Hierarchical Clustering

Some slides by Serafim Batzoglou

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From expression profiles to distances Expression

levels,

“Raw Data”

experiments

gene

s

In some situation the input for clustering is only the similarities / distances 10 20 30 40 50 60

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From the Raw Data matrix we compute the similarity matrix S. Sij reflects the similarity of the expression patterns of gene i and gene j.

experiments experiments

ABDBM © Ron Shamir

More generally • In K-means and SOM the input was a

vector for each item (e.g. a dot in Rn) • Here we have a matrix of pairwise

distances between items, and we wish to cluster the items.

• A distance based clustering alg

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An Alternative view of Clustering Form a tree-hierarchy of the input elements satisfying: • More similar elements are placed closer along the tree. •Or: Tree distances reflect element similarity •Note: No explicit partition into clusters.

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Partitioning vs Hierarchical Representations

“dendrogram”

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Hierarchical Representations (2)

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2.8

4.5 5.0

Ultrametric: rooted tree, all root-leaf distances are equal

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UPGMA Clustering (unweighted pair group method using arithmetic averages)

• Approach: Form a tree; closer species according to input distances should be closer in the tree

• Build the tree bottom up, each time merging two smaller trees

• All leaves are at same distance from the root

Hierarchical Clustering: UPGMA Sokal & Michener 58, Lance & Williams 67

UPGMA (unweighted pair group method using arithmetic averages) Given two disjoint clusters Ci, Cj, 1 dij = ––––––––– Σ{p ∈Ci, q ∈Cj}dpq |Ci| × |Cj| If Ck = Ci ∪ Cj, then distance from Ck to another cluster Cl is: dil |Ci| + djl |Cj| dkl = –––––––––––––– |Ci| + |Cj|

Algorithm: UPGMA

Initialization: Assign each xi into its own cluster Ci Define one leaf per sequence, height 0 Iteration: Find two clusters Cr, Cs s.t. drs is min Define a new cluster Ct = Cr ∪ Cs Define node Ars connecting Cr, Cs, height

drs/2

Delete Cr, Cs

dit=dti=(|Cr |•dir+ |Cs| • dis)/(|Cr |+ |Cs| ) length(Cr, Ars) = height(Ars) - height(Cr) length(Cs,Ars) = height(Ars) - height(Cs) Termination: When all sequences belong to one cluster

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Time: Naïve: O(n3); Can show O(n2 logn) (ex.); O(n2) (harder ex.)

Thm: If the input distances match an ultrametric tree – UPGMA finds it.

http://lectures.molgen.m

pg.de/Phylogeny/Ultram

etric/ 11 ABDBM © Ron Shamir

Robert R. Sokal (1926- 2012) Ph.D. 1952, University of Chicago. Was at Dept. of Ecology and Evolution, SUNY Stony Brook Member of the National Academy of Sciences & American Academy of Sciences. Promoted the use of statistics in biology and co-founded the field of numerical taxonomy. Together with P.H.A. Sneath, authored the two defining texts in this field. Along with F. James Rohlf, authored the very popular biostatistics book, Biometry. Editor of the American Naturalist, president of several learned societies.

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Results (2) • 10 major groups with similar patterns of co-

occurrence, confirming that specific groups of phenotypes co-occur within families.

• certain malformations co-occur in more than one group, e.g. TGA,AVSD.

• Some differences from a proposed taxonomy (Houyel 11)

• (Also: co-occurrence of defects in families is caused by shared susceptibility genes.)

• A starting point for further biomed research

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Variants on hierarchical clustering • Input: Distance matrix Dij; • Initially each element is a cluster. • Find min element Drs in D; merge clusters r,s • Delete elts. r,s, add new elt. t with updated weights • Repeat • Variants:

– Average linkage: UPGMA – Single linkage: Dit= min(Dir, Dis) – Max linkage Dit= max(Dir, Dis)

• Sometimes the number of clusters is needed.

Methods abound.

• Sometimes leaf order matters and not only topology. 16 ABDBM © Ron Shamir

Hierarchical clustering of GE data Eisen et al., PNAS 1998

• Growth response: Starved human fibroblast cells, added serum

• Monitored levels of 8600 genes over 13 time-points • tij - level of target gene i in condition j; • rij – same for reference • Dij= log(tij/rij) • D*ij= [Dij –E(Di)]/std(Di) • Similarity of genes k,l: Skl=(ΣjD*kj •D*lj)/Ncond • Applied average linkage method • Ordered leaves by increasing subtree weight:

average expression level, time of maximal induction, other criteria

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Observations • Distinct measurements of same genes

cluster together • Genes of similar function cluster

together • Many cluster-function specific

insights • Interpretation is a REAL biological

challenge

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Yeast GE data

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Mike Eisen & Pat Brown

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More on hierarchical methods (2) • The methods described above –

agglomerative (bottom up) • An alternative approach: Divisive (top down) • Advantages:

– gives a single coherent global picture – Intuitive for biologists (from phylogeny)

• Disadvantages: – no single partition; no specific clusters – Forces all elements to fit a tree

• There are other methods that do not assume an ultrametric solution, notably Neighbor Joining. In genomics still UPGMA rules.

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Hierarchical Clustering & Congenital Heart Defects

Ellsoe et al. (Soren Brunak lab) European Heart Journal (2017)

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CHD • Congenital heart defects (CHD)

affect almost 1% of all live born children

• Number of adults with CHD is increasing

• Recurrence patterns in families are poorly understood

• Do cases in the same family tend to have similar types of malformations?

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Study • 1163 families, 3080 family members

with clinical diagnosis (avg 2.65 CHD cases /family)

• Each case is identified as having one or more of 41 different types of CHD lesions: AVD, BSD, VSD,…

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Concordant & discordant disease pairs

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Concordant: (ASD,ASD), (ASD,VSD)… Discordant: (BAV,BAV)

Gender ratio, concordance & discordance

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Scoring pairs of defects • N(A,B) – # families with A, B • N(A,¬B) – # families with A, not B • N(¬ A,B) – # families with B, not A • N(¬ A, ¬ B) – # families with none • The odds ratio (OR) between phenotypes A and B:

• OR(A,B) = N(A,B) N(¬ A, ¬ B)/N(A,¬B)N(¬ A,B) • ??! • Perhaps OR(A,B) = N(A,B)/[N(A,¬B)+N(¬ A,B)]

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ABDBM © Ron Shamir

Results

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