Distance Metrics

Synonyms

Euclidean Distance; Manhattan Distance

Definition

The Euclidean distance is the direct measure between two points in some spatial space. These points can be represented in any n-dimensional space. Formally, the Euclidean distance can be mathematically expressed as:

$$\displaystyle\begin{array}{rcl} \sqrt{ (a_{1} - b_{1})^{2} + (a_{2} - b_{3})^{2} +\ldots +(a_{n} - b_{n})}& &{}\end{array}$$

(1)

where a and b are two points in some spatial space and n is the dimension.

The Manhattan distance can be mathematically described as:

$$\displaystyle\begin{array}{rcl} \left \vert x_{1} - x2\right \vert + \left \vert y1 - y2\right \vert & &{}\end{array}$$

(2)

where A and B are the following points (x₁, y₁) and (x₂, y₂), respectively. Notice that it does not matter which order the difference is taken from because of the absolute value condition.

Main Text

The Euclidean distance can be measured at a various number of dimensions. For dimensions above three, other feature sets corresponding...