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Geometric median, the Glossary

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.^[1]

54 relations: Alfred Weber, Andrew Vázsonyi, Annals of Operations Research, Annals of Statistics, Arg max, Association for Computing Machinery, Cartesian coordinate system, Center of mass, Central tendency, Centroid, Closed-form expression, Collinearity, Computational Geometry (journal), Convex function, Coplanarity, Discrete & Computational Geometry, Equivariant map, Estimator, Euclidean distance, Euclidean space, Evangelista Torricelli, Fermat point, Fréchet mean, Geometry, Giovanni Fagnano, Iteratively reweighted least squares, Journal of Symbolic Computation, K-medians clustering, Location parameter, Mathematical Programming, Mathematics Magazine, Maximum and minimum, Median, Medoid, Metric space, Model of computation, Norm (mathematics), Operations Research (journal), Optimal facility location, Pierre de Fermat, Quadrilateral, Radon's theorem, Riemannian manifold, Robust statistics, Rotation (mathematics), Second-order cone programming, Similarity (geometry), Steiner tree problem, Straightedge and compass construction, Time complexity, ... Expand index (4 more) »
Descriptive statistics

Alfred Weber

Carl David Alfred Weber (30 July 1868 – 2 May 1958) was a German economist, geographer, sociologist and theoretician of culture whose work was influential in the development of modern economic geography.

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Andrew Vázsonyi

Andrew Vázsonyi (1916–2003), also known as Endre Weiszfeld and Zepartzatt Gozinto) was a Hungarian mathematician and operations researcher. He is known for Weiszfeld's algorithm for minimizing the sum of distances to a set of points, and for founding The Institute of Management Sciences....

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Annals of Operations Research

Annals of Operations Research is a peer-reviewed academic journal published by Springer Science+Business Media.

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Annals of Statistics

The Annals of Statistics is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics.

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Arg max

In mathematics, the arguments of the maxima (abbreviated arg max or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a function output value is maximized and minimized, respectively.

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Association for Computing Machinery

The Association for Computing Machinery (ACM) is a US-based international learned society for computing.

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Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

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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 weighted relative position of the distributed mass sums to zero.

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Central tendency

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.

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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. Geometric median and centroid are means.

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Closed-form expression

In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (and integer powers) and function composition.

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Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

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Computational Geometry (journal)

Computational Geometry, also known as Computational Geometry: Theory and Applications, is a peer-reviewed mathematics journal for research in theoretical and applied computational geometry, its applications, techniques, and design and analysis of geometric algorithms.

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Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points.

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Coplanarity

In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all.

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Discrete & Computational Geometry

Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer.

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Equivariant map

In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces).

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Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished.

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Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.

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Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

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Evangelista Torricelli

Evangelista Torricelli (15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo.

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Fermat point

In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices.

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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. Geometric median and Fréchet mean are means.

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Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

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Giovanni Fagnano

Giovanni Francesco Fagnano dei Toschi (born 31 January 1715 in Senigallia, died 14 May 1797 in Senigallia) was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician.

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Iteratively reweighted least squares

The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form of a ''p''-norm: \mathop_ \sum_^n \big| y_i - f_i (\boldsymbol\beta) \big|^p, by an iterative method in which each step involves solving a weighted least squares problem of the form:C.

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Journal of Symbolic Computation

The Journal of Symbolic Computation is a peer-reviewed monthly scientific journal covering all aspects of symbolic computation published by Academic Press and then by Elsevier.

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In statistics, k-medians clustering is a cluster analysis algorithm.

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Location parameter

In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution.

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Mathematical Programming

Mathematical Programming is a peer-reviewed scientific journal that was established in 1971 and is published by Springer Science+Business Media.

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Mathematics Magazine

Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.

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Maximum and minimum

In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. Geometric median and maximum and minimum are mathematical optimization.

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The median of a set of numbers is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. Geometric median and median are means.

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Medoid

Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Geometric median and Medoid are means.

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Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

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Model of computation

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input.

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Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.

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Operations Research (journal)

Operations Research is a bimonthly peer-reviewed academic journal covering operations research that is published by the Institute for Operations Research and the Management Sciences.

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Optimal facility location

The study of facility location problems (FLP), also known as location analysis, is a branch of operations research and computational geometry concerned with the optimal placement of facilities to minimize transportation costs while considering factors like avoiding placing hazardous materials near housing, and competitors' facilities.

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Pierre de Fermat

Pierre de Fermat (between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

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Quadrilateral

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices).

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Radon's theorem

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that:Any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect.

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Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.

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Robust statistics

Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect.

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Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form where the problem parameters are f \in \mathbb^n, \ A_i \in \mathbb^, \ b_i \in \mathbb^, \ c_i \in \mathbb^n, \ d_i \in \mathbb, \ F \in \mathbb^, and g \in \mathbb^p.

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Similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other.

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Steiner tree problem

In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. Geometric median and Steiner tree problem are geometric algorithms.

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Straightedge and compass construction

In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.

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References

[1] https://en.wikipedia.org/wiki/Geometric_median

Also known as 1-median, Fermat-Weber point, Spatial median, Weiszfeld algorithm, Weiszfeld's algorithm.

, Tohoku Mathematical Journal, Translation (geometry), Univariate, Weber problem.

Geometric median, the Glossary

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See also

Descriptive statistics

References