symmetric.KL {flowMatch} | R Documentation

## Symmetrized Kullback-Leibler divergence

### Description

Compute the Symmetrized Kullback-Leibler divergence between a pair of normally distributed clusters.

### Usage

    symmetric.KL(mean1, mean2, cov1, cov2)

### Arguments

`mean1` | mean vector of length `p` for cluster 1, where `p` is the dimension of the clusters.
`mean2` | mean vector of length `p` for cluster 2.
`cov1` | `pxp` covariance matrix for cluster 1.
`cov2` | `pxp` covariance matrix for cluster 2.

### Details

Consider two `p`-dimensional, normally distributed clusters with centers μ1, μ2 and covariance matrices Σ1, Σ2. We compute the KL divergence `d12` between the clusters as follows:

d12 = 1/4 * ( t(μ2 - μ1) * ( Σ1^(-1) + Σ2^(-1) ) * (μ2 - μ1) + trace(Σ1/Σ2 + Σ2/Σ1) + 2p )

The dimension of the clusters must be same.

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.

### Value

`symmetric.KL` returns a numeric value measuring the Symmetrized Kullback-Leibler divergence between a pair of normally distributed clusters.

### Author(s)

Ariful Azad

### References

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`, `dist.cluster`

### Examples

    ## ------------------------------------------------
    ## 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)
    mean1 = get.center(get.clusters(clustSample)[[1]])
    mean2 = get.center(get.clusters(clustSample)[[2]])
    cov1 = get.cov(get.clusters(clustSample)[[1]])
    cov2 = get.cov(get.clusters(clustSample)[[2]])
    n1 = get.size(get.clusters(clustSample)[[1]])
    n2 = get.size(get.clusters(clustSample)[[2]])
    symmetric.KL(mean1, mean2, cov1, cov2)

[Package flowMatch version 1.22.0 Index]