Minkowski distance
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Minkowski distance is a distance/ similarity measurement between two points in the normed vector space (N dimensional real space) and is a generalization of the Euclidean distance and the Manhattan distance.
Minkowski distance in N-D space
In a N dimensional space, a point is represented as (x1, x2, ..., xN).
Consider two points P1 and P2:
P1: (X1, X2, ..., XN)
P2: (Y1, Y2, ..., YN)
Then, the Minkowski distance between P1 and P2 is given as:
$$ \sqrt[p]{{(x1-y1)}^p\ +\ {(x2-y2)}^p\ +\ ...\ +\ {(xN-yN)}^p}
$$
Euclidean distance from Minkowski distance
When p = 2, Minkowshi distance is same as Euclidean distance.
Manhattan distance from Minkowski distance
When p = 1, Minkowshi distance is same as Manhattan distance.
Visualize
Unit circles (path represents points with same Minkowshi distance) with various values of p (Minkowski distance):
Applications
Applications of Minkowshi Distance are:
Euclidean distance and Manhattan distance are same as Minkowshi Distance, hence, the applications of the previous two distance metrics are applications of Minkowshi Distance.