Briefly about Cluster Analysis

### The concept

Assume a data set defined in two dimensions, as presented below:

Let the horizontal and vertical distribution represent the values along the two dimensions. The relations between data a to f could be reflected according to Euclidean distance measure in one dimensions as shown below,

reflecting the hierarchical clustering between the data in terms of measured distances.
### Distances

Assume the following two data sets (x and y):

The distance between the two datasets could in cluster analysis be calculated on the basis of a number of measuring methods. The most important methods are the following:

#### Euclidean distance

#### Squared Euclidean distance

#### Manhattan distance

#### Chebychev or Chessboard distance

distance(x, y) = max(abs(xi-yi))

The Chessboard distance between the above data sets is

An numerical example:

If a = 1, b = 2, c = 3, d = 4, e = 5, f = 6, then the Chessboard distance is 3.

#### Bray-Curtis distance

The Bray-Curtis distance between the above data sets is