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# Documentation

### Contents

Agglomerative hierarchical cluster tree

## Description

Z = linkage(X) returns a matrix Z that encodes a tree of hierarchical clusters of the rows of the real matrix X.

Z = linkage(X,method) creates the tree using the specified method, where method describes how to measure the distance between clusters.

Z = linkage(X,method,metric) performs clustering using the distance measure metric to compute distances between the rows of X.

Z = linkage(X,method,pdist_inputs) passes parameters to the pdist function, which is the function that computes the distance between rows of X.

Z = linkage(X,method,metric,'savememory',value) uses a memory-saving algorithm when value is 'true', and uses the standard algorithm when value is 'false'.

Z = linkage(Y) uses a vector representation Y of a distance matrix. Y can be a distance matrix as computed by pdist, or a more general dissimilarity matrix conforming to the output format of pdist.

Z = linkage(Y,method) creates the tree using the specified method, where method describes how to measure the distance between clusters.

## Input Arguments

X

Matrix with two or more rows. The rows represent observations, the columns represent categories or dimensions.

method

Algorithm for computing distance between clusters.

MethodDescription
'average'

Unweighted average distance (UPGMA)

'centroid'

Centroid distance (UPGMC), appropriate for Euclidean distances only

'complete'

Furthest distance

'median'

Weighted center of mass distance (WPGMC), appropriate for Euclidean distances only

'single'

Shortest distance

'ward'

Inner squared distance (minimum variance algorithm), appropriate for Euclidean distances only

'weighted'

Weighted average distance (WPGMA)

Default: 'single'

metric

Any distance metric that the pdist function accepts.

MetricDescription
'euclidean'

Euclidean distance (default).

'seuclidean'

Standardized Euclidean distance. Each coordinate difference between rows in X is scaled by dividing by the corresponding element of the standard deviation S=nanstd(X). To specify another value for S, use D=pdist(X,'seuclidean',S).

'cityblock'

City block metric.

'minkowski'

Minkowski distance. The default exponent is 2. To specify a different exponent, use D = pdist(X,'minkowski',P), where P is a scalar positive value of the exponent.

'chebychev'

Chebychev distance (maximum coordinate difference).

'mahalanobis'

Mahalanobis distance, using the sample covariance of X as computed by nancov. To compute the distance with a different covariance, use D = pdist(X,'mahalanobis',C), where the matrix C is symmetric and positive definite.

'cosine'

One minus the cosine of the included angle between points (treated as vectors).

'correlation'

One minus the sample correlation between points (treated as sequences of values).

'spearman'

One minus the sample Spearman's rank correlation between observations (treated as sequences of values).

'hamming'

Hamming distance, which is the percentage of coordinates that differ.

'jaccard'

One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.

custom distance function

A distance function specified using @:
D = pdist(X,@distfun)

A distance function must be of form

`d2 = distfun(XI,XJ)`

taking as arguments a 1-by-n vector XI, corresponding to a single row of X, and an m2-by-n matrix XJ, corresponding to multiple rows of X. distfun must accept a matrix XJ with an arbitrary number of rows. distfun must return an m2-by-1 vector of distances d2, whose kth element is the distance between XI and XJ(k,:).

Default: 'euclidean'

pdist_inputs

A cell array of parameters accepted by the pdist function. For example, to set the metric to minkowski and use an exponent of 5, set pdist_inputs to {'minkowski',5}.

savememory

A string, either 'on' or 'off'. When applicable, the 'on' setting causes linkage to construct clusters without computing the distance matrix. savememory is applicable when:

• linkage is 'centroid', 'median', or 'ward'

• distance is 'euclidean' (default)

When savememory is 'on', linkage run time is proportional to the number of dimensions (number of columns of X). When savememory is 'off', linkage memory requirement is proportional to N2, where N is the number of observations. So choosing the best (least-time) setting for savememory depends on the problem dimensions, number of observations, and available memory. The default savememory setting is a rough approximation of an optimal setting.

Default: 'on' when X has 20 columns or fewer, or the computer does not have enough memory to store the distance matrix; otherwise 'off'

Y

A vector of distances with the same format as the output of the pdist function:

• A row vector of length m(m–1)/2, corresponding to pairs of observations in a matrix X with m rows

• Distances arranged in the order (2,1), (3,1), ..., (m,1), (3,2), ..., (m,2), ..., (m,m–1))

Y can be a more general dissimilarity matrix conforming to the output format of pdist.

## Output Arguments

 Z Z is a (m – 1)-by-3 matrix, where m is the number of observations in the original data. Columns 1 and 2 of Z contain cluster indices linked in pairs to form a binary tree. The leaf nodes are numbered from 1 to m. Leaf nodes are the singleton clusters from which all higher clusters are built. Each newly-formed cluster, corresponding to row Z(I,:), is assigned the index m+I. Z(I,1:2) contains the indices of the two component clusters that form cluster m+I. There are m-1 higher clusters which correspond to the interior nodes of the clustering tree. Z(I,3) contains the linkage distances between the two clusters merged in row Z(I,:). For example, suppose there are 30 initial nodes and at step 12 cluster 5 and cluster 7 are combined. Suppose their distance at that time is 1.5. Then Z(12,:) will be [5, 7, 1.5]. The newly formed cluster will have index 12 + 30 = 42. If cluster 42 appears in a later row, it means the cluster created at step 12 is being combined into some larger cluster.

## Examples

expand all

### Compare Cluster Assignments to Clusters

```load fisheriris
```

Compute four clusters of the Fisher iris data using Ward linkage and ignoring species information.

```Z = linkage(meas,'ward','euclidean');
c = cluster(Z,'maxclust',4);
```

See how the cluster assignments correspond to the three species.

```crosstab(c,species)
```
```ans =

0    25     1
0    24    14
0     1    35
50     0     0

```

Display the first five rows of Z.

```firstfive = Z(1:5,:)
```
```firstfive =

102.0000  143.0000         0
8.0000   40.0000    0.1000
1.0000   18.0000    0.1000
10.0000   35.0000    0.1000
129.0000  133.0000    0.1000

```

Create a dendrogram plot of Z .

```dendrogram(Z)
```

### Cluster Data and Plot the Result

Randomly generate the sample data with 20000 observations.

```rng default; % For reproducibility
X = rand(20000,3);
```

Create a hierarchical cluster tree using Ward's linkage.

```Z = linkage(X,'ward','euclidean','savememory','on');
```

If you set savememory to 'off' , you can get an out-of-memory error if your machine doesn't have enough memory to hold the distance matrix.

Cluster data into four groups and plot the result.

```c = cluster(Z,'maxclust',4);
scatter3(X(:,1),X(:,2),X(:,3),10,c)
```

expand all

The following notation is used to describe the linkages used by the various methods:

• Cluster r is formed from clusters p and q.

• nr is the number of objects in cluster r.

• xri is the ith object in cluster r.

• Single linkage, also called nearest neighbor, uses the smallest distance between objects in the two clusters:

$d\left(r,s\right)=\mathrm{min}\left(dist\left({x}_{ri},{x}_{sj}\right)\right),i\in \left(i,...,{n}_{r}\right),j\in \left(1,...,{n}_{s}\right)$

• Complete linkage, also called furthest neighbor, uses the largest distance between objects in the two clusters:

$d\left(r,s\right)=\mathrm{max}\left(dist\left({x}_{ri},{x}_{sj}\right)\right),i\in \left(1,...,{n}_{r}\right),j\in \left(1,...,{n}_{s}\right)$

• Average linkage uses the average distance between all pairs of objects in any two clusters:

$d\left(r,s\right)=\frac{1}{{n}_{r}{n}_{s}}\sum _{i=1}^{{n}_{r}}\sum _{j=1}^{{n}_{s}}dist\left({x}_{ri},{x}_{sj}\right)$

• Centroid linkage uses the Euclidean distance between the centroids of the two clusters:

$d\left(r,s\right)={‖{\overline{x}}_{r}-{\overline{x}}_{s}‖}_{2}$

where

${\overline{x}}_{r}=\frac{1}{{n}_{r}}\sum _{i=1}^{{n}_{r}}{x}_{ri}$

• Median linkage uses the Euclidean distance between weighted centroids of the two clusters,

$d\left(r,s\right)={‖{\stackrel{˜}{x}}_{r}-{\stackrel{˜}{x}}_{s}‖}_{2}$

where ${\stackrel{˜}{x}}_{r}$ and ${\stackrel{˜}{x}}_{s}$ are weighted centroids for the clusters r and s. If cluster r was created by combining clusters p and q, ${\stackrel{˜}{x}}_{r}$ is defined recursively as

${\stackrel{˜}{x}}_{r}=\frac{1}{2}\left({\stackrel{˜}{x}}_{p}+{\stackrel{˜}{x}}_{q}\right)$

• Ward's linkage uses the incremental sum of squares; that is, the increase in the total within-cluster sum of squares as a result of joining two clusters. The within-cluster sum of squares is defined as the sum of the squares of the distances between all objects in the cluster and the centroid of the cluster. The sum of squares measure is equivalent to the following distance measure d(r,s), which is the formula linkage uses:

$d\left(r,s\right)=\sqrt{\frac{2{n}_{r}{n}_{s}}{\left({n}_{r}+{n}_{s}\right)}}{‖{\overline{x}}_{r}-{\overline{x}}_{s}‖}_{2},$

where

• ${‖‖}_{2}$ is Euclidean distance

• ${\overline{x}}_{r}$ and ${\overline{x}}_{s}$ are the centroids of clusters r and s

• nr and ns are the number of elements in clusters r and s

In some references the Ward linkage does not use the factor of 2 multiplying nrns. The linkage function uses this factor so the distance between two singleton clusters is the same as the Euclidean distance.

• Weighted average linkage uses a recursive definition for the distance between two clusters. If cluster r was created by combining clusters p and q, the distance between r and another cluster s is defined as the average of the distance between p and s and the distance between q and s:

$d\left(r,s\right)=\frac{\left(d\left(p,s\right)+d\left(q,s\right)\right)}{2}$

### Tips

• Computing linkage(Y) can be slow when Y is a vector representation of the distance matrix. For the 'centroid', 'median', and 'ward' methods, linkage checks whether Y is a Euclidean distance. Avoid this time-consuming check by passing in X instead of Y.

• The centroid and median methods can produce a cluster tree that is not monotonic. This occurs when the distance from the union of two clusters, r and s, to a third cluster is less than the distance between r and s. In this case, in a dendrogram drawn with the default orientation, the path from a leaf to the root node takes some downward steps. To avoid this, use another method. The following image shows a nonmonotonic cluster tree.

In this case, cluster 1 and cluster 3 are joined into a new cluster, while the distance between this new cluster and cluster 2 is less than the distance between cluster 1 and cluster 3. This leads to a nonmonotonic tree.

• You can provide the output Z to other functions including dendrogram to display the tree, cluster to assign points to clusters, inconsistent to compute inconsistent measures, and cophenet to compute the cophenetic correlation coefficient.