\sigma_y^{(0)}} \right),$$ where $$\begin{aligned} \sigma_x^{(0)} & = & \sqrt{{\frac{1}{n}} \sum_{i=1}^{n}x_i^2}; \nonumber \\ \sigma_y^{(0)} & = & \sqrt{{\frac{1}{n}} \sum_{i=1}^{n}y_i^2}. \nonumber\end{aligned}$$ This is the same expression as for the regular Pearson correlation coefficient, except that the sample means $\bar{x}, \bar{y}$ are set equal to zero. The uncentered correlation may be appropriate if there is a zero reference state. For instance, in the case of gene expression data given in terms of log-ratios, a log-ratio equal to zero corresponds to the green and red signal being equal, which means that the experimental manipulation did not affect the gene expression.
The distance corresponding to the uncentered correlation coefficient is defined as $$d_{\mbox{U}} \equiv 1 - r_{\mbox{U}},$$ where $r_{\mbox{U}}$ is the uncentered correlation. As the uncentered correlation coefficient lies between -1 and 1, the corresponding distance lies between 0 and 2.
The uncentered correlation is equal to the cosine of the angle of the two data vectors in $n$-dimensional space, and is often referred to as such.
As for the regular Pearson correlation, we can define a distance measure using the absolute value of the uncentered correlation: $$d_{\mbox{AU}} \equiv 1 - \left|r_{\mbox{U}}\right|,$$ where $r_{\mbox{U}}$ is the uncentered correlation coefficient. As the absolute value of the uncentered correlation coefficient lies between 0 and 1, the corresponding distance lies between 0 and 1 as well.
Geometrically, the absolute value of the uncentered correlation is equal to the cosine between the supporting lines of the two data vectors (i.e., the angle without taking the direction of the vectors into consideration).
The Spearman rank correlation is an example of a non-parametric similarity measure, and tends to be more robust against outliers than the Pearson correlation.
To calculate the Spearman rank correlation, we replace each data value by their rank if we would order the data in each vector by their value. We then calculate the Pearson correlation between the two rank vectors instead of the data vectors.
As in the case of the Pearson correlation, we can define a distance measure corresponding to the Spearman rank correlation as $$d_{\mbox{S}} \equiv 1 - r_{\mbox{S}},$$ where $r_{\mbox{S}}$ is the Spearman rank correlation.
Kendall’s $\tau$ is another example of a non-parametric similarity measure. It is similar to the Spearman rank correlation, but instead of the ranks themselves only the relative ranks are used to calculate $\tau$ (see Snedecor & Cochran @snedecor1989).
We can define a distance measure corresponding to Kendall’s $\tau$ as $$d_{\mbox{K}} \equiv 1 - \tau.$$ As Kendall’s $\tau$ is always between -1 and 1, the corresponding distance will be between 0 and 2.
For most of the distance functions available in Bio.Cluster, a weight
vector can be applied. The weight vector contains weights for the items
in the data vector. If the weight for item $i$ is $w_i$, then that item
is treated as if it occurred $w_i$ times in the data. The weight do not
have to be integers. For the Spearman rank correlation and Kendall’s
$\tau$, weights do not have a well-defined meaning and are therefore not
implemented.
The distance matrix is a square matrix with all pairwise distances
between the items in data, and can be calculated by the function
distancematrix in the Bio.Cluster module:
In [1]:
from Bio.Cluster import distancematrix
matrix = distancematrix(data)
where the following arguments are defined:
data (required)\
Array containing the data for the items.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
weight (default: None)\
The weights to be used when calculating distances. If
weight==None, then equal weights are assumed.
transpose (default: 0)\
Determines if the distances between the rows of data are to be
calculated (transpose==0), or between the columns of data
(transpose==1).
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
To save memory, the distance matrix is returned as a list of 1D arrays. The number of columns in each row is equal to the row number. Hence, the first row has zero elements. An example of the return value is
[array([]),
array([1.]),
array([7., 3.]),
array([4., 2., 6.])]This corresponds to the distance matrix $$\left( \begin{array}{cccc} 0 & 1 & 7 & 4 \\ 1 & 0 & 3 & 2 \\ 7 & 3 & 0 & 6 \\ 4 & 2 & 6 & 0 \end{array} \right).$$
The centroid of a cluster can be defined either as the mean or as the
median of each dimension over all cluster items. The function
clustercentroids in Bio.Cluster can be used to calculate either:
In [2]:
from Bio.Cluster import clustercentroids
cdata, cmask = clustercentroids(data)
where the following arguments are defined:
data (required)\
Array containing the data for the items.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
clusterid (default: None)\
Vector of integers showing to which cluster each item belongs. If
clusterid is None, then all items are assumed to belong to the
same cluster.
method (default: 'a')\
Specifies whether the arithmetic mean (method=='a') or the median
(method=='m') is used to calculate the cluster center.
transpose (default: 0)\
Determines if the centroids of the rows of data are to be
calculated (transpose==0), or the centroids of the columns of
data (transpose==1).
This function returns the tuple (cdata, cmask). The centroid data are
stored in the 2D Numerical Python array cdata, with missing data
indicated by the 2D Numerical Python integer array cmask. The
dimensions of these arrays are
$\left(\textrm{number of clusters}, \textrm{number of columns}\right)$
if transpose is 0, or
$\left(\textrm{number of rows}, \textrm{number of clusters}\right)$ if
transpose is 1. Each row (if transpose is 0) or column (if
transpose is 1) contains the averaged data corresponding to the
centroid of each cluster.
Given a distance function between items, we can define the distance between two clusters in several ways. The distance between the arithmetic means of the two clusters is used in pairwise centroid-linkage clustering and in $k$-means clustering. In $k$-medoids clustering, the distance between the medians of the two clusters is used instead. The shortest pairwise distance between items of the two clusters is used in pairwise single-linkage clustering, while the longest pairwise distance is used in pairwise maximum-linkage clustering. In pairwise average-linkage clustering, the distance between two clusters is defined as the average over the pairwise distances.
To calculate the distance between two clusters, use
In [3]:
from Bio.Cluster import clusterdistance
distance = clusterdistance(data)
where the following arguments are defined:
data (required)\
Array containing the data for the items.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
weight (default: None)\
The weights to be used when calculating distances. If
weight==None, then equal weights are assumed.
index1 (default: 0)\
A list containing the indices of the items belonging to the
first cluster. A cluster containing only one item $i$ can be
represented either as a list [i], or as an integer i.
index2 (default: 0)\
A list containing the indices of the items belonging to the
second cluster. A cluster containing only one items $i$ can be
represented either as a list [i], or as an integer i.
method (default: 'a')\
Specifies how the distance between clusters is defined:
'a': Distance between the two cluster centroids (arithmetic
mean);
'm': Distance between the two cluster centroids (median);
's': Shortest pairwise distance between items in the two
clusters;
'x': Longest pairwise distance between items in the two
clusters;
'v': Average over the pairwise distances between items in the
two clusters.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
transpose (default: 0)\
If transpose==0, calculate the distance between the rows of
data. If transpose==1, calculate the distance between the
columns of data.
Partitioning algorithms divide items into $k$ clusters such that the sum
of distances over the items to their cluster centers is minimal. The
number of clusters $k$ is specified by the user. Three partitioning
algorithms are available in Bio.Cluster:
$k$-means clustering
$k$-medians clustering
$k$-medoids clustering
These algorithms differ in how the cluster center is defined. In $k$-means clustering, the cluster center is defined as the mean data vector averaged over all items in the cluster. Instead of the mean, in $k$-medians clustering the median is calculated for each dimension in the data vector. Finally, in $k$-medoids clustering the cluster center is defined as the item which has the smallest sum of distances to the other items in the cluster. This clustering algorithm is suitable for cases in which the distance matrix is known but the original data matrix is not available, for example when clustering proteins based on their structural similarity.
The expectation-maximization (EM) algorithm is used to find this partitioning into $k$ groups. In the initialization of the EM algorithm, we randomly assign items to clusters. To ensure that no empty clusters are produced, we use the binomial distribution to randomly choose the number of items in each cluster to be one or more. We then randomly permute the cluster assignments to items such that each item has an equal probability to be in any cluster. Each cluster is thus guaranteed to contain at least one item.
We then iterate:
Calculate the centroid of each cluster, defined as either the mean, the median, or the medoid of the cluster;
Calculate the distances of each item to the cluster centers;
For each item, determine which cluster centroid is closest;
Reassign each item to its closest cluster, or stop the iteration if no further item reassignments take place.
To avoid clusters becoming empty during the iteration, in $k$-means and $k$-medians clustering the algorithm keeps track of the number of items in each cluster, and prohibits the last remaining item in a cluster from being reassigned to a different cluster. For $k$-medoids clustering, such a check is not needed, as the item that functions as the cluster centroid has a zero distance to itself, and will therefore never be closer to a different cluster.
As the initial assignment of items to clusters is done randomly, usually a different clustering solution is found each time the EM algorithm is executed. To find the optimal clustering solution, the $k$-means algorithm is repeated many times, each time starting from a different initial random clustering. The sum of distances of the items to their cluster center is saved for each run, and the solution with the smallest value of this sum will be returned as the overall clustering solution.
How often the EM algorithm should be run depends on the number of items being clustered. As a rule of thumb, we can consider how often the optimal solution was found; this number is returned by the partitioning algorithms as implemented in this library. If the optimal solution was found many times, it is unlikely that better solutions exist than the one that was found. However, if the optimal solution was found only once, there may well be other solutions with a smaller within-cluster sum of distances. If the number of items is large (more than several hundreds), it may be difficult to find the globally optimal solution.
The EM algorithm terminates when no further reassignments take place. We noticed that for some sets of initial cluster assignments, the EM algorithm fails to converge due to the same clustering solution reappearing periodically after a small number of iteration steps. We therefore check for the occurrence of such periodic solutions during the iteration. After a given number of iteration steps, the current clustering result is saved as a reference. By comparing the clustering result after each subsequent iteration step to the reference state, we can determine if a previously encountered clustering result is found. In such a case, the iteration is halted. If after a given number of iterations the reference state has not yet been encountered, the current clustering solution is saved to be used as the new reference state. Initially, ten iteration steps are executed before resaving the reference state. This number of iteration steps is doubled each time, to ensure that periodic behavior with longer periods can also be detected.
The $k$-means and $k$-medians algorithms are implemented as the function
kcluster in Bio.Cluster:
In [4]:
from Bio.Cluster import kcluster
clusterid, error, nfound = kcluster(data)
where the following arguments are defined:
data (required)\
Array containing the data for the items.
nclusters (default: 2)\
The number of clusters $k$.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
weight (default: None)\
The weights to be used when calculating distances. If
weight==None, then equal weights are assumed.
transpose (default: 0)\
Determines if rows (transpose is 0) or columns (transpose is
1) are to be clustered.
npass (default: 1)\
The number of times the $k$-means/-medians clustering algorithm is
performed, each time with a different (random) initial condition. If
initialid is given, the value of npass is ignored and the
clustering algorithm is run only once, as it behaves
deterministically in that case.
method (default: a)\
describes how the center of a cluster is found:
method=='a': arithmetic mean ($k$-means clustering);
method=='m': median ($k$-medians clustering).
For other values of method, the arithmetic mean is used.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]). Whereas all eight distance measures
are accepted by kcluster, from a theoretical viewpoint it is best
to use the Euclidean distance for the $k$-means algorithm, and the
city-block distance for $k$-medians.
initialid (default: None)\
Specifies the initial clustering to be used for the EM algorithm. If
initialid==None, then a different random initial clustering is
used for each of the npass runs of the EM algorithm. If
initialid is not None, then it should be equal to a 1D array
containing the cluster number (between 0 and nclusters-1) for
each item. Each cluster should contain at least one item. With the
initial clustering specified, the EM algorithm is deterministic.
This function returns a tuple (clusterid, error, nfound), where
clusterid is an integer array containing the number of the cluster to
which each row or cluster was assigned, error is the within-cluster
sum of distances for the optimal clustering solution, and nfound is
the number of times this optimal solution was found.
The kmedoids routine performs $k$-medoids clustering on a given set of
items, using the distance matrix and the number of clusters passed by
the user:
In [5]:
from Bio.Cluster import kmedoids
clusterid, error, nfound = kmedoids(distance)
where the following arguments are defined: , nclusters=2, npass=1, initialid=None)|
`distance` (required)\ The matrix containing the distances between the items; this matrix can be specified in three ways:
distance = array([[0.0, 1.1, 2.3],
[1.1, 0.0, 4.5],
[2.3, 4.5, 0.0]])- as a 1D Numerical Python array containing consecutively the
distances in the left-lower part of the distance matrix:distance = array([1.1, 2.3, 4.5])- as a list containing the rows of the left-lower part of the
distance matrix:distance = [array([]|,
array([1.1]),
array([2.3, 4.5])
]These three expressions correspond to the same distance matrix.
nclusters (default: 2)\
The number of clusters $k$.
npass (default: 1)\
The number of times the $k$-medoids clustering algorithm is
performed, each time with a different (random) initial condition. If
initialid is given, the value of npass is ignored, as the
clustering algorithm behaves deterministically in that case.
initialid (default: None)\
Specifies the initial clustering to be used for the EM algorithm. If
initialid==None, then a different random initial clustering is
used for each of the npass runs of the EM algorithm. If
initialid is not None, then it should be equal to a 1D array
containing the cluster number (between 0 and nclusters-1) for
each item. Each cluster should contain at least one item. With the
initial clustering specified, the EM algorithm is deterministic.
This function returns a tuple (clusterid, error, nfound), where
clusterid is an array containing the number of the cluster to which
each item was assigned, error is the within-cluster sum of distances
for the optimal $k$-medoids clustering solution, and nfound is the
number of times the optimal solution was found. Note that the cluster
number in clusterid is defined as the item number of the item
representing the cluster centroid.
Hierarchical clustering methods are inherently different from the $k$-means clustering method. In hierarchical clustering, the similarity in the expression profile between genes or experimental conditions are represented in the form of a tree structure. This tree structure can be shown graphically by programs such as Treeview and Java Treeview, which has contributed to the popularity of hierarchical clustering in the analysis of gene expression data.
The first step in hierarchical clustering is to calculate the distance matrix, specifying all the distances between the items to be clustered. Next, we create a node by joining the two closest items. Subsequent nodes are created by pairwise joining of items or nodes based on the distance between them, until all items belong to the same node. A tree structure can then be created by retracing which items and nodes were merged. Unlike the EM algorithm, which is used in $k$-means clustering, the complete process of hierarchical clustering is deterministic.
Several flavors of hierarchical clustering exist, which differ in how
the distance between subnodes is defined in terms of their members. In
Bio.Cluster, pairwise single, maximum, average, and centroid linkage
are available.
In pairwise single-linkage clustering, the distance between two nodes is defined as the shortest distance among the pairwise distances between the members of the two nodes.
In pairwise maximum-linkage clustering, alternatively known as pairwise complete-linkage clustering, the distance between two nodes is defined as the longest distance among the pairwise distances between the members of the two nodes.
In pairwise average-linkage clustering, the distance between two nodes is defined as the average over all pairwise distances between the items of the two nodes.
In pairwise centroid-linkage clustering, the distance between two nodes is defined as the distance between their centroids. The centroids are calculated by taking the mean over all the items in a cluster. As the distance from each newly formed node to existing nodes and items need to be calculated at each step, the computing time of pairwise centroid-linkage clustering may be significantly longer than for the other hierarchical clustering methods. Another peculiarity is that (for a distance measure based on the Pearson correlation), the distances do not necessarily increase when going up in the clustering tree, and may even decrease. This is caused by an inconsistency between the centroid calculation and the distance calculation when using the Pearson correlation: Whereas the Pearson correlation effectively normalizes the data for the distance calculation, no such normalization occurs for the centroid calculation.
For pairwise single-, complete-, and average-linkage clustering, the distance between two nodes can be found directly from the distances between the individual items. Therefore, the clustering algorithm does not need access to the original gene expression data, once the distance matrix is known. For pairwise centroid-linkage clustering, however, the centroids of newly formed subnodes can only be calculated from the original data and not from the distance matrix.
The implementation of pairwise single-linkage hierarchical clustering is based on the SLINK algorithm (R. Sibson, 1973), which is much faster and more memory-efficient than a straightforward implementation of pairwise single-linkage clustering. The clustering result produced by this algorithm is identical to the clustering solution found by the conventional single-linkage algorithm. The single-linkage hierarchical clustering algorithm implemented in this library can be used to cluster large gene expression data sets, for which conventional hierarchical clustering algorithms fail due to excessive memory requirements and running time.
The result of hierarchical clustering consists of a tree of nodes, in
which each node joins two items or subnodes. Usually, we are not only
interested in which items or subnodes are joined at each node, but also
in their similarity (or distance) as they are joined. To store one node
in the hierarchical clustering tree, we make use of the class Node,
which defined in Bio.Cluster. An instance of Node has three
attributes:
left
right
distance
Here, left and right are integers referring to the two items or
subnodes that are joined at this node, and distance is the distance
between them. The items being clustered are numbered from 0 to
$\left(\textrm{number of items} - 1\right)$, while clusters are numbered
from -1 to $-\left(\textrm{number of items}-1\right)$. Note that the
number of nodes is one less than the number of items.
To create a new Node object, we need to specify left and right;
distance is optional.
In [6]:
from Bio.Cluster import Node
Node(2, 3)
Out[6]:
In [7]:
Node(2, 3, 0.91)
Out[7]:
The attributes left, right, and distance of an existing Node
object can be modified directly:
In [8]:
node = Node(4, 5)
node.left = 6
node.right = 2
node.distance = 0.73
node
Out[8]:
An error is raised if left and right are not integers, or if
distance cannot be converted to a floating-point value.
The Python class Tree represents a full hierarchical clustering
solution. A Tree object can be created from a list of Node objects:
In [9]:
from Bio.Cluster import Node, Tree
nodes = [Node(1, 2, 0.2), Node(0, 3, 0.5), Node(-2, 4, 0.6), Node(-1, -3, 0.9)]
tree = Tree(nodes)
print(tree)
The Tree initializer checks if the list of nodes is a valid
hierarchical clustering result:
In [11]:
nodes = [Node(1, 2, 0.2), Node(0, 2, 0.5)]
try:
Tree(nodes)
raise Exception("Should not arrive here")
except ValueError:
print("This tree is problematic")
Individual nodes in a Tree object can be accessed using square
brackets:
In [12]:
nodes = [Node(1, 2, 0.2), Node(0, -1, 0.5)]
tree = Tree(nodes)
tree[0]
Out[12]:
In [13]:
tree[1]
Out[13]:
In [14]:
tree[-1]
Out[14]:
As a Tree object is read-only, we cannot change individual nodes in a
Tree object. However, we can convert the tree to a list of nodes,
modify this list, and create a new tree from this list:
In [15]:
tree = Tree([Node(1, 2, 0.1), Node(0, -1, 0.5), Node(-2, 3, 0.9)])
print(tree)
In [16]:
nodes = tree[:]
nodes[0] = Node(0, 1, 0.2)
nodes[1].left = 2
tree = Tree(nodes)
print(tree)
This guarantees that any Tree object is always well-formed.
To display a hierarchical clustering solution with visualization
programs such as Java Treeview, it is better to scale all node distances
such that they are between zero and one. This can be accomplished by
calling the scale method on an existing Tree object:
In [12]:
tree.scale()
This method takes no arguments, and returns None.
After hierarchical clustering, the items can be grouped into $k$
clusters based on the tree structure stored in the Tree object by
cutting the tree:
In [13]:
clusterid = tree.cut(nclusters=1)
where nclusters (defaulting to 1) is the desired number of clusters
$k$. This method ignores the top $k-1$ linking events in the tree
structure, resulting in $k$ separated clusters of items. The number of
clusters $k$ should be positive, and less than or equal to the number of
items. This method returns an array clusterid containing the number of
the cluster to which each item is assigned.
To perform hierarchical clustering, use the treecluster function in
Bio.Cluster.
In [14]:
from Bio.Cluster import treecluster
tree = treecluster(data)
where the following arguments are defined:
data\
Array containing the data for the items.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
weight (default: None)\
The weights to be used when calculating distances. If
weight==None, then equal weights are assumed.
transpose (default: 0)\
Determines if rows (transpose==0) or columns (transpose==1) are
to be clustered.
method (default: 'm')\
defines the linkage method to be used:
method=='s': pairwise single-linkage clustering
method=='m': pairwise maximum- (or complete-) linkage
clustering
method=='c': pairwise centroid-linkage clustering
method=='a': pairwise average-linkage clustering
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
To apply hierarchical clustering on a precalculated distance matrix,
specify the distancematrix argument when calling treecluster
function instead of the data argument:
In [17]:
from Bio.Cluster import treecluster
tree = treecluster(distancematrix=distance)
In this case, the following arguments are defined:
distancematrix\
The distance matrix, which can be specified in three ways:
distance = array([[0.0, 1.1, 2.3],
[1.1, 0.0, 4.5],
[2.3, 4.5, 0.0]])- as a 1D Numerical Python array containing consecutively the
distances in the left-lower part of the distance matrix:distance = array([1.1, 2.3, 4.5])- as a list containing the rows of the left-lower part of the
distance matrix:distance = [array([]),
array([1.1]),
array([2.3, 4.5])These three expressions correspond to the same distance matrix. As
`treecluster` may shuffle the values in the distance matrix as part
of the clustering algorithm, be sure to save this array in a
different variable before calling `treecluster` if you need
it later.
method\
The linkage method to be used:
method=='s': pairwise single-linkage clustering
method=='m': pairwise maximum- (or complete-) linkage
clustering
method=='a': pairwise average-linkage clustering
While pairwise single-, maximum-, and average-linkage clustering can be calculated from the distance matrix alone, pairwise centroid-linkage cannot.
When calling treecluster, either data or distancematrix should be
None.
This function returns a Tree object. This object contains
$\left(\textrm{number of items} - 1\right)$ nodes, where the number of
items is the number of rows if rows were clustered, or the number of
columns if columns were clustered. Each node describes a pairwise
linking event, where the node attributes left and right each contain
the number of one item or subnode, and distance the distance between
them. Items are numbered from 0 to
$\left(\textrm{number of items} - 1\right)$, while clusters are numbered
-1 to $-\left(\textrm{number of items}-1\right)$.
Self-Organizing Maps (SOMs) were invented by Kohonen to describe neural networks (see for instance Kohonen, 1997 @kohonen1997). Tamayo (1999) first applied Self-Organizing Maps to gene expression data @tamayo1999.
SOMs organize items into clusters that are situated in some topology. Usually a rectangular topology is chosen. The clusters generated by SOMs are such that neighboring clusters in the topology are more similar to each other than clusters far from each other in the topology.
The first step to calculate a SOM is to randomly assign a data vector to each cluster in the topology. If rows are being clustered, then the number of elements in each data vector is equal to the number of columns.
An SOM is then generated by taking rows one at a time, and finding which cluster in the topology has the closest data vector. The data vector of that cluster, as well as those of the neighboring clusters, are adjusted using the data vector of the row under consideration. The adjustment is given by $$\Delta \underline{x}_{\textrm{cell}} = \tau \cdot \left(\underline{x}_{\textrm{row}} - \underline{x}_{\textrm{cell}} \right).$$ The parameter $\tau$ is a parameter that decreases at each iteration step. We have used a simple linear function of the iteration step: $$\tau = \tau_{\textrm{init}} \cdot \left(1 - {i \over n}\right),$$ $\tau_{\textrm{init}}$ is the initial value of $\tau$ as specified by the user, $i$ is the number of the current iteration step, and $n$ is the total number of iteration steps to be performed. While changes are made rapidly in the beginning of the iteration, at the end of iteration only small changes are made.
All clusters within a radius $R$ are adjusted to the gene under consideration. This radius decreases as the calculation progresses as $$R = R_{\textrm{max}} \cdot \left(1 - {i \over n}\right),$$ in which the maximum radius is defined as $$R_{\textrm{max}} = \sqrt{N_x^2 + N_y^2},$$ where $\left(N_x, N_y\right)$ are the dimensions of the rectangle defining the topology.
The function somcluster implements the complete algorithm to calculate
a Self-Organizing Map on a rectangular grid. First it initializes the
random number generator. The node data are then initialized using the
random number generator. The order in which genes or microarrays are
used to modify the SOM is also randomized. The total number of
iterations in the SOM algorithm is specified by the user.
To run somcluster, use
In [19]:
from Bio.Cluster import somcluster
clusterid, celldata = somcluster(data)
where the following arguments are defined:
data (required)\
Array containing the data for the items.
mask (default: None)\
Array of integers showing which data are missing. If mask[i,j]==0,
then data[i,j] is missing. If mask==None, then all data
are present.
weight (default: None)\
contains the weights to be used when calculating distances. If
weight==None, then equal weights are assumed.
transpose (default: 0)\
Determines if rows (transpose is 0) or columns (transpose is
1) are to be clustered.
nxgrid, nygrid (default: 2, 1)\
The number of cells horizontally and vertically in the rectangular
grid on which the Self-Organizing Map is calculated.
inittau (default: 0.02)\
The initial value for the parameter $\tau$ that is used in the
SOM algorithm. The default value for inittau is 0.02, which was
used in Michael Eisen’s Cluster/TreeView program.
niter (default: 1)\
The number of iterations to be performed.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
This function returns the tuple (clusterid, celldata):
clusterid:\
An array with two columns, where the number of rows is equal to the
number of items that were clustered. Each row contains the $x$ and
$y$ coordinates of the cell in the rectangular SOM grid to which the
item was assigned.
celldata:\
An array with dimensions
$\left(\verb|nxgrid|, \verb|nygrid|, \textrm{number of columns}\right)$
if rows are being clustered, or
$\left(\verb|nxgrid|, \verb|nygrid|, \textrm{number of rows}\right)$
if columns are being clustered. Each element [ix][iy] of this
array is a 1D vector containing the gene expression data for the
centroid of the cluster in the grid cell with coordinates
[ix][iy].
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) @yeung2001.
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 $n \times 3$ 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 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 principal component analysis, 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 points in the ellipsoidal cloud with respect to their centroid.
The function pca below first uses the singular value decomposition to
calculate the eigenvalues and eigenvectors of the data matrix. The
singular value decomposition is implemented as a translation in C of the
Algol procedure svd @golub1971, which uses Householder
bidiagonalization and a variant of the QR algorithm. The principal
components, the coordinates of each data vector along the principal
components, and the eigenvalues corresponding to the principal
components are then evaluated and returned in decreasing order of the
magnitude of the eigenvalue. If data centering is desired, the mean
should be subtracted from each column in the data matrix before calling
the pca routine.
To apply Principal Component Analysis to a rectangular matrix data,
use
In [17]:
from Bio.Cluster import pca
columnmean, coordinates, components, eigenvalues = pca(data)
This function returns a tuple
columnmean, coordinates, components, eigenvalues:
columnmean\
Array containing the mean over each column in data.
coordinates\
The coordinates of each row in data with respect to the
principal components.
components\
The principal components.
eigenvalues\
The eigenvalues corresponding to each of the principal components.
The original matrix data can be recreated by calculating
columnmean + dot(coordinates, components).
Cluster/TreeView are GUI-based codes for clustering gene expression
data. They were originally written by Michael
Eisen while at Stanford University. Bio.Cluster
contains functions for reading and writing data files that correspond to
the format specified for Cluster/TreeView. In particular, by saving a
clustering result in that format, TreeView can be used to visualize the
clustering results. We recommend using Alok Saldanha’s
http://jtreeview.sourceforge.net/Java TreeView program,
which can display hierarchical as well as $k$-means clustering results.
An object of the class Record contains all information stored in a
Cluster/TreeView-type data file. To store the information contained in
the data file in a Record object, we first open the file and then read
it:
In [20]:
from Bio import Cluster
handle = open("mydatafile.txt")
record = Cluster.read(handle)
handle.close()
This two-step process gives you some flexibility in the source of the data. For example, you can use
In [19]:
import gzip # Python standard library
handle = gzip.open("mydatafile.txt.gz")
to open a gzipped file, or
In [20]:
import urllib # Python standard library
handle = urllib.urlopen("http://somewhere.org/mydatafile.txt")
to open a file stored on the Internet before calling read.
The read command reads the tab-delimited text file mydatafile.txt
containing gene expression data in the format specified for Michael
Eisen’s Cluster/TreeView program. For a description of this file format,
see the manual to Cluster/TreeView. It is available at Michael Eisen’s
lab website and at
our
website.
A Record object has the following attributes:
data\
The data array containing the gene expression data. Genes are stored
row-wise, while microarrays are stored column-wise.
mask\
This array shows which elements in the data array, if any,
are missing. If mask[i,j]==0, then data[i,j] is missing. If no
data were found to be missing, mask is set to None.
geneid\
This is a list containing a unique description for each gene (i.e.,
ORF numbers).
genename\
This is a list containing a description for each gene (i.e.,
gene name). If not present in the data file, genename is set to
None.
gweight\
The weights that are to be used to calculate the distance in
expression profile between genes. If not present in the data file,
gweight is set to None.
gorder\
The preferred order in which genes should be stored in an
output file. If not present in the data file, gorder is set to
None.
expid\
This is a list containing a description of each microarray, e.g.
experimental condition.
eweight\
The weights that are to be used to calculate the distance in
expression profile between microarrays. If not present in the data
file, eweight is set to None.
eorder\
The preferred order in which microarrays should be stored in an
output file. If not present in the data file, eorder is set to
None.
uniqid\
The string that was used instead of UNIQID in the data file.
After loading a Record object, each of these attributes can be
accessed and modified directly. For example, the data can be
log-transformed by taking the logarithm of record.data.
To calculate the distance matrix between the items stored in the record, use
In [21]:
matrix = record.distancematrix()
where the following arguments are defined:
transpose (default: 0)\
Determines if the distances between the rows of data are to be
calculated (transpose==0), or between the columns of data
(transpose==1).
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
This function returns the distance matrix as a list of rows, where the number of columns of each row is equal to the row number (see section [subsec:distancematrix]).
To calculate the centroids of clusters of items stored in the record, use
In [22]:
cdata, cmask = record.clustercentroids()
clusterid (default: None)\
Vector of integers showing to which cluster each item belongs. If
clusterid is not given, then all items are assumed to belong to
the same cluster.
method (default: 'a')\
Specifies whether the arithmetic mean (method=='a') or the median
(method=='m') is used to calculate the cluster center.
transpose (default: 0)\
Determines if the centroids of the rows of data are to be
calculated (transpose==0), or the centroids of the columns of
data (transpose==1).
This function returns the tuple cdata, cmask; see section
[subsec:clustercentroids] for a description.
To calculate the distance between clusters of items stored in the record, use
In [23]:
distance = record.clusterdistance()
where the following arguments are defined:
index1 (default: 0)\
A list containing the indices of the items belonging to the
first cluster. A cluster containing only one item $i$ can be
represented either as a list [i], or as an integer i.
index2 (default: 0)\
A list containing the indices of the items belonging to the
second cluster. A cluster containing only one item $i$ can be
represented either as a list [i], or as an integer i.
method (default: 'a')\
Specifies how the distance between clusters is defined:
'a': Distance between the two cluster centroids (arithmetic
mean);
'm': Distance between the two cluster centroids (median);
's': Shortest pairwise distance between items in the two
clusters;
'x': Longest pairwise distance between items in the two
clusters;
'v': Average over the pairwise distances between items in the
two clusters.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
transpose (default: 0)\
If transpose==0, calculate the distance between the rows of
data. If transpose==1, calculate the distance between the
columns of data.
To perform hierarchical clustering on the items stored in the record, use
In [24]:
tree = record.treecluster()
where the following arguments are defined:
transpose (default: 0)\
Determines if rows (transpose==0) or columns (transpose==1) are
to be clustered.
method (default: 'm')\
defines the linkage method to be used:
method=='s': pairwise single-linkage clustering
method=='m': pairwise maximum- (or complete-) linkage
clustering
method=='c': pairwise centroid-linkage clustering
method=='a': pairwise average-linkage clustering
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
transpose\
Determines if genes or microarrays are being clustered. If
transpose==0, genes (rows) are being clustered. If transpose==1,
microarrays (columns) are clustered.
This function returns a Tree object. This object contains
$\left(\textrm{number of items} - 1\right)$ nodes, where the number of
items is the number of rows if rows were clustered, or the number of
columns if columns were clustered. Each node describes a pairwise
linking event, where the node attributes left and right each contain
the number of one item or subnode, and distance the distance between
them. Items are numbered from 0 to
$\left(\textrm{number of items} - 1\right)$, while clusters are numbered
-1 to $-\left(\textrm{number of items}-1\right)$.
To perform $k$-means or $k$-medians clustering on the items stored in the record, use
In [25]:
clusterid, error, nfound = record.kcluster()
where the following arguments are defined:
nclusters (default: 2)\
The number of clusters $k$.
transpose (default: 0)\
Determines if rows (transpose is 0) or columns (transpose is
1) are to be clustered.
npass (default: 1)\
The number of times the $k$-means/-medians clustering algorithm is
performed, each time with a different (random) initial condition. If
initialid is given, the value of npass is ignored and the
clustering algorithm is run only once, as it behaves
deterministically in that case.
method (default: a)\
describes how the center of a cluster is found:
method=='a': arithmetic mean ($k$-means clustering);
method=='m': median ($k$-medians clustering).
For other values of method, the arithmetic mean is used.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
This function returns a tuple (clusterid, error, nfound), where
clusterid is an integer array containing the number of the cluster to
which each row or cluster was assigned, error is the within-cluster
sum of distances for the optimal clustering solution, and nfound is
the number of times this optimal solution was found.
To calculate a Self-Organizing Map of the items stored in the record, use
In [26]:
clusterid, celldata = record.somcluster()
where the following arguments are defined:
transpose (default: 0)\
Determines if rows (transpose is 0) or columns (transpose is
1) are to be clustered.
nxgrid, nygrid (default: 2, 1)\
The number of cells horizontally and vertically in the rectangular
grid on which the Self-Organizing Map is calculated.
inittau (default: 0.02)\
The initial value for the parameter $\tau$ that is used in the
SOM algorithm. The default value for inittau is 0.02, which was
used in Michael Eisen’s Cluster/TreeView program.
niter (default: 1)\
The number of iterations to be performed.
dist (default: 'e', Euclidean distance)\
Defines the distance function to be used
(see [sec:distancefunctions]).
This function returns the tuple (clusterid, celldata):
clusterid:\
An array with two columns, where the number of rows is equal to the
number of items that were clustered. Each row contains the $x$ and
$y$ coordinates of the cell in the rectangular SOM grid to which the
item was assigned.
celldata:\
An array with dimensions
$\left(\verb|nxgrid|, \verb|nygrid|, \textrm{number of columns}\right)$
if rows are being clustered, or
$\left(\verb|nxgrid|, \verb|nygrid|, \textrm{number of rows}\right)$
if columns are being clustered. Each element [ix][iy] of this
array is a 1D vector containing the gene expression data for the
centroid of the cluster in the grid cell with coordinates
[ix][iy].
To save the clustering result, use
In [27]:
record.save(jobname, geneclusters, expclusters)
where the following arguments are defined:
jobname\
The string jobname is used as the base name for names of the files
that are to be saved.
geneclusters\
This argument describes the gene (row-wise) clustering result. In
case of $k$-means clustering, this is a 1D array containing the
number of the cluster each gene belongs to. It can be calculated
using kcluster. In case of hierarchical clustering, geneclusters
is a Tree object.
expclusters\
This argument describes the (column-wise) clustering result for the
experimental conditions. In case of $k$-means clustering, this is a
1D array containing the number of the cluster each experimental
condition belongs to. It can be calculated using kcluster. In case
of hierarchical clustering, expclusters is a Tree object.
This method writes the text file jobname.cdt, jobname.gtr,
jobname.atr, jobname*.kgg, and/or jobname*.kag for subsequent
reading by the Java TreeView program. If geneclusters and
expclusters are both None, this method only writes the text file
jobname.cdt; this file can subsequently be read into a new Record
object.
This is an example of a hierarchical clustering calculation, using
single linkage clustering for genes and maximum linkage clustering for
experimental conditions. As the Euclidean distance is being used for
gene clustering, it is necessary to scale the node distances genetree
such that they are all between zero and one. This is needed for the Java
TreeView code to display the tree diagram correctly. To cluster the
experimental conditions, the uncentered correlation is being used. No
scaling is needed in this case, as the distances in exptree are
already between zero and two. The example data cyano.txt can be found
in the data subdirectory.
In [28]:
from Bio import Cluster
handle = open("cyano.txt")
record = Cluster.read(handle)
handle.close()
genetree = record.treecluster(method='s')
genetree.scale()
exptree = record.treecluster(dist='u', transpose=1)
record.save("cyano_result", genetree, exptree)
This will create the files cyano_result.cdt, cyano_result.gtr, and
cyano_result.atr.
Similarly, we can save a $k$-means clustering solution:
In [29]:
from Bio import Cluster
handle = open("cyano.txt")
record = Cluster.read(handle)
handle.close()
(geneclusters, error, ifound) = record.kcluster(nclusters=5, npass=1000)
(expclusters, error, ifound) = record.kcluster(nclusters=2, npass=100, transpose=1)
record.save("cyano_result", geneclusters, expclusters)