Understanding Euclidean Distance: From Theory to Practice

R example

R provides several ways to calculate Euclidean distance. We'll create a custom function and compare it with the dist() function from the stats library.

euclidean_distance <- function(point1, point2) {
  sqrt(sum((point1 - point2)^2))
}

# 2D example
point_a <- c(1, 2)
point_b <- c(4, 6)
distance_2d <- euclidean_distance(point_a, point_b)
print(paste("2D Euclidean distance:", round(distance_2d, 2)))

# 3D example
point_c <- c(1, 2, 3)
point_d <- c(4, 6, 8)
distance_3d <- euclidean_distance(point_c, point_d)
print(paste("3D Euclidean distance:", round(distance_3d, 2)))

# Using the dist() function from stats
distance_builtin <- stats::dist(rbind(point_c, point_d), method = "euclidean")
print(paste("3D Euclidean distance (built-in):", round(as.numeric(distance_builtin), 2)))

Running this R code should produce output like:

[1] "2D Euclidean distance: 5"
[1] "3D Euclidean distance: 7.07"
[1] "3D Euclidean distance (built-in): 7.07"

Our custom euclidean_distance function uses R's vectorized operations, making it concise and efficient. The dist() function from stats returns the same result, validating our custom function. Both methods return a matrix.

Applications of Euclidean Distance

Euclidean distance is a straightforward way to measure how far apart things are. It’s used in various areas to handle problems involving space and distance.

K-nearest neighbors (KNN)

In the k-nearest neighbors algorithm, Euclidean distance helps find the nearest neighbors to a point. This helps decide how to classify new data—like deciding if an email is spam based on what similar emails look like, or recommending products that are similar to what a customer already likes.

K-means clustering

In k-means clustering, Euclidean distance helps sort data points into groups by connecting each point to the nearest center of a cluster. This helps in organizing data into categories that share similarities, useful in customer segmentation or during research to group similar subjects together.

Multidimensional scaling (MDS)

Multidimensional scaling uses Euclidean distance to simplify complex data into something easier to visualize and understand. It takes data that usually has many details (dimensions) and reduces it so that it’s simpler to look at and analyze, helping to spot trends and patterns more clearly.

Image processing

In tasks like detecting edges in images or recognizing objects, Euclidean distance measures how much pixel colors change, which helps in outlining objects or identifying important features in an image. This is helpful in things like medical imaging to identify diseases, or in security systems to recognize faces or objects.

Robotics

For robots, like drones or self-driving cars, Euclidean distance helps calculate the simplest route from one point to another. This helps robots and other automated systems move efficiently and safely, avoiding obstacles and calculating the easiest paths to their destinations.

Comparison with Other Distance Metrics

Euclidean distance is one of many ways to measure how far apart points are, but different situations call for different methods. Here’s how it stacks up against other common distance metrics:

Manhattan distance

Also known as "city block" distance, Manhattan distance measures the total sum of the absolute differences along each dimension. Imagine you're walking through a city's grid-like streets; the distance you'd travel block by block is your Manhattan distance. This method is particularly handy in environments that mimic a grid, like navigating city streets or in some types of games. It's also useful when you’re dealing with very high-dimensional data, where Euclidean distance can become less reliable. To explore more, check out our tutorial on Manhattan distance.

Cosine distance

Cosine distance looks at the angle between two points or vectors. Instead of focusing on how long the line is between them, it considers how they are oriented relative to each other. This makes it especially useful in fields like text analysis or recommendation systems, where the direction of the data (like word counts in articles or user preferences) matters more than the magnitude (how much). For a deeper understanding, see our article on cosine distance.

Chebyshev distance

Chebyshev distance is another way to measure distance, focusing on the greatest difference along any dimension. It’s like playing chess: the king moves to the square that is the farthest away in the minimum number of moves, irrespective of whether it's horizontal, vertical, or diagonal. This metric is particularly useful in scenarios where you need to consider only the most significant of multiple differences. Learn more by reading our tutorial on Chebyshev distance.

Limitations of Euclidean Distance

While Euclidean distance is widely used due to its intuitive nature and straightforward calculation, it does have some notable limitations. Understanding these can help in choosing the right distance measure or in adjusting the data to mitigate these issues.

Scale sensitivity

Euclidean distance can be disproportionately affected by the scale of the features. For instance, in a dataset containing income and age, income typically spans a much larger range (perhaps thousands or tens of thousands) compared to age (usually ranging just up to about 100). This disparity can lead to income overwhelming the distance calculation, skewing results towards its scale.

Mitigation: Normalizing or standardizing the data can help balance the scales of different features, ensuring that no single feature unduly influences the distance calculation.

Curse of dimensionality

The curse of dimensionality refers to various phenomena that emerge as the number of dimensions in a dataset increases. One of these is that the concept of 'proximity' or 'distance' becomes less meaningful—distances tend to converge, making it hard to distinguish between near and far points effectively.

Mitigation: Techniques like principal component analysis (PCA) or t-distributed stochastic neighbor embedding (t-SNE) reduce the dimensionality of the data. These methods help by distilling the essential features with the most variance, which can help to simplify the data.

Outlier sensitivity

Euclidean distance calculations can also be heavily influenced by outliers. In high-dimensional spaces, a single outlier can drastically alter distances, making some data points appear far more similar or different than they truly are. 

In fact, the sensitivity of linear regression to outliers is related to the concept of Euclidean distance. This is because, in linear regression, the model minimizes the sum of squared residuals, which are the differences between observed and predicted values. Doing this essentially measures the Euclidean distance between the observed data points and the regression line. Outliers can disproportionately affect this distance because the square of larger deviations increases more rapidly than smaller ones. 

Mitigation: Using more robust metrics that are less sensitive to outliers can help, such as the Manhattan distance for certain types of data. Additionally, preprocessing data to identify and treat outliers—either by adjusting them or removing them—can prevent them from skewing the distance calculations.

Alternative Approach: Considering weighted Euclidean distance is another strategy. This variation of Euclidean distance assigns different weights to different dimensions or features, potentially down-weighting those prone to outliers or noise.

Conclusion

As we've explored, Euclidean distance is a fundamental metric in many analytical and technological fields, providing a straightforward way to measure the straight-line distance between points. Understanding and utilizing Euclidean distance can enhance the accuracy and effectiveness of many applications, from machine learning algorithms to spatial analysis.

I encourage you to experiment with Euclidean distance in your projects and explore further learning opportunities through courses like Designing Machine Learning Workflows in Python and the Anomaly Detection in Python Course.