Fréchet mean

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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.[1][2] On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

Definition[edit]

Let (M, d) be a complete metric space. Let x1, x2, …, xN be random 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:

The Karcher means are then those points, m of M, which locally minimise Ψ:[2]

If there is an m of M that globally minimises Ψ, then it is Fréchet mean.

Sometimes, the xi are assigned weights wi, the Fréchet variance is calculated at a weighted sum,

Examples of Fréchet means[edit]

Arithmetic mean and median[edit]

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, using the square root of the distance.[3]

Geometric mean[edit]

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:

.

Harmonic mean[edit]

On the positive real numbers, the metric (distance function):

can be defined. The harmonic mean is the corresponding Fréchet mean.[citation needed]

Power means[edit]

Given a non-zero real number , the power mean can be obtained as a Fréchet mean by introducing the metric[citation needed]

f-mean[edit]

Given an invertible function , the f-mean can be defined as the Fréchet mean obtained by using the metric:[citation needed]

This is sometimes called the generalised f-mean or quasi-arithmetic mean.

Weighted means[edit]

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.

References[edit]

  1. ^ Grove, Karsten; Karcher, Hermann (1973), How to conjugate C1-close group actions, Math.Z. 132 (PDF), Springer, p. 11-20 .
  2. ^ a b Nielsen, Frank; Bhatia, Rajendra (2012), Matrix Information Geometry, Springer, p. 171, ISBN 9783642302329 .
  3. ^ Nielsen & Bhatia (2012), p. 136.