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Chebyshev Centre
In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case). Mathematical representation There exist several alternative representations for the Chebyshev center. Consider the set Q and denote its Chebyshev center by \hat. \hat can be computed by solving: : \min_ \left\ with respect to the Euclidean norm \, \cdot\, , or alternatively by solving: : \operatorname \max_ \left\, x - \hat x \right\, ^2. Despite these properties, finding the Chebyshev center may be a hard numerical optimization problem. For example, in the second representation above, the inner maximization i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constrained Least Squares
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. This means, the unconstrained equation \mathbf \boldsymbol = \mathbf must be fit as closely as possible (in the least squares sense) while ensuring that some other property of \boldsymbol is maintained. There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below: * Equality constrained least squares: the elements of \boldsymbol must exactly satisfy \mathbf \boldsymbol = \mathbf (see Ordinary least squares). * Stochastic (linearly) constrained least squares: the elements of \boldsymbol must satisfy \mathbf \boldsymbol = \mathbf + \mathbf , where \mathbf is a vector of random variables such that \operatorname(\mathbf ) = \mathbf and \operatorname(\mathbf \mathbf ^) = \tau^\mathbf. This effectively imposes a prior distribution for \boldsymbol and is therefore equivalent to Bayesian li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Methods
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available.C. Lon Enloe, Elizabeth Garnett, Jonathan Miles, ''Physical Science: What the Technology Professional Needs to Know'' (2000), p. 47. Typically, estimation involves "using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter".Raymond A. Kent, "Estimation", ''Data Construction and Data Analysis for Survey Research'' (2001), p. 157. The sample provides information that can be projected, through various formal or informal processes, to determine a range most likely to describe the missing information. An estimate that turns out to be incorrect will be an overestimate if the estimate exceeds the actual result and an underestimate if the estimate fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the '' barycenter'' or ''center of mass'' coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin. In physics, if variations in gravity are considered, then a '' center of gravity'' can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. History The term "centroid" was coined in 1814. It is used as a substitute for the older terms "center of grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centre (geometry)
In geometry, a centre (English in the Commonwealth of Nations, Commonwealth English) or center (American English) () of an shape, object is a point (geometry), point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups, then a centre is a fixed point of all the isometries that move the object onto itself. Circles, spheres, and segments The centre of a circle is the point equidistant from the points on the edge. Similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends. Symmetric objects For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions. So the centre of a square (geometry), square, rectangle, rhombus or parallelogram is where the diagonals intersect, this is (among other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscribed Circle
In geometry, a circumscribed circle for a set of points is a circle passing through each of them. Such a circle is said to ''circumscribe'' the points or a polygon formed from them; such a polygon is said to be ''inscribed'' in the circle. * Circumcircle, the circumscribed circle of a triangle, which always exists for a given triangle. * Cyclic polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points. All triangles are cyclic polygons. * Cyclic quadrilateral, a special case of a cyclic polygon. See also * Smallest-circle problem, the related problem of finding the circle with minimal radius containing an arbitrary set of points, not necessarily passing through them. * Inscribed figure {{sia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smallest-circle Problem
The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, least bounding circle problem, smallest enclosing circle problem) is a computational geometry problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. The corresponding problem in ''n''-dimensional space, the smallest bounding sphere problem, is to compute the smallest ''n''-sphere that contains all of a given set of points. The smallest-circle problem was initially proposed by the English mathematician James Joseph Sylvester in 1857. The smallest-circle problem in the plane is an example of a facility location problem (the 1-center problem) in which the location of a new facility must be chosen to provide service to a number of customers, minimizing the farthest distance that any customer must travel to reach the new facility. Both the smallest circle problem in the plane, and the smallest bounding sphere problem in any hig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounding Sphere
In mathematics, given a non-empty set of objects of finite extension in d-dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is a d-dimensional solid sphere containing all of these objects. Used in computer graphics and computational geometry, a bounding sphere is a special type of bounding volume. There are several fast and simple bounding sphere construction algorithms with a high practical value in real-time computer graphics applications. In statistics and operations research, the objects are typically points, and generally the sphere of interest is the minimal bounding sphere, that is, the sphere with minimal radius among all bounding spheres. It may be proven that such a sphere is unique: If there are two of them, then the objects in question lie within their intersection. But an intersection of two non-coinciding spheres of equal radius is contained in a sphere of smaller radius. The problem of computing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Semi-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior (topology)
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem. Definitions Interior point If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.) This definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
