Let (M, d) be a complete metric space. 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 locally minimise Ψ:[2]
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,
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, if the definition of the function Ψ is generalized to the non-quadratic
where , 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 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:
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.
fréchet, mean, mathematics, statistics, generalization, centroids, metric, spaces, giving, single, representative, point, central, tendency, cluster, points, named, after, maurice, fréchet, karcher, mean, renaming, riemannian, center, mass, construction, devel. In mathematics and statistics the Frechet 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 Frechet Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher 1 2 On the real numbers the arithmetic mean median geometric mean and harmonic mean can all be interpreted as Frechet means for different distance functions Contents 1 Definition 2 Examples of Frechet means 2 1 Arithmetic mean and median 2 2 Geometric mean 2 3 Harmonic mean 2 4 Power means 2 5 f mean 2 6 Weighted means 3 See also 4 ReferencesDefinition EditLet M d be a complete metric space Let x1 x2 xN be points in M For any point p in M define the Frechet variance to be the sum of squared distances from p to the xi PS p i 1 N d 2 p x i displaystyle Psi p sum i 1 N d 2 left p x i right The Karcher means are then those points m of M which locally minimise PS 2 m arg min p M i 1 N d 2 p x i displaystyle m mathop text arg min p in M sum i 1 N d 2 left p x i right If there is an m of M that globally minimises PS then it is Frechet mean Sometimes the xi are assigned weights wi Then the Frechet variance is calculated as a weighted sum PS p i 1 N w i d 2 p x i m arg min p M i 1 N w i d 2 p x i displaystyle Psi p sum i 1 N w i d 2 left p x i right m mathop text arg min p in M sum i 1 N w i d 2 left p x i right Examples of Frechet means EditArithmetic mean and median Edit For real numbers the arithmetic mean is a Frechet mean using the usual Euclidean distance as the distance function The median is also a Frechet mean if the definition of the function PS is generalized to the non quadratic PS p i 1 N d a p x i displaystyle Psi p sum i 1 N d alpha left p x i right where a 1 displaystyle alpha 1 and the Euclidean distance is the distance function d 3 In higher dimensional spaces this becomes the geometric median Geometric mean Edit 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 Frechet mean Indeed f x e x displaystyle f x mapsto e x is then an isometry from the euclidean space to this hyperbolic space and must respect the Frechet mean the Frechet mean of the x i displaystyle x i is the image by f displaystyle f of the Frechet mean in the Euclidean sense of the f 1 x i displaystyle f 1 x i i e it must be f 1 n i 1 n f 1 x i exp 1 n i 1 n log x i x 1 x n n displaystyle f left frac 1 n sum i 1 n f 1 left x i right right exp left frac 1 n sum i 1 n log x i right sqrt n x 1 cdots x n Harmonic mean Edit On the positive real numbers the metric distance function d H x y 1 x 1 y displaystyle d operatorname H x y left frac 1 x frac 1 y right can be defined The harmonic mean is the corresponding Frechet mean citation needed Power means Edit Given a non zero real number m displaystyle m the power mean can be obtained as a Frechet mean by introducing the metric citation needed d m x y x m y m displaystyle d m left x y right left x m y m right f mean Edit Given an invertible and continuous function f displaystyle f the f mean can be defined as the Frechet mean obtained by using the metric citation needed d f x y f x f y 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 Edit The general definition of the Frechet mean that includes the possibility of weighting observations can be used to derive weighted versions for all of the above types of means See also EditCircular mean Frechet distance M estimator Geometric medianReferences Edit Grove Karsten Karcher Hermann 1973 How to conjugate C1 close group actions Math Z 132 Mathematische Zeitschrift 132 1 11 20 doi 10 1007 BF01214029 a b Nielsen Frank Bhatia Rajendra 2012 Matrix Information Geometry Springer p 171 ISBN 9783642302329 Nielsen amp Bhatia 2012 p 136 Retrieved from https en wikipedia org w index php title Frechet mean amp oldid 1129711017, wikipedia, wiki, book, books, library,