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Correlation Function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations. Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations. Correlation functions us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Convolution Correlation
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and to what degree. Where characteristics are different, the differences may then be evaluated to determine which thing is best suited for a particular purpose. The description of similarities and differences found between the two things is also called a comparison. Comparison can take many distinct forms, varying by field: To compare things, they must have characteristics that are similar enough in relevant ways to merit comparison. If two things are too different to compare in a useful way, an attempt to compare them is colloquially referred to in English as "comparing apples and oranges." Comparison is widely used in society, in science and in the arts. General usage Comparison is a natural activity, which even animals engage in when dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Does Not Imply Causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The idea that "correlation implies causation" is an example of a questionable-cause logical fallacy, in which two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known by the Latin phrase ''cum hoc ergo propter hoc'' ('with this, therefore because of this'). This differs from the fallacy known as ''post hoc ergo propter hoc'' ("after this, therefore because of this"), in which an event following another is seen as a necessary consequence of the former event, and from conflation, the errant merging of two events, ideas, databases, etc., into one. As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not necessarily imply that the resulting co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals. Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance. Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Osterwalder-Schrader Axioms
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions (named after Julian Schwinger) and they are real-analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms (named after Konrad Osterwalder and Robert Schrader).Osterwalder, K., and Schrader, R.: "Axioms for Euclidean Green’s functions," ''Comm. Math. Phys.'' 31 (1973), 83–112; 42 (1975), 281–305. Schwinger functions are also referred to as Euclidean correlation functions. Osterwalder–Schrader axioms Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitian scalar field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (relativity), events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Albert Einstein, Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wick Rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. This transformation is also used to find solutions to problems in quantum mechanics and other areas. Overview Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature convention) :ds^2 = -\left(dt^2\right) + dx^2 + dy^2 + dz^2 and the four-dimensional Euclidean metric :ds^2 = d\tau^2 + dx^2 + dy^2 + dz^2 are equivalent if one permits the coordinate to take on imaginary values. The Minkowski metric becomes Euclidean when is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates , and substituting sometimes yields a problem in real Euclidean coordinate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Positivity
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions (named after Julian Schwinger) and they are real-analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms (named after Konrad Osterwalder and Robert Schrader).Osterwalder, K., and Schrader, R.: "Axioms for Euclidean Green’s functions," ''Comm. Math. Phys.'' 31 (1973), 83–112; 42 (1975), 281–305. Schwinger functions are also referred to as Euclidean correlation functions. Osterwalder–Schrader axioms Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitian scalar field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Integral Formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are ''coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itō Calculus
Itō may refer to: *Itō (surname), a Japanese surname *Itō, Shizuoka, Shizuoka Prefecture, Japan *Ito District, Wakayama Prefecture, Japan See also * Itô's lemma, used in stochastic calculus *Itoh–Tsujii inversion algorithm, in field theory *Itô calculus, an extension of calculus to stochastic processes, named after Kiyoshi Itô *Ito (other) *ITO (other) Ito may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan * Itō, Shizuo ..., for the three-letter acronym {{DEFAULTSORT:Ito es:Ito fr:Ito nl:Ito ja:いとう pt:Ito ru:Ито ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
