✍🏻Math for the Data Scientist: another 3 Distance Measures, Part 2
• Manhattan Distance, also called a taxi or city block measure, calculates the distance between vectors with real values. Then Manhattan distance refers to the distance between two vectors on a uniform grid if they can only move at right angles. No diagonal movement is used when calculating the distance. While Manhattan distance seems to be acceptable for multidimensional data, it is a less intuitive measure than Euclidean distance. A measure is more likely to give a higher distance value than Euclidean distance, since it is not the shortest possible distance. However, if the dataset has discrete and / or binary attributes, the Manhattan distance works well because it takes into account real paths within the possible values.
• Chebyshev distance is defined as the greatest difference between two vectors in any coordinate dimension, it is simply the maximum distance along one axis. This measure is also often called the distance of the chessboard, since the minimum number of moves required for the king to move from one square to another is equal to the distance of Chebyshev. This distance is usually used in very specific use cases, making it difficult to use it as a universal measure of distance, as opposed to Euclidean distance or cosine similarity. Therefore, the Chebyshev distance is only recommended in certain cases. For example, to determine the minimum number of moves in games that allow unlimited 8-sided movement. Also, the Chebyshev distance is often used in warehouse logistics, for example, to determine the time required for an overhead crane to move an object.
• Minkowski distance is a more complex measure used in normalized vector space (n-dimensional real space), where distances can be represented as a vector with length. When using this measure, there is a zero vector that has zero length and all others are positive, the vector can be multiplied by a number (scalar coefficient), and the shortest distance between two points is a straight line. You can also use the p parameter here to control distance metrics similar to other measures, for example, p = 1 for Manhattan distance, p = 2 for Euclidean, and p = ∞ for Chebyshev distance. Therefore, in order to work with the Minkowski distance, you need to understand the purpose, advantages and disadvantages of the Manhattan, Euclidean and Chebyshev measures. Finding the correct value for p can be computationally inefficient, gives flexibility in the distance metric, and can be a huge advantage if correctly selected.