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Nearest Neighbour Distribution
In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution,A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. ''Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004'', pages 1–75, 2007. nearest-neighbor distribution function or nearest neighbor distribution is a mathematical function that is defined in relation to mathematical objects known as point processes, which are often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both.D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003. More specifically, nearest neighbor func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Function
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Space
In mathematics, a space is a set (sometimes known as a ''universe'') endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Moment Measure
In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of factorial moments, which are useful for studying non-negative integer-valued random variables.D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003. The first factorial moment measure of a point process coincides with its first moment measure or ''intensity measure'', which gives the expected or average number of points of the point process located in some region of space. In general, if the number of points in some region is considered as a random variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-parametric Statistics
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated. Definitions The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others: The first meaning of ''nonparametric'' involves techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: * Methods which are ''distribution-free'', which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. * Statistics defined to be a function on a sample, without dependency on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Measure
In probability and statistics, a moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of (raw) moments of random variables, hence arise often in the study of point processes and related fields.D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. . Probability and its Applications (New York). Springer, New York, second edition, 2008. An example of a moment measure is the first moment measure of a point process, often called mean measure or intensity measure, which gives the expected or average number of points of the point process being located in some region of space.F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Network ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Point Process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. Definition Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let E be a separable complete metric space and let \mathcal E be its Borel \sigma -algebra. (The most common example of a separable complete metric space is \R^n .) As a transition kernel A random measure \zeta is a ( a.s.) locally finite transition kernel from an abstract probability space (\Omega, \mathcal A, P) to (E, \mathcal E) . Being a transition kernel means that *For any fixed B \in \mathcal \mathcal E , the mapping : \omega \mapsto \zeta(\omega,B) :is measurable from (\Omega, \mathcal A) to (\R, \mathcal B(\R)) *For every fixed \omega \in \Omega , the mapping : B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
