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Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The Pearson product-moment correlation coefficient, correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables. A distinction must be made between (1) the covariance of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, income, disease, temperature, radioactivity, architectural design, etc.); one that is heterogeneous is distinctly nonuniform in at least one of these qualities. Etymology and spelling The words ''homogeneous'' and ''heterogeneous'' come from Medieval Latin ''homogeneus'' and ''heterogeneus'', from Ancient Greek ὁμογενής (''homogenēs'') and ἑτερογενής (''heterogenēs''), from ὁμός (''homos'', "same") and ἕτερος (''heteros'', "other, another, different") respectively, followed by γένος (''genos'', "kind"); -ous is an adjectival suffix. Alternate spellings omitting the last ''-e-'' (and the associated pronunciations) are common, but mistaken: ''homogenous'' is strictly a biological/pathological term whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ornstein–Uhlenbeck Process
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck. The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over time, the process tends to drift towards its mean function: such a process is called ''mean-reverting''. The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move bac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MIT Press
The MIT Press is the university press of the Massachusetts Institute of Technology (MIT), a private research university in Cambridge, Massachusetts. The MIT Press publishes a number of academic journals and has been a pioneer in the Open Access movement in academic publishing. History MIT Press traces its origins back to 1926 when MIT published a lecture series entitled ''Problems of Atomic Dynamics'' given by the visiting German physicist and later Nobel Prize winner, Max Born. In 1932, MIT's publishing operations were first formally instituted by the creation of an imprint called Technology Press. This imprint was founded by James R. Killian, Jr., at the time editor of MIT's alumni magazine and later to become MIT president. Technology Press published eight titles independently, then in 1937 entered into an arrangement with John Wiley & Sons in which Wiley took over marketing and editorial responsibilities. In 1961, the centennial of MIT's founding charter, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropy
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Process
In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. More formally, the joint probability distribution of the process remains the same when shifted in time. This implies that the process is statistically consistent across different time periods. Because many statistical procedures in time series analysis assume stationarity, non-stationary data are frequently transformed to achieve stationarity before analysis. A common cause of non-stationarity is a trend in the mean, which can be due to either a unit root or a deterministic trend. In the case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. With a deterministic trend, the process is called trend-stationary, and shocks have only transitory effects, with the variable tending towards a determin ... [...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\Big[\big(Z(x)-\mathbb[Z(x)]\big)\big(Z(y)-\mathbb[Z(y)]\big) \Big].\, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
