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.
Let x1, x2, …, xN be 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 minimise Ψ:[2] If there is a unique m of M that strictly minimises Ψ, then it is Fréchet mean.
Then, the Fréchet variances and the Fréchet mean are defined using weighted sums: 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 where
On the positive real numbers, the (hyperbolic) distance function
, i.e. it must be: On the positive real numbers, the metric (distance function): can be defined.
[citation needed] Given a non-zero real number
, the power mean can be obtained as a Fréchet mean by introducing the metric[citation needed] Given an invertible and continuous 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.
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.