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 means are a closely related construction named after Hermann Karcher.[1] 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]

The Fréchet mean (), is the point, x, that minimizes the Fréchet function, in cases where such a unique minimizer exists. The value at a point p, of the Fréchet function associated to a random point X on a complete metric space (M, d) is the expected squared distance from p to X. In particular, the Fréchet mean of a set of discrete random points xi is the minimizer m of the weighted sum of squared distances from an arbitrary point to each point of positive probability (weight), assuming this minimizer is unique. In symbols:

m = \mathop{\mathrm{arg\ min}}_{p\in M} \sum_{i=1}^N w_i d^2(p,x_i).

A Karcher mean is a local minimum of the same function.[1]

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.[2]

Geometric mean[edit]

On the positive real numbers, the (hyperbolic) distance function  d(x,y)= | \log(x) - \log(y) | can be defined. The geometric mean is the corresponding Fréchet mean. Indeed  f:x\mapsto e^x is then an isometry from the euclidean space to this "hyperbolic" space and must respect the Fréchet mean: the Fréchet mean of the x_i is the image by f of the Fréchet mean (in the Euclidean sense) of the  f^{-1}(x_i), i.e. it must be:

 f\left( \frac{1}{n}\sum_{i=1}^n f^{-1}(x_i)  \right) = \exp \left( \frac{1}{n}\sum_{i=1}^n\log x_i \right) = \sqrt[n]{x_1 \cdots x_n}.

Harmonic mean[edit]

On the positive real numbers, the metric (distance function)  d_H(x,y) = \left| \frac{1}{x} - \frac{1}{y} \right| can be defined. The harmonic mean is the corresponding Fréchet mean.[citation needed]

Power means[edit]

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

d_m(x,y)=| x^m - y^m |.

f-mean[edit]

Given an invertible function f, the f-mean can be defined as the Fréchet mean obtained by using the metric d_f(x,y)= | f(x)-f(y)|.[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. ^ a b Nielsen, Frank; Bhatia, Rajendra (2012), Matrix Information Geometry, Springer, p. 171, ISBN 9783642302329 .
  2. ^ Nielsen & Bhatia (2012), p. 136.