Clustering

BIOL309: Clustering & classiﬁcation
Paul Gardner
October 3, 2017
Paul Gardner BIOL309: Clustering & classiﬁcation

Multivariate data
We collect some data, for each item we measure certain
features. E.g. Fisher’s famous iris data set, for 150 plants he
measured:
Sepal width
Sepal length
Petal width
Petal length
For some data sets, we may know that each item belongs to a
particular class (e.g. in the iris data, we also know what
species the plant is)
The features can be thought of as explanatory/predictor
variables and the class can be thought of as the categorical
response variable

Two main styles of analysis (1)
1. Classiﬁcation (Supervised learning)
We have a dataset where we know what the right answer is
for each item (e.g. we know what species each plant is)
We want to come up with a good procedure for classifying
new items (e.g. deciding what species a new plant is given
weve got the same measurements as before)
this procedure can be trained on our existing data, where we
know the right answers, and then applied to novel data.

Two main styles of analysis (2)
2. Clustering (Unsupervised learning)
In this case we have also collected some data, i.e. we have
some items, and for each of them we have measured some
features
BUT, we dont know what groups, if any, the items belong to
We want to ﬁnd ways of viewing the data that will reveal if it
has any interesting structure.
If our data is low dimensional this might not be too tricky,
e.g. start with scatter plots or other exploratory data analysis
methods.

Common techniques
Classiﬁcation (supervised learning)
Linear discrimination analysis (LDA)
Quadratic discrimination (QDA)
Logistic regression
Classiﬁcation trees
Clustering (unsupervised learning)
Principle components analysis (PCA)
Multi-dimensional scaling (MDS)
PCA & MDS are often called “ordination” by ecologists
K-means clustering
Hierarchical clustering (trees)

Clustering examples
mRNA expression microarray data
which genes have similar expression proﬁles?
which samples have similar expression proﬁles?
Determining species boundaries
environmental sequencing projects (metagenomics): how many
species are in my sample?
delineating viruses

Most datasets have more than two dimensions
Scatter plots in > 3
dimensions are not a
good idea
plot(iris[,1:2])
plot(iris[,1:3])
#...
#OR:
pairs(iris[,1:4])
Sepal.Length
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Petal.Length
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Petal.Width

Dimensionality reduction (Ecology: “Ordination”)
Principal component analysis
Multi-dimensional scaling
High-dimensional data is hard on the brain, but there are
techniques for coercing data into 2 dimensions in ways that
preserve useful structure

Principal components analysis
The method depends on the units used, so normally you
would begin by centering all the variables (features) and
rescaling them to have variance 1.
Pick new axes (linear combinations of variables) starting with
the directions of greatest variability.

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The ﬁrst linear principal component of a set of data. The line
minimizes the total squared distance from each point to its
orthogonal projection onto the line.

Principal components analysis in 3D
Sepal.Length
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iris.pca = prcomp(iris[,1:4], scale = TRUE)
plot(iris.pca)
plot(iris.pca$x[,1:2],
col=as.numeric(iris$Species))
biplot(iris.pca,col=c("blue",’red’))
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−3 −2 −1 0 1 2 3
−2−1012
PC1
PC2
setosa
versicolor
virginica
−0.2 −0.1 0.0 0.1 0.2
−0.2−0.10.00.10.2
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iris.pca
Variances
0.00.51.01.52.02.5

Biplots
−0.2 −0.1 0.0 0.1 0.2
−0.2−0.10.00.10.2
PC1
PC2
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Biplots
“Any matrix of rank two can be displayed as a biplot which
consists of a vector for each row and a vector for each
column, chosen so that any element of the matrix is exactly
the inner product of the vectors corresponding to its row and
to its column.”
The biplot provides a useful tool of data analysis and allows
the visual appraisal of the structure of large data matrices.
It is especially revealing in principal component analysis,
where the biplot can show inter-unit distances and indicate
clustering of units as well as display variances and correlations
of the variables.
Gabriel (1971) The Biplot Graphic Display of Matrices with Application to Principal Component Analysis.
Biometrika.

Principal components are linear combinations of of the
features
>iris.pca
Standard deviations:
[1] 1.7083611 0.9560494 0.3830886 0.1439265
Rotation:
PC1 PC2 PC3 PC4
Sepal.Length 0.5210659 -0.37741762 0.7195664 0.2612863
Sepal.Width -0.2693474 -0.92329566 -0.2443818 -0.1235096
Petal.Length 0.5804131 -0.02449161 -0.1421264 -0.8014492
Petal.Width 0.5648565 -0.06694199 -0.6342727 0.5235971
>summary(iris.pca)
Importance of components:
PC1 PC2 PC3 PC4
Standard deviation 1.7084 0.9560 0.38309 0.14393
Proportion of Variance 0.7296 0.2285 0.03669 0.00518
Cumulative Proportion 0.7296 0.9581 0.99482 1.00000

Dissimilarities
A distance is a function d on a set of objects that maps from
S ∗ S → non-negative reals
1. Non-negative d(x, y) for all x, y in S
2. Symmetric d(x, y) = d(y, x)
3. Triangle inequality d(a, b) + d(b, c) ≥ d(a, c)
4. d(x, y) = 0 if and only if x = y
A dissimilarity may only obey the ﬁrst 2 conditions

Dissimilarities
Euclidean distance
DEuc(i, j) = (yi,1 − yj,1)2 + (yi,2 − yj,2)2 + (yi,3 − yj,3)2 + ...
Bray-Curtis dissimilarity
DBC (i, j) =
n−1
k=0 |yi,k − yj,k|
n−1
k=0 (yi,k + yj,k)
= 1 − 2 ∗
|A ∩ B|
|A| + |B|

MDS
Multi-dimensional scaling
Given a NxN matrix of dissimilarities as input, MDS seeks to
ﬁnd a “map” or 2D representation (i.e. in the plane) such
that the (Euclidean) distances in the map are as similar as
possible to the input dissimilarities.
The stress measures how well this has been achieved. Low
stress means the match between the 2D distances in the map
and the original input distances is good.
E.g. minimise stress = i,j (dinput(i, j) − dMDS (i, j))2

MDS in R
iris.dist <- dist(iris[,1:4])
iris.mds <- cmdscale(iris.dist)
plot(iris.mds,col=as.numeric(iris$Species))
legend("topleft",legend=unique(iris$Species),
fill=unique(as.numeric(iris$Species)) )
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iris.mds[,1]
iris.mds[,2]
setosa
versicolor
virginica

Clustering
The goal is to partition the data into groups such that the
pairwise dissimilarity between members of the same group is
small, and the pairwise dissimilarity between members of
diﬀerent groups is large
E.g. we might try to minimise the within group scatter
W = groups k=1:N pairs i,j in group k d(i, j)
For any reasonable sized problem there will be too many
possible groups to do an exact search

Frequently used for phylogenetics & clustering genes
Genetic or morphological distances
Dissimilarity between expression proﬁles
Interactions between molecules

k-means clustering
A very popular iterative clustering method – highly
generalizable
1. Pick a starting solution where each item is assigned to one of
k groups.
2. Find the mean value for each group m1, m2, . . . , mk
3. Reallocate each item x to the group where the distance from it
to the mean d(x, mi ) is smallest
4. Repeat steps 2 and 3 until convergence
NB. a random starting point means that each solution may be
diﬀerent.
Random number generators depend on a “seed”, can be set in
R with “set.seed(1)”
the default seed is created from the current time and the
process ID when one is required.

Irises
setosa, virginica, versicolor
Images by: Radomi Binek, Frank Mayﬁeld & D. Gordon E. Robertson

Example: k-means, iris data with k=3
iris.mds <- cmdscale(iris.dist) #for viz below
iris.kmeans <- kmeans(iris.dist, 3)
par(mfrow=c(1,2))
plot(iris.mds,col=as.numeric(iris$Species),
main="true species", pch=19)
legend("topleft", legend = unique(iris$Species),
fill = unique(as.numeric(iris$Species)))
plot(iris.mds,col=as.numeric(iris.kmeans$cluster),
main="k-means", pch=19)
#repeat to look at the stability of the clustering
#fix by running set.seed(1) each time
#accuracy?
table(iris$Species, iris.kmeans$cluster)

Example: k-means, iris data with k=3
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true species
iris.mds[,1]
iris.mds[,2]
setosa
versicolor
virginica
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k−means
iris.mds[,1]
iris.mds[,2]

Key question is how to pick k
Choose a number of clusters so that adding another cluster
doesn’t give much better modeling of the data.
k1 <- kmeans(iris.dist, 1)
#tot.withinss: Total within-cluster sum of squares
k.score<-c(k1$tot.withinss,
k2$tot.withinss,k3$tot.withinss,
k4$tot.withinss,k5$tot.withinss)
plot(k.score, xlab="number of clusters")
lines(k.score)

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1 2 3 4 5
1000020000300004000050000
number of clusters
iris.kmeans.score

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iris.mds[,2]
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iris.mds[,2]
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iris.mds[,2]

Hierarchical clustering
Agglomerative (bottom up) methods successively merge
clusters starting with each observation in its own cluster
Divisive (top down) methods successively split clusters into
two parts starting with every observation in the same cluster
(less commonly used)
Both methods produce results that can be visualised by a
rooted bifurcating tree
These trees can be used to deﬁne clusters by cutting them at
a given height.

Agglomerative criteria
The key thing is having a rule to decide which two clusters to
merge.
Single linkage (a.k.a. nearest neighbour, a.k.a. a “friends of
friends” clustering strategy) picks the two clusters that
minimize the minimum distance between cluster members
Complete linkage picks the two clusters that minimizes the
maximum distance
Group Average (a.k.a. UPGMA, Unweighted Pair Group
Method with Arithmetic mean) picks the two clusters the
minimize the average distance between cluster members

Agglomerative hierarchical clustering
1. Calculate a matrix of dissimilarities (dij ) between all pairs of
objects.
2. The ﬁrst cluster is formed between the two objects with the
smallest dissimilarity.
3. The dissimilarities between this cluster and the remaining
objects are then recalculated.
4. A second cluster is formed between the two objects with the
next smallest dissimilarity.
5. The procedure continues until all objects are linked in clusters.

Clustering example
#add human readable names to the iris dataset:
rownames(iris) <- paste(iris$Species, 1:150, sep="")
#average linkage clustering
iris.hc <- hclust(iris.dist, method="average")
#try "single" & "complete" too
plot(iris.hc)
cuts <- cutree(iris.hc, k=3) #to assign clusters
table(iris$Species, cuts)

Clustering example
method=average (= UPGMA)
cuts
1 2 3
setosa 50 0 0
versicolor 0 50 0
virginica 0 14 36
method=single
cuts
1 2 3
setosa 50 0 0
versicolor 0 50 0
virginica 0 48 2
method=complete
cuts
1 2 3
setosa 50 0 0
versicolor 0 23 27
virginica 0 49 1
setosa42setosa15setosa16setosa33setosa34setosa37setosa21setosa32setosa44setosa24setosa27setosa36setosa5setosa38setosa50setosa8setosa40setosa28setosa29setosa41setosa1setosa18setosa45setosa6setosa19setosa17setosa11setosa49setosa47setosa20setosa22setosa23setosa14setosa43setosa9setosa39setosa12setosa25setosa7setosa13setosa2setosa46setosa26setosa10setosa35setosa30setosa31setosa3setosa4setosa48virginica105virginica129virginica133virginica112virginica104virginica117virginica138virginica111virginica148virginica113virginica140virginica142virginica146virginica116virginica137virginica149virginica101virginica125virginica121virginica144virginica141virginica145virginica109virginica135virginica110virginica118virginica132virginica119virginica106virginica123virginica136virginica108virginica131virginica103virginica126virginica130versicolor61versicolor99versicolor58versicolor94versicolor66versicolor76versicolor55versicolor59versicolor78versicolor77versicolor87versicolor51versicolor53versicolor86versicolor52versicolor57versicolor74versicolor79versicolor64versicolor92versicolor72versicolor75versicolor98virginica120versicolor69versicolor88virginica115virginica122virginica114virginica102virginica143virginica150versicolor71virginica128virginica139virginica147virginica124virginica127versicolor73versicolor84virginica134virginica107versicolor63versicolor68versicolor83versicolor93versicolor62versicolor95versicolor100versicolor89versicolor96versicolor97versicolor67versicolor85versicolor56versicolor91versicolor65versicolor80versicolor60versicolor54versicolor90versicolor70versicolor81versicolor82
01234
Cluster Dendrogram
hclust (*, "average")
iris.dist
Height

Further reading:
Ch 17&18 of: Quinn & Keough, Experimental Design and
Data Analysis for Biologists
Hastie, Tibshirani & Friedman, The elements of statistical
learning
Blog post: PCA, 3D Visualization, and Clustering in R
http://planspace.org/2013/02/03/pca-3d-visualization-and-
clustering-in-r/
These slides are based upon a course run by Prof. Barbara
Holland (University of Tasmania)
Altman & Krzywinski (2017) Points of Signiﬁcance:
Clustering. Nature Methods.

Not tested: Spurious correlations
See also: http://www.tylervigen.com/spurious-correlations and
https://en.wikipedia.org/wiki/Correlation does not imply causation

Not tested: Misuse of statistics
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−4 −2 0 2 4
−2024
x
y
R2
= 5.92e−07
p = 0.94
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
−4 −2 0 2 4
−2024
x
y
R2
= 0.351
p = 0

The End

Clustering

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