# Dissimilarity between a pair of clusters

### Description

Calculate the dissimilarity between a pair of cell populations (clusters) from the distributions of the clusters.

### Usage

1 | ```
dist.cluster(cluster1,cluster2, dist.type = 'Mahalanobis')
``` |

### Arguments

`cluster1 ` |
an object of class |

`cluster2 ` |
an object of class |

`dist.type` |
character, indicating the method with which the dissimilarity between a pair of clusters is computed. Supported dissimilarity measures are: 'Mahalanobis', 'KL' and 'Euclidean'. |

### Details

Consider two `p`

-dimensional, normally distributed clusters with centers *μ1, μ2* and covariance matrices *Σ1, Σ2*. Assume the size of the clusters are `n1`

and `n2`

respectively. We compute the dissimilarity `d12`

between the clusters as follows:

If dist.type='Mahalanobis': we compute the dissimilarity

`d12`

with the Mahalanobis distance between the distributions of the clusters.*Σ = ( (n1-1) * Σ1 + (n2-1) * Σ2) / (n1+n2-2)**d12 = sqrt( t(μ1-μ2) * Σ^(-1) * (μ1-μ2))*If dist.type='KL': we compute the dissimilarity

`d12`

with the Symmetrized Kullback-Leibler divergence between the distributions of the clusters. Note that KL-divergence is not symmetric in its original form. We converted it symmetric by averaging both way KL divergence. The symmetrized KL-divergence is not a metric because it does not satisfy triangle inequality.*d12 = 1/4 * ( t(μ2 - μ1) * ( Σ1^(-1) + Σ2^(-1) ) * (μ2 - μ1) + trace(Σ1/Σ2 + Σ2/Σ1) + 2p )*If dist.type='Euclidean': we compute the dissimilarity

`d12`

with the Euclidean distance between the centers of the clusters.*d12 =sqrt(∑(μ1-μ2)^2 )*

The dimension of the clusters must be same.

### Value

`dist.cluster`

returns a numeric value denoting the dissimilarities between a pair of cell populations (clusters).

### Author(s)

Ariful Azad

### References

McLachlan, GJ (1999) Mahalanobis distance; Journal of Resonance 4(6), 20–26.

Abou–Moustafa, Karim T and De La Torre, Fernando and Ferrie, Frank P (2010) Designing a Metric for the Difference between Gaussian Densities; Brain, Body and Machine, 57–70.

### See Also

`mahalanobis.dist, symmetric.KL, dist.matrix`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
## ------------------------------------------------
## load data and retrieve a sample
## ------------------------------------------------
library(healthyFlowData)
data(hd)
sample = exprs(hd.flowSet[[1]])
## ------------------------------------------------
## cluster sample using kmeans algorithm
## ------------------------------------------------
km = kmeans(sample, centers=4, nstart=20)
cluster.labels = km$cluster
## ------------------------------------------------
## Create ClusteredSample object
## and compute mahalanobis distance between two clsuters
## ------------------------------------------------
clustSample = ClusteredSample(labels=cluster.labels, sample=sample)
clust1 = get.clusters(clustSample)[[1]]
clust2 = get.clusters(clustSample)[[2]]
dist.cluster(clust1, clust2, dist.type='Mahalanobis')
dist.cluster(clust1, clust2, dist.type='KL')
dist.cluster(clust1, clust2, dist.type='Euclidean')
``` |

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