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Cross-correlation Matrix
The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms. Definition For two random vectors \mathbf = (X_1,\ldots,X_m)^ and \mathbf = (Y_1,\ldots,Y_n)^, each containing random elements whose expected value and variance exist, the cross-correlation matrix of \mathbf and \mathbf is defined by and has dimensions m \times n. Written component-wise: :\operatorname_ = \begin \operatorname _1 Y_1& \operatorname _1 Y_2& \cdots & \operatorname _1 Y_n\\ \\ \operatorname _2 Y_1& \operatorname _2 Y_2& \cdots & \operatorname _2 Y_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname _m Y_1& \operatorname _m Y_2& \cdots & \operatorname _m Y_n\\ \\ \end The random vectors \mathbf and \mathbf need not have the same dimension, and either might be a scalar value. Example For example, if \mathbf = \lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Vector
In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of ''an unspecified person'' from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. Formally, a multivariate random variable is a column vector \mathbf = (X_1,\dots,X_n)^\maths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Function (statistical Mechanics)
In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function. Correlation functions describe how microscopic variables, such as spin and density, at different positions or times are related. More specifically, correlation functions measure quantitatively the extent to which microscopic variables fluctuate together, on average, across space and/or time. Keep in mind that correlation doesn’t automatically equate to causation. So, even if there’s a non-zero correlation between two points in space or time, it doesn’t mean there is a direct causal link between them. Sometimes, a correlation can exist without any causal relationship. This could be purely coincidental or due to other underlying factors, known as confounding variables, which cause both points to covary (statistically). A classic example of spatial correlation can be seen in ferromagnetic and antiferromagnetic materials. In these ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Analysis
Spatial analysis is any of the formal Scientific technique, techniques which study entities using their topological, geometric, or geographic properties, primarily used in Urban design, Urban Design. Spatial analysis includes a variety of techniques using different analytic approaches, especially ''spatial statistics''. It may be applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, or to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is geospatial analysis, the technique applied to structures at the human scale, most notably in the analysis of geographic data. It may also applied to genomics, as in Spatial transcriptomics, transcriptomics data, but is primarily for spatial data. Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Correlation
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways. If ''X'' and ''Y'' are two random variables, with means (expected values) ''μX'' and ''μY'' and standard deviations ''σX'' and ''σY'', respectively, then their covariance and correlation are as follows: ; covariance :\text_ = \sigma_ = E X-\mu_X)\,(Y-\mu_Y)/math> ; correlation :\text_ = \rho_ = E X-\mu_X)\,(Y-\mu_Y)(\sigma_X \sigma_Y)\,, so that \rho_ = \sigma_ / (\sigma_X \sigma_Y) where ''E'' is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If ''Y'' always takes on the same values as ''X'', we have the covariance of a variable with itself (i.e. \sigma_), which is called the variance and is more commonly denoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guang Gong
Guang Gong (born 1956) studied applied math whose research topics include lightweight cryptography and algebraic coding theory for wireless communication. Educated in Sichuan China, she works at the University of Waterloo in Canada as a professor in the department of Electrical and Computer Engineering and University Research Chair. Education After a 2-year program at Xichang Normal Vocational School, Gong was given a master's degree in applied mathematics in 1985 at the Northwest Institute of Telecommunication Engineering, now Xidian University. She completed her PhD in 1990 at the University of Electronic Science and Technology of China. She was a postdoctoral researcher with the Fondazione Ugo Bordoni in Italy. From 1996 to 1998 she was affiliated working with Solomon W. Golomb. She moved to the University of Waterloo as an adjunct and later associate professor, and has been full professor there since 2004. She was given a University Research Chair in 2018. Books Gong's b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radial Distribution Function
In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If a given particle is taken to be at the origin ''O'', and if \rho = N/V is the average number density of particles, then the local time-averaged density at a distance r from ''O'' is \rho g(r). This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below. In simplest terms it is a measure of the probability of finding a particle at a distance of r away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r and r+dr away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and the blue particles are those whose centers are with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Distortion Theory
Rate or rates may refer to: Finance * Rate (company), an American residential mortgage company formerly known as Guaranteed Rate * Rates (tax), a type of taxation system in the United Kingdom used to fund local government * Exchange rate, rate at which one currency will be exchanged for another Mathematics and science * Rate (mathematics), a specific kind of ratio, in which two measurements are related to each other (often with respect to time) * Rate function, a function used to quantify the probabilities of a rare event * Reaction rate, in chemistry the speed at which reactants are converted into products Military * Naval rate, a junior enlisted member of a navy * Rating system of the Royal Navy, a former method of indicating a British warship's firepower People * Ed Rate (1899–1990), American football player * José Carlos Rates (1879–1945), General Secretary of the Portuguese Communist Party * Peter of Rates (died 60 AD), traditionally considered to be the first bisho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual Statistical dependence, dependence between the two variables. More specifically, it quantifies the "Information content, amount of information" (in Units of information, units such as shannon (unit), shannons (bits), Nat (unit), nats or Hartley (unit), hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of Entropy (information theory), entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the Pearson correlation coefficient, correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is from the product of the marginal distributions of X and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Function (quantum Field Theory)
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements, although they are not themselves observables. This is because they need not be gauge invariant, nor are they unique, with different correlation functions resulting in the same S-matrix and therefore describing the same physics. They are closely related to correlation functions between random variables, although they are nonetheless different objects, being defined in Minkowski spacetime and on quantum operators. Definition For a scalar field theory with a single field \phi(x) and a vacuum state , \Omega\rangle at every event in spacetime, the -point correlation function is the vacuum expectation value of the time-ordered products of field operators in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Function (astronomy)
In astronomy, a correlation function describes the distribution of objects (often stars or galaxies) in the universe. By default, "correlation function" refers to the two-point autocorrelation function. The two-point autocorrelation function is a function of one variable (distance); it describes the excess probability of finding two galaxies separated by this distance (excess over and above the probability that would arise if the galaxies were simply scattered independently and with uniform probability). It can be thought of as a "clumpiness" factor - the higher the value for some distance scale, the more "clumpy" the universe is at that distance scale. The following definition (from Peebles 1980) is often cited: : ''Given a random galaxy in a location, the correlation function describes the probability that another galaxy will be found within a given distance.'' However, it can only be correct in the statistical sense that it is averaged over a large number of galaxies cho ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
