Manhattan Distance

Definition

Manhattan distance is a metric used to measure the distance between two points in a grid-like path, defined as the sum of the absolute differences of their Cartesian coordinates. This metric is especially relevant in hierarchical clustering, as it allows for the quantification of how similar or dissimilar objects are based on their coordinates in a multi-dimensional space, influencing how clusters are formed and analyzed.

5 Must Know Facts For Your Next Test

1. Manhattan distance is sometimes referred to as 'taxicab' or 'city block' distance because it represents how a taxi would navigate through a grid of streets.
2. In a two-dimensional space, if point A has coordinates (x1, y1) and point B has coordinates (x2, y2), the Manhattan distance is calculated as |x1 - x2| + |y1 - y2|.
3. This distance metric is particularly useful for high-dimensional data where objects are represented by multiple attributes, allowing for clearer cluster separation.
4. Unlike Euclidean distance, Manhattan distance can be more robust to outliers since it focuses on absolute differences rather than squared differences.
5. In hierarchical clustering, choosing Manhattan distance can lead to different cluster formations compared to other distance metrics, impacting the interpretation of data relationships.

Review Questions

• How does Manhattan distance influence the formation of clusters in hierarchical clustering?
  • Manhattan distance influences cluster formation by providing a unique way to measure similarity based on the grid-like path between points. By summing up the absolute differences in their coordinates, it determines how closely related objects are in terms of their attributes. This metric can lead to different clusters than other methods like Euclidean distance, affecting how groups are identified and interpreted within the dataset.
• Discuss the advantages and disadvantages of using Manhattan distance compared to Euclidean distance in hierarchical clustering.
  • Manhattan distance has advantages such as being less sensitive to outliers and providing a clear measure for high-dimensional data where dimensions may vary widely. However, it may not capture diagonal relationships as effectively as Euclidean distance. The choice between these metrics depends on the nature of the data; for instance, if the data is more aligned along axes, Manhattan might be preferred, whereas Euclidean could be better for more uniformly distributed data.
• Evaluate how selecting different distance metrics like Manhattan or Euclidean can impact the outcomes of hierarchical clustering and interpretations drawn from the dendrograms produced.
  • Selecting different distance metrics significantly affects the resulting clusters formed during hierarchical clustering and can alter the interpretations of dendrograms. For example, using Manhattan distance may produce clusters that reflect distinct groupings based on attribute differences, while Euclidean distance could merge clusters due to its sensitivity to variance among dimensions. This impacts decisions made based on cluster analysis, as different metrics could lead to contrasting insights about the relationships among data points and their overall structure.

Related terms

Euclidean Distance: A metric that calculates the straight-line distance between two points in Euclidean space, often used for measuring similarity in clustering.

Dendrogram: A tree-like diagram that illustrates the arrangement of the clusters formed during hierarchical clustering, showing how they are related to one another.

Clustering Algorithm: A method used to group similar data points together based on certain metrics, such as Manhattan or Euclidean distance, to identify patterns or structures in the data.