- Green Line: Manhattan Distance
- Red Line: Euclidean Distance

Visual Representation of the Difference Between Manhattan and Euclidean Distances

The Manhattan distance between two points is the sum of the absolute differences of their corresponding Cartesian coordinates. For two points $$x$$ and $$y$$ in a $$k$$-dimensional space, the distance is calculated as: $$d(x, y) = \sum_{i=1}^k |x_i - y_i|$$

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Often times, statistical methods rely on measuring the distance between quantitative data points. However, there are various ways to measure this distance. These include:

1. Euclidean distance 
2. Manhattan distance
3. Hamming Distance

Measuring Distance Between Quantitative Data Points

The reference below has a good reference to KNN, as well as different ways in which distance is measured.

Brownlee, J. (2019, August 12). K-Nearest Neighbors for Machine Learning. Retrieved March 23, 2020, from [machinelearningmastery.com/k-nearest-neighbors-for-machine-learning/](https://machinelearningmastery.com/k-nearest-neighbors-for-machine-learning/) 

K-Nearest Neighbors for Machine Learning (Reference)

Euclidean distance is the straight-line distance between two points in a Euclidean space. For two points $$x$$ and $$y$$ in a $$k$$-dimensional space, the distance is calculated as the square root of the sum of the squared differences between their coordinates:

$$d(x, y) = \\sqrt{ \\sum_{i=1}^k (x_i - y_i)^2 }$$

Euclidean Distance

Manhattan Distance

The Hamming distance is a metric useful for comparing categorical variables. It measures distance by counting the number of positions at which corresponding elements differ. For each position, if the two elements are the same, the difference is 0; if they are different, the difference is 1. Its formula is given by: $$D_H = \sum_{i=1}^k |x_i - y_i|$$ where $$|x_i - y_i| = 0$$ if $$x_i = y_i$$, and $$|x_i - y_i| = 1$$ if $$x_i \neq y_i$$.

Hamming Distance

We need to specify a distance metric in our future space, so we know how to properly select the nearby neighbors. In the fruit classification example, we use the simple straight line, or Euclidean distance to measure the distance between points.

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