Distance metrics

L-p metrics

Manhattan distance (L1)

Given two vectors $p$ and $q$, such that

0

we define the Manhattan distance as:

1

Euclidean distance (L2)

In general, for points given by Cartesian coordinates in $n$-dimensional Euclidean space, the distance is

2

Cluster distances

Within-cluster sum of squares (WCSS)

Given a set of observations ($x_1, x_2,\dots,x_n$), where each observation is a $d$-dimensional real vector, $k$-means clustering aims to partition the $n$ observations into $k$ ($\leq n$) sets $S=\lbrace S_1, S_2, \dots, S_k\rbrace$ so as to minimize the within-cluster sum of squares (WCSS) (i.e. variance). Formally, the objective is to find:

3

where $\mu_i$ is the mean of points in $S_i$. This is equivalent to minimizing the pairwise squared deviations of points in the same cluster:

4

The equivalence can be deduced from identity

5

Because the total variance is constant, this is equivalent to maximizing the sum of squared deviations between points in different clusters (between-cluster sum of squares, BCSS) which follows from the law of total variance.

Dunn index

A full explanation is available at Dunn index.

Gower distance

A full explanation with examples is available at Gower distance.