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Canadian Bioinformatics Workshops
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Module #: Title of Module
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Module 2: Clustering,
Classification and Feature
Selection
Sohrab Shah
Centre for Translational and Applied Genomics
Molecular Oncology Breast Cancer Research Program
BC Cancer Agency
sshah@bccrc.ca
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Module Overview
• Introduction to clustering
– distance metrics
– hierarchical, partitioning and model based clustering
• Introduction to classification
– building a classifier
– avoiding overfitting
– cross validation
• Feature Selection in clustering and classification
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Introduction to clustering
• What is clustering?
– unsupervised learning
– discovery of patterns in data
– class discovery
• Grouping together “objects” that are most similar (or
least dissimilar)
– objects may be genes, or samples, or both
• Example question: Are there samples in my cohort
that can be subgrouped based on molecular
profiling?
– Do these groups have correlation to clinical outcome?
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Distance metrics
• In order to perform clustering, we need to have a way
to measure how similar (or dissimilar) two objects are
• Euclidean distance:
p
xy 
(x
i
 y i )2
i1
• Manhattan distance:

p
dxy   x i  y i
dissimilar
similar
i1
• 1-correlation

– proportional to Euclidean
distance, but invariant to range of
measurement from one sample to the next
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Distance metrics compared
Euclidean
Manhattan
1-Pearson
Conclusion: distance matters!
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Other distance metrics
• Hamming distance for ordinal, binary or
categorical data:
p
dxy   I(x i  y i )
i1

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Approaches to clustering
• Partitioning methods
– K-means
– K-medoids (partitioning around medoids)
– Model based approaches
• Hierarchical methods
– nested clusters
• start with pairs
• build a tree up to the root
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Partitioning methods
• Anatomy of a partitioning based method
– data matrix
– distance function
– number of groups
• Output
– group assignment of every object
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Partitioning based methods
• Choose K groups
– initialise group centers
• aka centroid, medoid
– assign each object to the
nearest centroid according to
the distance metric
– reassign (or recompute)
centroids
– repeat last 2 steps until
assignment stabilizes
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K-medoids in action
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K-means vs K-medoids
K-means
K-medoids
Centroids are the ‘mean’ of
the clusters
Centroids are an actual
object that minimizes the
total within cluster distance
Centroids need to be
recomputed every iteration
Centroid can be determined
from quick look up into the
distance matrix
Initialisation difficult as
notion of centroid may be
unclear before beginning
Initialisation is simply K
randomly selected objects
kmeans
pam
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Partitioning based methods
Advantages
Disadvantages
Number of groups is well
defined
Have to choose the number
of groups
A clear, deterministic
Sometimes objects do not
assignment of an object to a fit well to any cluster
group
Simple algorithms for
inference
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Can converge on locally
optimal solutions and often
require multiple restarts with
random initializations
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Agglomerative hierarchical clustering
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Hierarchical clustering
• Anatomy of hierarchical clustering
– distance matrix
– linkage method
• Output
– dendrogram
• a tree that defines the relationships between objects and
the distance between clusters
• a nested sequence of clusters
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Linkage methods
single
complete
distance between centroids
average
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Linkage methods
• Ward (1963)
– form partitions that minimizes the loss associated with each
grouping
– loss defined as error sum of squares (ESS)
– consider 10 objects with scores (2, 6, 5, 6, 2, 2, 2, 2, 0, 0, 0)
ESSOnegroup = (2 -2.5)2 + (6 -2.5)2 + ....... + (0 -2.5)2 = 50.5
On the other hand, if the 10 objects are classified according to their scores into four sets,
{0,0,0}, {2,2,2,2}, {5}, {6,6}
The ESS can be evaluated as the sum of squares of four separate error sums of squares:
ESSOnegroup = ESSgroup1 + ESSgroup2 + ESSgroup3 + ESSgroup4 = 0.0
Thus, clustering the 10 scores into 4 clusters results in no loss of information.
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Linkage methods in action
• clustering based on single linkage
•
•
single <- hclust(dist(t(exprMatSub),method="euclidean"), method=”single");
plot(single);
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Linkage methods in action
• clustering based on complete linkage
•
•
complete <- hclust(dist(t(exprMatSub),method="euclidean"), method="complete");
plot(complete)
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Linkage methods in action
• clustering based on centroid linkage
•
•
centroid <- hclust(dist(t(exprMatSub),method="euclidean"), method=”centroid");
plot(centroid);
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Linkage methods in action
• clustering based on average linkage
•
•
average <- hclust(dist(t(exprMatSub),method="euclidean"), method=”average");
plot(average);
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Linkage methods in action
• clustering based on Ward linkage
•
•
ward <- hclust(dist(t(exprMatSub),method="euclidean"), method=”ward");
plot(ward);
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Linkage methods in action
Conclusion: linkage matters!
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Hierarchical clustering analyzed
Advantages
Disadvantages
There may be small clusters Clusters might not be
nested inside large ones
naturally represented by a
hierarchical structure
No need to specify number
groups ahead of time
Its necessary to ‘cut’ the
dendrogram in order to
produce clusters
Flexible linkage methods
Bottom up clustering can
result in poor structure at
the top of the tree. Early
joins cannot be ‘undone’
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Model based approaches
• Assume the data are ‘generated’ from a mixture of K distributions
– What cluster assignment and parameters of the K distributions best
explain the data?
• ‘Fit’ a model to the data
• Try to get the best fit
• Classical example: mixture of Gaussians (mixture of normals)
• Take advantage of probability theory and well-defined distributions in
statistics
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Model based clustering: array
CGH
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Model based clustering of aCGH
Problem: patient cohorts often exhibit molecular heterogeneity making rarer shared
CNAs hard to detect
Approach: Cluster the data by extending the profiling to the multi-group setting
A mixture of HMMs: HMM-Mix
Group g
Sparse profiles
…
…
Distribution of calls in a group
Profile
State c
CNA calls
State k
Patient p
Shah et al (Bioinformatics, 2009)
Raw data
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Advantages of model based approaches
• In addition to clustering patients into groups, we output
a ‘model’ that best represents the patients in a group
• We can then associate each model with clinical
variables and simply output a classifier to be used on
new patients
• Choosing the number of groups becomes a model
selection problem (ie the Bayesian Information
Criterion)
– see Yeung et al Bioinformatics (2001)
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Clustering 106 follicular lymphoma
patients with HMM-Mix
Initialisation
Profiles
Converged
Clinical
 Recapitulates known FL subgroups
 Subgroups have clinical relevance
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Feature selection
• Most features (genes, SNP probesets, BAC clones) in high
dimensional datasets will be uninformative
– examples: unexpressed genes, housekeeping genes, ‘passenger
alterations’
• Clustering (and classification) has a much higher chance of
success if uninformative features are removed
• Simple approaches:
– select intrinsically variable genes
– require a minimum level of expression in a proportion of samples
– genefilter package (Bioonductor): Lab1
• Return to feature selection in the context of classification
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Advanced topics in clustering
•
•
•
•
Top down clustering
Bi-clustering or ‘two-way’ clustering
Principal components analysis
Choosing the number of groups
– model selection
• AIC, BIC
• Silhouette coefficient
• The Gap curve
• Joint clustering and feature selection
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What Have We Learned?
• There are three main types of clustering approaches
– hierarchical
– partitioning
– model based
• Feature selection is important
– reduces computational time
– more likely to identify well-separated groups
• The distance metric matters
• The linkage method matters in hierarchical clustering
• Model based approaches offer principled probabilistic methods
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Module Overview
• Clustering
• Classification
• Feature Selection
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Classification
• What is classificiation?
– Supervised learning
– discriminant analysis
• Work from a set of objects with predefined classes
– ie basal vs luminal or good responder vs poor responder
• Task: learn from the features of the objects: what is the
basis for discrimination?
• Statistically and mathematically heavy
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Classification
poor response
poor response
poor response
learn a classifier
good response
good response
good response
new patient
What is the most likely response?
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Example: DLBCL subtypes
Wright et al,
PNAS (2003)
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DLBCL subtypes
Wright et al, PNAS (2003)
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Classification approaches
• Wright et al PNAS (2003)
• Weighted features in a linear predictor score:
• aj: weight of gene j determined by t-test statistic
• Xj: expression value of gene j
• Assume there are 2 distinct distributions of LPS:
1 for ABC, 1 for GCB
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Wright et al, DLBCL, cont’d
• Use Bayes’ rule to determine a probability that a sample
comes from group 1:
•
: probability density function that represents
group 1
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Learning the classifier, Wright et al
• Choosing the genes (feature selection):
– use cross validation
– Leave one out cross validation
•
•
•
•
•
Pick a set of samples
Use all but one of the samples as training, leaving one out for test
Fit the model using the training data
Can the classifier correctly pick the class of the remaining case?
Repeat exhaustively for leaving out each sample in turn
– Repeat using different sets and numbers of genes based on tstatistic
– Pick the set of genes that give the highest accuracy
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Overfitting
• In many cases in biology, the number of features is much larger
than the number of samples
• Important features may not be represented in the training data
• This can result in overfitting
– when a classifier discriminates well on its training data, but does
not generalise to orthogonally derived data sets
• Validation is required in at least one external cohort to believe
the results
• example: the expression subtypes for breast cancer have been
repeatedly validated in numerous data sets
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Overfitting
• To reduce the problem of overfitting, one can use
Bayesian priors to ‘regularize’ the parameter
estimates of the model
• Some methods now integrate feature selection and
classification in a unified analytical framework
– see Law et al IEEE (2005): Sparse Multinomial Logistic
Regression (SMLR): http://www.cs.duke.edu/~amink/software/smlr/
• Cross validation should always be used in training a
classifier
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Evaluating a classifier
• The receiver operator characteristic curve
– plots the true positive rate vs the false positive rate
• Given ground truth and a
probabilistic classifier
– for some number of probability
thresholds
– compute the TPR
– proportion of positives that
were predicted as true
– compute the FPR
– number of false predictions
over the total number of
predictions
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Other methods for classification
•
•
•
•
•
Support vector machines
Linear discriminant analysis
Logistic regression
Random forests
See:
– Ma and Huang Briefings in Bioinformatics (2008)
– Saeys et al Bioinformatics (2007)
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Questions?
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Lab: Clustering and feature
selection
• Get familiar clustering tools and plotting
–
–
–
–
Feature selection methods
Distance matrices
Linkage methods
Partition methods
• Try to reproduce some of the figures from
Chin et al using the freely available data
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Module 2: Lab
Coffee break
Back at: 15:00
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