# Dens_bw: Inter-cluster density - Internal Measure In clv: Cluster Validation Techniques

## Description

Function computes inter-cluster density.

## Usage

 `1` ```clv.DensBw(data, clust, scatt.obj, dist="euclidean") ```

## Arguments

 `data` `matrix` or `data.frame` where columns correspond to variables and rows to observations `clust` integer `vector` with information about cluster id the object is assigned to. If vector is not integer type, it will be coerced with warning. `scatt.obj` object returned by `clv.Scatt` function. `dist` chosen metric: "euclidean" (default value), "manhattan", "correlation"

## Details

The definition of inter-cluster density is given by equation:

`Dens_bw` = 1/(|C|*(|C|-1)) * sum{forall i in 1:|C|} sum{forall j in 1:|C| and j != i} density(u(i,j))/max{density(v(i)), density(v(j))}

where:

 |C| - number of clusters, v(i), v(j) - centers of clusters i and j, u(i,j) - middle point of the line segment defined by the clusters' centers v(i), v(j), density(x) - see below.

Let define function f(x,u):

 f(x,u) = 0 if dist(x,u) > stdev (stdev is defined in `clv.Scatt`) f(x,u) = 1 otherwise

Function f is used in definition of density(u):

density(u) = sum{forall i in 1:n(i,j)} f(xi,u)

where n(i,j) is the number of objects which belongs to clusters i and j and xi is such object.

This value is used by `clv.SDbw`.

## Value

As result `Dens_bw` value is returned.

## Author(s)

Lukasz Nieweglowski

`clv.SD` and `clv.SDbw`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```# load and prepare data library(clv) data(iris) iris.data <- iris[,1:4] # cluster data agnes.mod <- agnes(iris.data) # create cluster tree v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree # compute Dens_bw index scatt <- clv.Scatt(iris.data, v.pred) dens.bw <- clv.DensBw(iris.data, v.pred, scatt) ```

### Example output

```Loading required package: cluster