In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, 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 surface of the figure. The same definition extends to any ...
s to
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
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. 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
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
,
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
,
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
, and
harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
can all be interpreted as Fréchet means for different distance functions.
Definition
Let (''M'', ''d'') be a complete metric space. 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'':
:
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 ''x''
''i'' are assigned weights ''w''
''i''. Then, the Fréchet variance is calculated as a weighted sum,
:
Examples of Fréchet means
Arithmetic mean and median
For real numbers, the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
is a Fréchet mean, using the usual Euclidean distance as the distance function.
The
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
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''.
p. 136
In higher-dimensional spaces, this becomes the
geometric median
In geometry, the geometric median of a discrete set of sample points 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 ...
.
Geometric mean
On the positive real numbers, the (hyperbolic) distance function
can be defined. The
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
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:
: