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:

$$ \sqrt{(a_1 - b_1)^2 + (a_2 - b_3)^2 + \dots + (a_n - b_n)} $$

(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:

$$ \left|x_1 - x2\right| + \left|y1 - y2\right| $$

(2)

where A and B are the following points $(x_1,y_1)$ and $(x_2,y_2)$, 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 to each point could be added as more dimensions within a data set. Thus, there can be an infinite number of...