# optimalleaforder

Optimal leaf ordering for hierarchical clustering

## Description

returns an optimal leaf ordering for the hierarchical binary cluster tree, `leafOrder`

= optimalleaforder(`tree`

,`D`

)`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.

returns the optimal leaf ordering using one or more name-value pair arguments.`leafOrder`

= optimalleaforder(`tree`

,`D`

,`Name,Value`

)

## 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('default') % 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`

.

leafOrder

`leafOrder = `*1×10*
1 4 9 10 2 5 8 3 7 6

### 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
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

**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*denote the distance and similarity between leaves

_{i,j}*i*and

*j*, respectively. The included similarity transformations are:

`'linear'` | Sim = max_{i,j}_{i,j} (d) – _{i,j }d_{i,j} |

`'inverse'` | Sim = 1/_{i,j}d_{i,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.

## See Also

`dendrogram`

| `linkage`

| `pdist`

**Introduced in R2012b**

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