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Fréchet mean

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Fréchet mean

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.

Contents

  • Definition 1
  • Examples of Fréchet Means 2
    • Arithmetic mean and median 2.1
    • Geometric mean 2.2
    • Harmonic mean 2.3
    • Power means 2.4
    • f-mean 2.5
    • Weighted means 2.6
  • References 3

Definition

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

Arithmetic mean and median

For real numbers, the arithmetic mean is a Fréchet mean, using as distance function the usual Euclidean distance. The median is also a Fréchet mean, using the square root of the distance.[2]

Geometric mean

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

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.

Power means

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

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

f-mean

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)|. This is sometimes called the Generalised f-mean or Quasi-arithmetic mean.

Weighted means

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

  1. ^ a b Nielsen, Frank; Bhatia, Rajendra (2012), Matrix Information Geometry, Springer, p. 171,  .
  2. ^ Nielsen & Bhatia (2012), p. 136.
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