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.
Best Answer
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