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 mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions. 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 mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions. Let (M, d) be a complete metric space. Let x1, x2, …, xN be random 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 locally minimise Ψ: If there is an m of M that globally minimises Ψ, then it is Fréchet mean. Sometimes, the xi are assigned weights wi. Then, the Fréchet variance is calculated as a weighted sum, 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. On the positive real numbers, the (hyperbolic) distance function d ( x , y ) = | log ⁡ ( x ) − log ⁡ ( y ) | {displaystyle d(x,y)=|log(x)-log(y)|} can be defined. The geometric mean is the corresponding Fréchet mean. Indeed f : x ↦ e x {displaystyle f:xmapsto 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 {displaystyle x_{i}} is the image by f {displaystyle f} of the Fréchet mean (in the Euclidean sense) of the f − 1 ( x i ) {displaystyle f^{-1}(x_{i})} , i.e. it must be: On the positive real numbers, the metric (distance function): can be defined. The harmonic mean is the corresponding Fréchet mean.