Minkowskietäetäisyys

Minkowskietäisyys, also known as Minkowski distance or L_p distance, is a metric in a vector space. It is a generalization of the Euclidean distance and the Manhattan distance. For two points x and y in an n-dimensional space, the Minkowski distance with parameter p is defined as the p-th root of the sum of the absolute differences of their corresponding coordinates raised to the power of p. Mathematically, it is expressed as d_p(x, y) = (sum_{i=1}^n |x_i - y_i|^p)^(1/p).

The parameter p can be any real number greater than or equal to 1. When p=1, the Minkowski distance is equivalent to the Manhattan distance (also known as L_1 distance or taxicab distance), which is the sum of the absolute differences of the coordinates. When p=2, it simplifies to the familiar Euclidean distance (L_2 distance), which is the shortest straight-line distance between two points. As p approaches infinity, the Minkowski distance converges to the Chebyshev distance (L_infinity distance), which is the maximum absolute difference between any pair of corresponding coordinates.

The Minkowski distance is widely used in various fields, including machine learning, pattern recognition, and data analysis, for tasks such as clustering, classification, and similarity measurement. The choice of the parameter p can significantly influence the results of these applications, depending on the specific characteristics of the data and the desired properties of the distance measure.