Fréchet mean

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

${\displaystyle \Psi (p)=\sum _{i=1}^{N}d^{\alpha }(p,x_{i}),}$

where ${\displaystyle \alpha =1}$, and the Euclidean distance is the distance function d.^[3] In higher-dimensional spaces, this becomes the geometric median.

Geometric mean

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

${\displaystyle f\left({\frac {1}{n}}\sum _{i=1}^{n}f^{-1}(x_{i})\right)=\exp \left({\frac {1}{n}}\sum _{i=1}^{n}\log x_{i}\right)={\sqrt[{n}]{x_{1}\cdots x_{n}}}}$.

Harmonic mean

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

${\displaystyle d_{\operatorname {H} }(x,y)=\left|{\frac {1}{x}}-{\frac {1}{y}}\right|}$

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

Power means

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

${\displaystyle d_{m}(x,y)=\left|x^{m}-y^{m}\right|}$

f-mean

Given an invertible and continuous function ${\displaystyle f}$, the f-mean can be defined as the Fréchet mean obtained by using the metric:^{[citation needed]}

${\displaystyle d_{f}(x,y)=\left|f(x)-f(y)\right|}$

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.