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

Visual Representation of the Difference Between Manhattan and Euclidean Distances

In order to find the Euclidean distance, one has to take the square root of the sum of the squared difference between two points. The formula for calculating the Euclidean distance is given by:

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


University of Michigan - Ann Arbor

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 power parameter p controls which distance metric is used in determining neighbors of a point. The default is p=2, which specifies Euclidean distance. Setting p=1 specifies Manhattan distance (aka city block distance), which calculates distance along a grid-like path. Manhattan distance is preferred for high-dimensional data.

"p" Parameter for KNN Classifier

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

The Manhattan distance is found by using the absolute value of the sum of the difference between two points. Its formula is given by:

		$\sum_{i = 1}^k {|x_i-y_i|}$ 

Manhattan Distance

If you are using categorical variables, then using the Hamming distance will be helpful. If two points are the same, then the difference between them is 0. If they are not the same, the difference between them is 1. Its formula is given by:

		$D_H =  \sum_{i=1}^k {|x_i - y_i|}$
		$if x = y \Rightarrow 0$
		$if x \neq y \Rightarrow 1$

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.

Specifying a distance metric for k-Nearest Neighbors algorithm

https://medium.com/@kunal_gohrani/different-types-of-distance-metrics-used-in-machine-learning-e9928c5e26c7#:~:text=Manhattan%20distance%20is%20usually%20preferred,similarity%20between%20two%20data%20points.

Different Types of Distance Metrics used in Machine Learning

Manhattan distance, aka city block distance, is the number of units one must traverse to get from point X to point Y along a grid-like path. In other words, each step involves a change in value across exactly one dimension. In machine learning, Manhattan distance is preferred (over higher values of p) for high-dimensionality data because of simpler computation and because of something called the "curse of high dimensionality."

More formally, let $x_i$ be the value of point X across the $i$th dimension, and let $y_i$ be the value of point Y across the $i$th dimension (NOT to be confused with the conventional (x,y) notation for a point in 2D). Then, in $n$ dimensions, the Manhattan distance between points X and Y is
$d(X,Y) = \sum_{i=1}^n|x_i - y_i|$.

(Note that this is an instantiation of the Minkowski distance formula with p=1.)

Manhattan Distance (p=1)

https://bib.dbvis.de/uploadedFiles/155.pdf

On the Surprising Behavior of Distance Metrics
in High Dimensional Space

The curse of high dimensionality is the empirical phenomenon that in high dimensional space, concepts of proximity, distance, and/or nearest neighbor lose qualitative meaning. This problem is more severe for Euclidean distance than for Manhattan distance, hence why Manhattan distance is preferred for high dimensional data.

Learn Before

Related

Learn After