Fréchet Mean
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the Fréchet mean is a generalization of
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
s to
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s, giving a single representative point or
central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
for a cluster of points. It is named after
Maurice Fréchet Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean * Maurice, Iowa, a city * Maurice, Louisiana, a village * Maurice River, a t ...
. Karcher mean is the renaming of the Riemannian
Center of Mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
construction developed by Karsten Grove and Hermann Karcher... On the real numbers, the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
,
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
,
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
, and
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
can all be interpreted as Fréchet means for different distance functions.


Definition

Let (''M'', ''d'') be a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. Let ''x''1, ''x''2, …, ''x''''N'' 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 ''x''''i'': :\Psi(p) = \sum_^N d^2(p, x_i) The Karcher means are then those points, ''m'' of ''M'', which minimise Ψ: :m = \mathop_ \sum_^N d^2(p, x_i) If there is a unique ''m'' of ''M'' that strictly minimises Ψ, then it is Fréchet mean.


Examples of Fréchet means


Arithmetic mean and median

For real numbers, the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
is a Fréchet mean, using the usual
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
as the distance function. The
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
is also a Fréchet mean, if the definition of the function Ψ is generalized to the non-quadratic :\Psi(p) = \sum_^N d^\alpha(p, x_i), where \alpha=1, and the Euclidean distance is the distance function ''d''.
p. 136
In higher-dimensional spaces, this becomes the
geometric median In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances or absolute ...
.


Geometric mean

On the positive real numbers, the (hyperbolic) distance function d(x,y)= , \log(x) - \log(y) , can be defined. The
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
is the corresponding Fréchet mean. Indeed 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 x_i is the image by f of the Fréchet mean (in the Euclidean sense) of the f^(x_i), i.e. it must be: : f\left( \frac\sum_^n f^(x_i)\right) = \exp \left( \frac \sum_^n\log x_i \right) = \sqrt /math>.


Harmonic mean

On the
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
, the
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
(distance function): : d_\operatorname(x,y) = \left, \frac - \frac \ can be defined. The
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
is the corresponding Fréchet mean.


Power means

Given a non-zero real number m, the power mean can be obtained as a Fréchet mean by introducing the metric : d_m(x, y) = \left, x^m - y^m \


f-mean

Given an invertible and continuous function f, the f-mean can be defined as the Fréchet mean obtained by using the metric: : d_f(x,y) = \left, f(x) - f(y)\ This is sometimes called the
generalised f-mean In mathematics and statistics, the quasi-arithmetic mean or generalised ''f''-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is a ...
or
quasi-arithmetic mean In mathematics and statistics, the quasi-arithmetic mean or generalised ''f''-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is a ...
.


Weighted means

The general definition of the Fréchet mean (which includes the possibility of weighting observations) can be used to derive weighted versions for all of the above types of means. For the arithmetic mean, the ''x''''i'' are assigned weights ''w''''i''. Then, the Fréchet variances and the Fréchet mean are defined as: :\Psi(p) = \sum_^N w_i\, d^2(p, x_i), \;\;\;\; m = \mathop_ \sum_^N w_i d^2(p, x_i).


See also

* Circular mean *
Fréchet distance In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. Intuitive definition Imagine a person traversing ...
*
M-estimator In statistics, M-estimators are a broad class of extremum estimators for which the objective function is a sample average. Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-estim ...
*
Geometric median In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances or absolute ...


References

{{DEFAULTSORT:Frechet Mean Means