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Covariance Operator
In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product \langle \cdot,\cdot\rangle , the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by :\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, \mathrm \mathbf (z) for all ''x'' and ''y'' in ''H''. The covariance operator ''C'' is then defined by :\mathrm(x, y) = \langle Cx, y \rangle (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint. When P is a centred Gaussian measure, ''C'' is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. Even more generally, for a probability measure P on a Banach space ''B'', the covariance of P is the bilinear form on the algebraic dual ''B''#, defined by :\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, ... [...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 probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (mathematics)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...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 probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples * The constant zero function, mapping every vector to zero, is trivially a linear functional. * Indexing int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold. Formal definition Given a probability space (\Omega, \mathcal, P), an ''X''-valued random field is a collection of ''X''-valued random variables indexed by elements in a topological space ''T''. That is, a random field ''F'' is a collection : \ where each F_t is an ''X''-valued random variable. Examples In its discrete version, a random field is a list of rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion proc ... [...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, a fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Function
In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a domain ''D'', a covariance function ''C''(''x'', ''y'') gives the covariance of the values of the random field at the two locations ''x'' and ''y'': :C(x,y):=\operatorname(Z(x),Z(y))=\mathbb\left \cdot\ \right\, The same ''C''(''x'', ''y'') is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that ''x'' and ''y'' refer to locations in time rather than in space), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, Cov(''Z''(''x''1), ''Y''(''x''2))). Admissibility For locations ''x''1, ''x''2, …, ''x''''N'' ∈ ''D'' the variance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are Compact operator, compact operators. In quantum mechanics, Mixed state (physics), mixed states are described by Density matrix, density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Definition Suppose H is a Hilbert s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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