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

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:

.

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 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

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

On the positive real numbers, the metric (distance function) can be defined. The harmonic mean is the corresponding Fréchet mean.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

Power means

Given a non-zero real number , the power mean can be obtained as a Fréchet mean by introducing the metric{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

.

f-mean

Given an invertible function , the f-mean can be defined as the Fréchet mean obtained by using the metric .{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 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. 1.0 1.1 {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  2. Template:Harvtxt, p. 136.