The Minkowski distance is a metric used to define the distance between two points in normed vector space. The beauty of this distance measure lies in its generalization of other distances such as the Euclidean distance (commonly used in space geometry) and the Manhattan distance (used in urban layouts).
Formula of Minkowski Distance Calculator
The formula for Minkowski distance is succinctly given by:
where:
- D(x, y) represents the Minkowski distance between points x and y.
- Σ (sigma) denotes summation across all dimensions (i = 1 to n).
- x and y are n-dimensional vectors.
- xi and yi are the corresponding elements in the ith dimension of vectors x and y, respectively.
- p is a parameter that determines the type of Minkowski distance:
- p = 1: Manhattan distance (sum of absolute differences)
- p = 2: Euclidean distance (square root of sum of squared differences)
- Other values of p (between 1 and infinity) are also possible
Practical Use-Case Table for Minkowski Distance
| pp-Value | Name of Distance | Formula | Common Applications |
|---|---|---|---|
| 1 | Manhattan Distance | ( D(x, y) = \sum_{i=1}^n | x_i – y_i |
| ∞ | Chebyshev Distance | ( D(x, y) = \max_i( | x_i – y_i |
| 0.5 | Not a proper metric | ( D(x, y) = (\sum_{i=1}^n | x_i- y_i |
Example of Minkowski Distance Calculator
Consider two points X=(7,−4,3) and Y=(17,6,2) in a 3-dimensional space, and we wish to find both the Manhattan and Euclidean distances. Using the Minkowski distance formula:
- For p=1 (Manhattan distance):
D(X, Y) = |7-17| + |-4-6| + |3-2| = 10 + 10 + 1 = 21
- For p=2 (Euclidean distance):
D(X, Y) = sqrt((7-17)² + (-4-6)² + (3-2)²) = sqrt(100 + 100 + 1) = sqrt(201) ≈ 14.177
Most Common FAQs
It is widely used in machine learning for clustering algorithms, where the choice of distance metric can significantly impact the clusters formed. It’s also essential in operations research and when performing logistical optimizations.
While Euclidean distance measures the shortest path (as the crow flies) between points, Manhattan distance measures the path between points navigated along axes at right angles. The Minkowski distance generalizes these by introducing the parameter p, which adjusts the calculation based on the path’s dimensions and geometry considered.
Theoretically, pp can be any positive value, but typically pp values of 1, 2, and ∞ are most useful, corresponding to Manhattan, Euclidean, and Chebyshev distances, respectively. Values of p less than 1 are not valid as they violate the triangle inequality, a fundamental property of distance metrics.