Lecture 13
Clustering
What is Clustering?
• Clustering is an EXPLORATORY statistical technique
used to break up a data set into smaller groups or
“clusters” with the idea that objects within a cluster are
similar and objects in different clusters are different.
• It uses different distance measures between units of a
group and across groups to decide which units fall in a
group.
Clustering: some thoughts
• Biologists really LOVE clustering, and believe that
clustering can produce “discoveries” of patterns.
• Statisticians for the most part are somewhat skeptical about
these methods.
• Clustering is called “unsupervised learning” by computer
scientists and “class discovery” by micro-array biologists.
• Though clustering DOES provide clusters or smaller
groups, keep in mind that to interpret them you need to
know the epidemiological and technical aspect of the
study.
Keep in mind
• Epidemiological aspects of the study: experiment or
observational study, and if the latter, knowledge of possible
confounders; relation between training and test sets; relation
between current data and future data to which results might be
applied,…
• Technical aspects of the study: tissue collection, storage and
processing procedures, microarray assays and analyses, focussing
on the role of time, place, personnel, reagents, and methods used,
including evidence of design, randomization, blinding, to avoid or
correct for possible confounding….
• In other contexts, and possibly in these, the results have been
driven by study inadequacies rather than by biology. Beware!
Clustering in MA
• Idea: to group observations that are “similar” based on
predefined criteria.
• Clustering can be applied to rows (genes) and/or columns
(arrays) of an expression data matrix.
• Clustering allows for reordering of the rows/columns of an
expression data matrix which is appropriate for
visualization.
• Fundamentally an exploratory tool, clustering is firmly
imbedded in many biologists’ minds as the statistical
method for the analysis of microarray gene expression
data. Worrying, but an opportunity!
Aim and End product of clustering
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Clustering leads to readily interpretable figures and can be helpful
for identifying patterns in time or space, especially artifacts!
Examples:
We can cluster cell samples (columns),e.g. 1) for identification
(profiles). Here, we might want to estimate the number of different
neuronal cell types in a set of samples, based on gene expression
measurements. 2) the identification of new / unknown tumor classes
using gene expression profiles.
• We can cluster genes (rows) , e.g. 1) using large numbers of yeast
experiments, to identify groups of co-regulated genes. These are
usually interpreted and subjected to further analysis. 2) to reduce
redundancy (cf. variable selection) in predictive models.
• BIOLOGIST like the latter, but it is often NOT practical to do.
Why Cluster Samples?
• There are very few formal theories about clustering though
intuitively the idea is:
• cluster the internal cohesion and external isolation.
• Time-course experiments are often clustered to see if there
are developmental similarities.
• Useful for visualization.
• Generally considered appropriate in typical clinical
experiments.
Clustering
• How is “closeness decided”?
• For clustering we generally need two ideas:
• Distance: the original distance used to measure the
distance between two points
• Linkage: condensation of each group of observations into a
single representative point
Clustering: preliminaries
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Distance or similarity measures:
Geometric distances
L1 (Manhattan): d1(x,y)=S |xi-yi|
L2(Euclidean, ruler distance): d2(x,y)= [S (xi-yi)2 ]1/2
[ (xi-yi)’ (xi-yi) ]1/2
Standardized ruler-distance [ (zi1-zi2)’ (zi1-zi2) ]1/2
Mahalanobis Distance: [ (xi-yi)’ S-1(xi-yi) ]1/2
Correlation distance: 1-r, where r is the correlation
coefficient.
• CAN HAVE WEIGHTED VERSIONS OF THESE.
Clustering: preliminaries
Linkage:
• Average Linkage: the distance between two groups of points is the
average of all pairwise distances.
• Median Linkage: the distance between two groups of points is the
median of all pairwise distances.
• Centroid method: the distance between two groups of points is the
distance between the centroids of the two groups.
• Single Linkage: the distance between two-groups is the smallest of all
pairwise distances.
• Complete Linkage: the distance between two-groups is the largest of
all pairwise distances.
Types of Clustering
• Hierarchical and Non-hierarchical methods:
• Non-Hierarchical (Partitioning): Have an initial set of
cluster seed points and then build clusters around the point,
using one of the distance measures. If the cluster is too
large, it can split into smaller ones.
• Hierarchical: Observed data points are grouped into
clusters in a nested sequence of groups.
Non-hierarchical: Partitioning methods
Partition the data into a pre-specified number k of mutually
exclusive and exhaustive groups.
Iteratively reallocate the observations to clusters until some
criterion is met, e.g. minimize within cluster sums of
squares.
Issues:Need to know the seeds and the number of clusters to
start off with. If one uses the computer the pick the seeds
the order of entry of the data may make a difference.
Hierarchical methods
• Hierarchical clustering methods produce a tree or
dendrogram often using single-link clustering methods
• They avoid specifying how many clusters are appropriate
by providing a partition for each k obtained from cutting
the tree at some level.
• The tree can be built in two distinct ways
- bottom-up: agglomerative clustering.**
- top-down: divisive clustering.
Partitioning vs. Hierarchical
• Partitioning:
• Hierarchical
Advantages
Advantages
• Optimal for certain criteria.
• Faster computation.
• Genes automatically assigned to
clusters
• Visual.
Disadvantages
• Need initial k;
• Often require long computation
times.
• All genes are forced into a
cluster.
Disadvantages
• Unrelated genes are eventually
joined
• Rigid, cannot correct later for
erroneous decisions made
earlier.
• Hard to define clusters.
Bottom-up- Agglomerative Method
• This is the most common used method and produces the
famous tree-diagram.
• VERY common in MA literature, first introduced by Eisen
(1998).
• Start with n mRNA sample (or g gene) clusters
• At each step, merge the two closest clusters using a
measure of between-cluster dissimilarity which reflects the
shape of the clusters
• The distance between clusters is defined by the method
used (e.g., if complete linkage, the distance is defined as
the distance between furthest pair of points in the two
clusters)
Example
• Suppose we have 5 genes with a distance matrix given by:
1
1
2
3
4
5
2
.31
3
.43
.48
4
.47
.47
.37
5
.23
.33
.46
.45
Example
• First we have 5 clusters:
• C0 = {[1],[2],[3],[4],[5]}
• Since 1 and 5 have the least distance they are combined
and C1 = {[1,5],[2],[3],[4]}
• And C2= {[1,5],[2],[3,4]}
• And C3= {[1,5,2],[3,4]}
• And C4= {[1,5,2,3,4]}
Dendograms
• The dendogram should be interpreted with care, remember
each branch of the dendogram is really like a mobile and
can rotate, without altering the mathematical structure of
the tree.
• Neighboring nodes are “close” ONLY if they lie on the
same branch.
• It has been proposed one should slice the tree and look at
the clusters produced therein. However, WHERE to cut
the tree is subjective and there is no consensus about this.
• Issue: mistakes made early have no way of being corrected
later in this approach.
Two-way Clustering
• Refers to methods that use samples and genes
simultaneously to extract information.
Some examples:
• - Block Clustering (Hartigan, 1972) which repeatedly
rearranges rows and columns to obtain the largest
reduction of total within block variance.
• - Plaid Models (Lazzeroni and Owen, 2002)
• - Friedman and Meulmann (2002) present an algorithm to
cluster samples based on the subsets of attributes, i.e. each
group of samples could have been characterized by
different gene sets.
How many clusters? A brief discussion
• Global Criteria:
1. Statistics based on within- and between-clusters matrices of sumsofsquares and cross-products (30 methods reviewed by Milligan &
Cooper, 1985).
2. Average silhouette (Kaufman & Rousseeuw, 1990).
3. Graph theory (e.g.: cliques in CAST) (Ben-Dor et al., 1999).
4. Model-based methods: EM algorithm for Gaussian mixtures, Fraley &
Raftery (1998, 2000) and McLachlan et al. (2001).
• Resampling methods:
1. Gap statistic (Tibshirani et al., 2000).
2. WADP (Bittner et al., 2000).
3. Clest (Dudoit & Fridlyand, 2001).
4. Boostrap (van der Laan & Pollard, 2001).
Some remarks on clustering- 1
• Simplistically, clustering cannot fail. That is, every
clustering method will return clusters, whether the data are
organized in clusters or not.
• Clustering helps to group / order information and is a
visualization tool for learning about the data. However,
clustering results do not provide any kind of “proof” of
anything.
Some remarks-II
• One of the more paradoxical aspects of clustering is that it gets used in
biology, even when class labels are available instead of using a
discrimination method.
• The idea is: it is somehow seen as less “biased” to demonstrate the
ability of the data to produce the class differences without using class
labels.
• When the inferred clusters largely coincide with the known classes,
this is thought to “validate” the class labels.
• The illogicality and inefficiency of this process does not seem to have
become widely appreciated. One sees different “classifiers” (e.g.
different gene sets) compared w.r.t their ability to separate known
classes, simply by inspecting the clustering they produce, rather than
by building classifiers.
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library(stats)
library(cluster)
#reading a tab-delimited file with 10 rows (condition1-10) and 200 #cols(200
genes)
my.data1=read.table("cluster.csv",header=TRUE,sep=",")
#removing the first column (with the row names from the dataset)
my.data=my.data1[,-1]
par(mfrow=c(1,3))
#clustering using euclidean,manhattan,correlation distance,
average,single,complete linkage
dist1=dist(t(my.data),method="euclidean")
hc1=hclust(dist1,"ave")
plot(hc1)
dist2=dist(t(my.data),method="manhattan")
hc2=hclust(dist2,"single")
plot(hc2)
dd <- as.dist((1 - cor((my.data))/2))
round(1000 * dd) # (prints more nicely)
hc3=hclust((dd),method="complete") # to see a dendrogram of clustered
variables
plot(hc3)
Cluster Dendrogram
Cluster Dendrogram
C8
C4
C6
0.9
dist1
hclust (*, "average")
C5
C9
C3
C7
0.8
dist2
hclust (*, "single")
C2
C10
C6
C7
C7
C10
C10
C8
C9
C3
C4
C6
820
C2
C11
C5
C11
C11
840
C8
C9
1.0
Height
C2
C4
C3
860
Height
90
80
70
Height
880
1.1
100
900
C5
1.2
110
920
Cluster Dendrogram
(dd)
hclust (*, "complete")