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Distance metrics in Machine Learnjng

1. 1 Amity Institute of Information Technology Ist Semester Machine Learning with Python Dr Neha Prerna Tigga

2. Minkowski Distance 2 Minkowski Distance is a generalization of several distance metrics in a geometric space. It is a metric used to measure the distance between two points in a normed vector space. The Minkowski distance between two points 𝑃=( , ,…, ) and 𝑄=( , ,…, ) in an n-dimensional space is defined as: Where: and are the coordinates of points 𝑃 and 𝑄, 𝑝 is a parameter that defines the type of distance.  

3. Types of Minkowski Distance Based on 𝑝 When 𝑝=1 : This becomes the Manhattan Distance (or L1-Norm), also known as Taxicab Distance. It measures the distance between points along the grid lines (like a taxi driving along city blocks). 3

4. When 𝑝=2 : This becomes the Euclidean Distance (or L2-Norm), the most common form of distance. It measures the straight-line distance between two points in Euclidean space. 4

5. When p→∞: This becomes the Chebyshev Distance (or L∞-Norm ). It measures the greatest difference between any single coordinate dimension of the two points. 5

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7. EXAMPLES Consider two points in a 2D space: Point P: (3, 5) Point Q: (1, 1) Calculate the Manhattan Distance , Euclidean Distance , and Chebyshev Distance between these two points. Points in 5D Space Point P: (2, 3, 7, 1, 5) Point Q: (4, 1, 6, 3, 2) 7