In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.
Let (M, d) be a complete metric space. Let x1, x2, …, xN be points in M. For any point p in M, define the Fréchet variance to be the sum of squared distances from p to the xi:
If there is an m of M that globally minimises Ψ, then it is Fréchet mean.
Sometimes, the xi are assigned weights wi. Then, the Fréchet variance is calculated as a weighted sum,
Examples of Fréchet means
Arithmetic mean and median
For real numbers, the arithmetic mean is a Fréchet mean, using the usual Euclidean distance as the distance function.
The median is also a Fréchet mean, if the definition of the function Ψ is generalized to the non-quadratic
On the positive real numbers, the (hyperbolic) distance function can be defined. The geometric mean is the corresponding Fréchet mean. Indeed is then an isometry from the euclidean space to this "hyperbolic" space and must respect the Fréchet mean: the Fréchet mean of the is the image by of the Fréchet mean (in the Euclidean sense) of the , i.e. it must be:
Given an invertible and continuous function , the f-mean can be defined as the Fréchet mean obtained by using the metric:
The general definition of the Fréchet mean that includes the possibility of weighting observations can be used to derive weighted versions for all of the above types of means.
- Grove, Karsten; Karcher, Hermann (1973), "How to conjugate C1-close group actions, Math.Z. 132", Mathematische Zeitschrift, 132 (1): 11–20, doi:10.1007/BF01214029.
- Nielsen, Frank; Bhatia, Rajendra (2012), Matrix Information Geometry, Springer, p. 171, ISBN 9783642302329.
- Nielsen & Bhatia (2012), p. 136.