Difference between Euclidean, Pearson, Geodesic and Mahalanobis distance metrics

Given a set of samples $$X$$. We are tasked to find an appropriate distance metric for $$X$$ from the given options which are

• Euclidean
• Pearson
• Geodesic and
• Mahalanobis distance metrics.

To solve this, I need an intuition as to what information each distance metric preserves and does not preserve. For example, geodesic distance metric preserves the curvature in the distribution of data, but I am not sure what the other distance metrics do.

Euclidean:

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

Pearson:

Pearson Correlation measures the similarity in shape between two profiles.

Geodesic:

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance.

Wikipedia for Geodesic distance

Mahalonobis:

The Mahalanobis distance is a measure of the distance between a point P and a distribution D. It is a multi-dimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D. This distance is zero if P is at the mean of D, and grows as P moves away from the mean along each principal component axis.

Wikipedia for Mahalonobis

• Could you show how these concepts can be applied to a dataset and how they might be directly related to or compared to one another? Currently, much of this post is not relevant in the context (of a "set of samples") and therefore might be a little misleading about the intuition. For instance, the appropriate sense of "geodesic" lies in the concepts of differential geometry, not graph theory (there is no graph naturally associated with an arbitrary dataset). "Similarities" are not distances, although often they can be related to them. Etc.
– whuber
Commented May 15, 2019 at 15:08