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Simple Point Process
A simple point process is a special type of point process in probability theory. In simple point processes, every point is assigned the weight one. Definition Let S be a locally compact second countable Hausdorff space and let \mathcal S be its Borel \sigma -algebra. A point process \xi , interpreted as random measure on (S, \mathcal S) , is called a simple point process if it can be written as : \xi =\sum_ \delta_ for an index set I and random elements X_i which are almost everywhere pairwise distinct. Here \delta_x denotes the Dirac measure on the point x . Examples Simple point processes include many important classes of point processes such as Poisson processes, Cox processes and binomial processes. Uniqueness If \mathcal I is a generating ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Process
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. , .Daley, D.J, Vere-Jones, D. (1988). ''An Introduction to the Theory of Point Processes''. Springer, New York. , . Point processes can be used for spatial data analysis,Diggle, P. (2003). ''Statistical Analysis of Spatial Point Patterns'', 2nd edition. Arnold, London. . which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. There are different mathematical interpretations of a point process, such as a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-countable Space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of open subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many " well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R''n'') with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis. Properties Second-countabilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Set
In mathematics, a Borel set is any set in a topological space that can be formed from 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 topological ... [...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 a (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 (E, \mathcal E) *For every fixed \omega \in \Omega , the mapping : B \mapsto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an '' (indexed) family'', often written as . Examples *An enumeration of a set gives an index set J \sub \N, where is the particular enumeration of . *Any countably infinite set can be (injectively) indexed by the set of natural numbers \N. *For r \in \R, the indicator function on is the function \mathbf_r\colon \R \to \ given by \mathbf_r (x) := \begin 0, & \mbox x \ne r \\ 1, & \mbox x = r. \end The set of all such indicator functions, \_ , is an uncountable set indexed by \mathbb. Other uses In computational complexity theory and cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.” The modern-day usage of “random element” frequently assumes the space of values is a topological vector space, often a Banach or Hilbert space with a specified natural sigma algebra of subsets. Definition Let (\Omega, \mathcal, P) be a probability space, and (E, \mathcal) a measurable space. A random element with values in ''E'' is a function which is (\mathcal, \mathcal)-measurable. That is, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. Definition A Dirac measure is a measure on a set (with any -algebra of subsets of ) defined for a given and any (measurable) set by :\delta_x (A) = 1_A(x)= \begin 0, & x \not \in A; \\ 1, & x \in A. \end where is the indicator function of . The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome in the sample space . We can also say that the measure is a single atom at ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. The Dirac measures are the extreme points of the convex set of probability measures on . The name is a back-formation from the Dirac delta func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random 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 Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cox Process
In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955. Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron), and also in financial mathematics where they produce a "useful framework for modeling prices of financial instruments in which credit risk is a significant factor." Definition Let \xi be a random measure. A random measure \eta is called a Cox process directed by \xi , if \mathcal L(\eta \mid \xi=\mu) is a Poisson process with intensity measure \mu . Here, \mathcal L(\eta \mid \xi=\mu) is the conditional distribution of \eta , given \ . Laplace transform If \eta is a Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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