optimalleaforder

Optimal leaf ordering for hierarchical clustering

Syntax

leafOrder = optimalleaforder(tree,D)

leafOrder = optimalleaforder(tree,D,Name,Value)

Description

leafOrder = optimalleaforder(tree,D) returns an optimal leaf ordering for the hierarchical binary cluster tree, tree, using the distances, D. An optimal leaf ordering of a binary tree maximizes the sum of the similarities between adjacent leaves by flipping tree branches without dividing the clusters.

leafOrder = optimalleaforder(tree,D,Name,Value) returns the optimal leaf ordering using one or more name-value pair arguments.

Examples

Plot Dendrogram with Optimal Leaf Order

Create a hierarchical binary cluster tree using linkage. Then, compare the dendrogram plot with the default ordering to a dendrogram with an optimal leaf ordering.

Generate sample data.

rng(0,"twister") % For reproducibility
X = rand(10,2);

Create a distance vector and a hierarchical binary clustering tree. Use the distances and clustering tree to determine an optimal leaf order.

D = pdist(X);
tree = linkage(D,"average");
leafOrder = optimalleaforder(tree,D);

Plot the dendrogram with the default ordering and the dendrogram with the optimal leaf ordering.

figure()
subplot(2,1,1)
dendrogram(tree)
title("Default Leaf Order")

subplot(2,1,2)
dendrogram(tree,reorder=leafOrder)
title("Optimal Leaf Order")

The order of the leaves in the bottom figure corresponds to the elements in leafOrder. The optimal leaf order flips tree branches to maximize the sum of the similarities between adjacent leaves.

leafOrder

leafOrder = 1×10

     1     4     9    10     2     5     8     3     7     6

Create a scatter plot of the sample data and label the points.

figure()
plot(X(:,1),X(:,2),".")
text(X(:,1),X(:,2),num2str((1:size(X,1))'))

In the default leaf ordering, points 1 and 4 are located next to points 3 and 8. In the optimal leaf ordering, points 1 and 4 are located next to point 9, which reflects their relative positions in the scatter plot.

Optimal Leaf Order Using Inverse Distance Similarity

Generate sample data.

rng('default') % For reproducibility
X = rand(10,2);

Create a distance vector and a hierarchical binary clustering tree.

D = pdist(X);
tree = linkage(D,'average');

Use the inverse distance similarity transformation to determine an optimal leaf order.

leafOrder = optimalleaforder(tree,D,'Transformation','inverse')

leafOrder = 1×10

     1     4     9    10     2     5     8     3     7     6

Input Arguments

tree — Hierarchical binary cluster tree

matrix returned by linkage

Hierarchical binary cluster tree, specified as an (M – 1)-by-3 matrix that you generate using linkage, where M is the number of leaves.

D — Distances

matrix | vector

Distances for determining similarities between leaves, specified as a matrix or vector of distances. For example, you can generate distances using pdist.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'Criteria','group','Transformation','inverse' specifies that the sum of similarities be maximized between every leaf and all other leaves in adjacent clusters, using an inverse similarity transformation.

Criteria — Optimization criterion

'adjacent' (default) | 'group'

Optimization criterion for determining an optimal leaf ordering, specified as the comma-separated pair consisting of 'criteria' and one of these values:

'adjacent'	Maximize the sum of similarities between adjacent leaves.
'group'	Maximize the sum of similarities between every leaf and all other leaves in the adjacent clusters at the same level of the dendrogram.

Example: 'Criteria','group'

Transformation — Method for transforming distances to similarities

'linear' (default) | 'inverse' | function handle

Method for transforming distances to similarities, specified as the comma-separated pair consisting of 'Transformation' and one of 'linear', 'inverse', or a function handle.

Let d_i,j and Sim_i,j denote the distance and similarity between leaves i and j, respectively. The included similarity transformations are:

'linear'	Simi,j = maxi,j (di,j ) – di,j
'inverse'	Simi,j = 1/di,j

To use a custom transformation function, specify a handle to a function that accepts a matrix of distances, D, and returns a matrix of similarities, S. The function should be monotonic decreasing in the range of distance values. S must have the same size as D, with S(i,j) being the similarity computed based on D(i,j).

Example: 'Transformation',@myTransform

Output Arguments

leafOrder — Optimal leaf order

vector

Optimal leaf order, returned as a length-M vector, where M is the number of leaves. leafOrder is a permutation of the vector 1:M, giving an optimal leaf ordering based on the specified distances and similarity transformation.

References

[1] Bar-Joseph, Z., Gifford, D.K., and Jaakkola, T.S. (2001). "Fast optimal leaf ordering for hierarchical clustering." Bioinformatics Vol. 17, Suppl 1:S22–9. PMID: 11472989.

Version History

Introduced in R2012b