Manhattan distance

Definition

Manhattan distance is a metric used to measure the distance between two points in a grid-like path based on the absolute differences of their coordinates. It is particularly useful in clustering algorithms because it captures the concept of distance in a way that reflects how we would navigate through a city with a rectangular street grid, moving only along axes and never diagonally.

5 Must Know Facts For Your Next Test

1. Manhattan distance is calculated by summing the absolute differences of their Cartesian coordinates: $$d = |x_1 - x_2| + |y_1 - y_2|$$.
2. This metric is particularly effective for high-dimensional data where Euclidean distance might not perform as well due to issues like the curse of dimensionality.
3. In K-means clustering, Manhattan distance can lead to different cluster shapes compared to Euclidean distance, often resulting in more compact clusters.
4. Hierarchical clustering can also utilize Manhattan distance, which helps in forming clusters based on the cumulative distances between points, reflecting real-world navigation.
5. Using Manhattan distance can make algorithms less sensitive to outliers compared to Euclidean distance, which could significantly skew results.

Review Questions

• How does Manhattan distance differ from Euclidean distance, and why might one be preferred over the other in clustering algorithms?
  • Manhattan distance differs from Euclidean distance by calculating the total grid-like travel between points instead of the straight-line distance. In clustering algorithms, Manhattan distance might be preferred when dealing with high-dimensional data or when we want to avoid sensitivity to outliers, as it provides a more robust measure of similarity based on axis-aligned distances.
• Discuss the implications of using Manhattan distance in K-means clustering compared to hierarchical clustering.
  • Using Manhattan distance in K-means clustering typically results in clusters that are more compact and axis-aligned, which can help accurately represent data distribution. In hierarchical clustering, this metric influences how distances between clusters are calculated, impacting the linkage criteria. Therefore, while K-means focuses on centroid-based partitions, hierarchical methods reveal a tree-like structure based on cumulative distances.
• Evaluate how Manhattan distance affects the performance and outcomes of clustering algorithms when applied to datasets with different characteristics.
  • Manhattan distance can significantly impact clustering performance depending on the dataset's characteristics. For example, in datasets where features have varying scales or where data points are not uniformly distributed, using Manhattan distance can help create clearer cluster boundaries. This metric's insensitivity to outliers also allows it to handle datasets with noise better than Euclidean distance. As such, selecting an appropriate distance metric is crucial for achieving meaningful and interpretable clustering results.

Related terms

Euclidean distance:

A measure of the straight-line distance between two points in Euclidean space, calculated using the Pythagorean theorem.

Clustering: The process of grouping a set of objects in such a way that objects in the same group are more similar to each other than to those in other groups.

Distance metric: A function that defines a distance between elements of a set, crucial for various machine learning algorithms including clustering and classification.