# Linkage methods for Cluster Variables

## Average

In average linkage, the distance between two clusters is the average distance between a variable in one cluster and a variable in the other cluster. The average distance is calculated with the following distance matrix:

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j
Nknumber of variables cluster k
Nlnumber of variables in cluster l
Nmnumber of variables in cluster m

## Centroid

In centroid linkage, the distance between two clusters is the distance between the cluster centroids or means. The distance is calculated with the following distance matrix:

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j
Nknumber of variables cluster k
Nlnumber of variables in cluster l
Nmnumber of variables in cluster m

## Complete

With the complete linkage method (also called furthest neighbor method), the distance between two clusters is the maximum distance between a variable in one cluster and a variable in the other cluster. The complete distance is calculated with the following distance matrix:

dmj = max (dkj, dlj)

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j

## McQuitty

With McQuitty's linkage method, the distance is calculated with the following distance matrix:

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j

## Median

In median linkage, the distance between two clusters is the median distance between a variable in one cluster and a variable in the other cluster. The median distance is calculated with the following distance matrix:

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dlj distance between clusters l and j
dkldistance between clusters k and l

## Single

With the single linkage method (also called nearest neighbor method), the distance between two clusters is the minimum distance between a variable in one cluster and a variable in the other cluster.

The distance is calculated with the following distance matrix:

dmj = min (dkj, dlj)

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j

## Ward

In Ward's linkage, the distance between two clusters is the sum of squared deviations from points to centroids. The objective of Ward's linkage is to minimize the within-cluster sum of squares. The distance is calculated with the following distance matrix:

###### Note

In Ward's linkage, the distance between two clusters can be larger than d(max), the maximum value in the original distance matrix, D. If this happens, the similarity will be negative.

### Notation

TermDescription
dmjdistance between clusters m and j
mmerged cluster that consists of clusters k and l, with m = (k,i)
dkjdistance between clusters k and j
dljdistance between clusters l and j
dkldistance between clusters k and l
Njnumber of variables in cluster j
Nknumber of variables in cluster k
Nlnumber of variables in cluster l
Nmnumber of variables in cluster m
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