**stochastic**- These sources are frequently described in terms of probability distributions to account for the stochastic nature of complex engineering systems. (wikipedia.org)
- Travel distance and residence time probability distributions are the key components of stochastic models for coarse sediment transport. (wiley.com)
- As in many fields with stochastic approaches to particle transport, the conceptual model used to represent sediment transport through time in gravel bed rivers equates the distribution of overall travel distance at time t of a single particle as the sum of a random number of randomly sized step lengths. (wiley.com)
- In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. (wikipedia.org)
- Within a probabilistic framework, we describe the characterization of uncertainties using mathematical statistics methods, the propagation of uncertainties through computational models using either Monte Carlo sampling or stochastic expansion methods, and the sensitivity analysis of uncertainties using variance- and differentiation-based methods. (ac.be)
- Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of stochastic processes. (wikipedia.org)
- Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators. (wikipedia.org)
- The first relation between supersymmetry and stochastic dynamics was established by Giorgio Parisi and Nicolas Sourlas who demonstrated that the application of the BRST gauge fixing procedure to Langevin SDEs, i.e., to SDEs with linear phase spaces, gradient flow vector fields, and additive noises, results in N=2 supersymmetric models. (wikipedia.org)

**displaystyle**- The asymmetric Laplace distribution, including the special case of μ = 0 {\displaystyle {\boldsymbol {\mu }}=\mathbf {0} } , is an example of a geometric stable distribution. (wikipedia.org)
- This learning framework is very general and can be applied to distributions over any space Ω {\displaystyle \Omega } on which a sensible kernel function (measuring similarity between elements of Ω {\displaystyle \Omega } ) may be defined. (wikipedia.org)
- Thus if x {\displaystyle x} is a function on Euclidean space Rd and n {\displaystyle n} is a natural number, then the convolution power is defined by x ∗ n = x ∗ x ∗ x ∗ ⋯ ∗ x ∗ x ⏟ n , x ∗ 0 = δ 0 {\displaystyle x^{*n}=\underbrace {x*x*x*\cdots *x*x} _{n},\quad x^{*0}=\delta _{0}} where * denotes the convolution operation of functions on Rd and δ0 is the Dirac delta distribution. (wikipedia.org)
- Equivalently, x ∗ n / σ n {\displaystyle x^{*n}/\sigma {\sqrt {n}}} tends weakly to the standard normal distribution. (wikipedia.org)
- end{cases}}} Here, w ( n ) {\displaystyle w(n)} is the size distribution for connected components, u 1 ( k ) = k + 1 μ 1 u ( k + 1 ) , {\displaystyle u_{1}(k)={\frac {k+1}{\mu _{1}}}u(k+1),} is the excess degree distribution, and u ( k ) {\displaystyle u(k)} denotes the degree distribution. (wikipedia.org)
- 0 {\displaystyle e^{-z/2}>0} for all non-negative z {\displaystyle z} ), in general elliptical distributions can be bounded or unbounded-such a distribution is bounded if g ( z ) = 0 {\displaystyle g(z)=0} for all z {\displaystyle z} greater than some value. (wikipedia.org)
- The Laplace transform of the phase type distribution is given by M ( s ) = α 0 + α ( s I − S ) − 1 S 0 , {\displaystyle M(s)=\alpha _{0}+{\boldsymbol {\alpha }}(sI-S)^{-1}\mathbf {S^{0}} ,} where I is the identity matrix. (wikipedia.org)

**mathematical statistics**- Characterization theorems in probability theory and mathematical statistics are such theorems that establish a connection between the type of the distribution of random variables or random vectors and certain general properties of functions in them. (wikipedia.org)

**Convolution**- Decisions of non-homogeneous convolution equations on the half and applications for building stability estimations in characterizations of probability distributions devoted one of the major works of the author. (wikipedia.org)
- Convolution equations in stability problems on characterizations of probability laws. (wikipedia.org)
- If x is the distribution function of a random variable on the real line, then the nth convolution power of x gives the distribution function of the sum of n independent random variables with identical distribution x. (wikipedia.org)
- Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined. (wikipedia.org)
- In applications to quantum field theory, the convolution exponential, convolution logarithm, and other analytic functions based on the convolution are constructed as formal power series in the elements of the algebra (Brouder, Frabetti & Patras 2008). (wikipedia.org)
- A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. (wikipedia.org)

**arbitrary distributions**- A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. (wikipedia.org)
- It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified by Ben Chrouda, El Oued & Ouerdiane (2002). (wikipedia.org)

**Poisson**- Discrete probability distributions, Poisson process. (boun.edu.tr)
- In probability, statistics and related fields, a Poisson point process or Poisson process (also called a Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object that consists of points randomly located on a mathematical space. (wikipedia.org)
- Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. (wikipedia.org)
- The Poisson point process is one of the most studied and used point processes, in both the field of probability and in more applied disciplines concerning random phenomena, due to its convenient properties as a mathematical model as well as being mathematically interesting. (wikipedia.org)

**finite**- It represents the limiting distribution for a sum of independent, identically distributed random variables with finite variance and covariance where the number of elements to be summed is itself an independent random variable distributed according to a geometric distribution. (wikipedia.org)
- The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finite moments. (wikipedia.org)
- Learning algorithms based on this framework exhibit good generalization ability and finite sample convergence, while often being simpler and more effective than information theoretic methods Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms. (wikipedia.org)
- This definition makes sense if x is an integrable function (in L1), a compactly supported distribution, or is a finite Borel measure. (wikipedia.org)
- Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). (wikipedia.org)

**Continuous**- Random variables, discrete and continuous distributions. (boun.edu.tr)
- Continuous probability distributions. (boun.edu.tr)
- The continuous phase-type distribution is the distribution of time from the above process's starting until absorption in the absorbing state. (wikipedia.org)
- The following probability distributions are all considered special cases of a continuous phase-type distribution: Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases. (wikipedia.org)

**Theory**- born July 10, 1953) is a Lithuanian mathematician who worked in probability theory and characterization of probability distributions and its stability. (wikipedia.org)
- In the mathematical theory of probability, multivariate Laplace distributions are extentions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. (wikipedia.org)
- Jean-François Mertens made some contributions to probability theory and published articles on elementary topology, but he was mostly active in economic theory. (wikipedia.org)
- The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. (wikipedia.org)
- Eugene Lukacs (Hungarian: Lukács Jenő, 14 August 1906 - 21 December 1987) was a Hungarian statistician born in Szombathely, notable for his work in characterization of distributions, stability theory, and being the author of Characteristic Functions, a classic textbook in the field. (wikipedia.org)
- From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics - the dynamical systems theory and topological field theories. (wikipedia.org)
- STS has a close relation with algebraic topology and its topological sector belongs to the class of models known as Witten-type topological or cohomological field theory. (wikipedia.org)
- Elliptical distributions are defined in terms of the characteristic function of probability theory. (wikipedia.org)

**Discrete**- It has a discrete time equivalent the discrete phase-type distribution. (wikipedia.org)

**density functions**- The flexible approach of QuakeML allows further extensions of the standard in order to represent waveform data, macroseismic information, probability density functions, slip distributions, shake maps, and others. (wikipedia.org)
- Some elliptical distributions are alternatively defined in terms of their density functions. (wikipedia.org)

**Stability**- In 1976-1978 Januškevičius trained in Steklov Institute of Mathematics in Moscow, where under the guidance of Professor Vladimir Zolotarev wrote and in 1978 defended his thesis "Investigation of stability in some problems of characterization of distributions" and received the Candidate of Sciences degree. (wikipedia.org)
- The main result of this thesis was the work of the author, which considered the stability for decompositions of probability distributions into components. (wikipedia.org)
- His doctoral thesis "Stability for characterizations of distributions" was published as a monograph. (wikipedia.org)
- His investigation of stability of characterizations of probability distributions R. Januškevičius has completed in the book in 2014. (wikipedia.org)
- On estimates of the stability measure for decompositions of probability distributions into components. (wikipedia.org)
- Stability for characterizations of distributions. (wikipedia.org)
- Stability characterizations of some probability distributions. (wikipedia.org)

**seismology**- It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering, seismology, time-frequency analysis for music signals, spectrograms in biology and speech processing, and engine design. (wikipedia.org)
- The Quake Markup Language (QuakeML) is a flexible, extensible and modular XML representation of seismological data (e.g. epicenter, hypocenter, magnitude) which is intended to cover a broad range of fields of application in modern seismology. (wikipedia.org)

**statistical**- Handbook of Exploration Geochemistry, Volume 2: Statistics and Data Analysis in Geochemical Prospecting aims to survey the techniques available for the quality control of laboratory data, storage and retrieval of field and laboratory information, statistical analysis of single- and multi-element data, and presentation of geochemical data as maps. (elsevier.com)
- We include some recent advances in the propagation and sensitivity analysis of uncertainties that are characterized by arbitrary probability distributions that may exhibit statistical dependence. (ac.be)
- The American Statistical Association has described him as "a living legend whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine. (wikipedia.org)
- As Head and later Director of the Research and Training School at the Indian Statistical Institute for a period of over 40 years, Rao developed research and training programs and produced several leaders in the field of Mathematics. (wikipedia.org)
- Elliptical distributions are also used in robust statistics to evaluate proposed multivariate-statistical procedures. (wikipedia.org)

**characteristic function**- Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings. (wikipedia.org)

**elliptical**- The symmetric multivariate Laplace distribution is elliptical. (wikipedia.org)
- In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. (wikipedia.org)
- In statistics, the normal distribution is used in classical multivariate analysis, while elliptical distributions are used in generalized multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). (wikipedia.org)
- The definition of elliptical distributions for real random-vectors has been extended to accommodate random vectors in Euclidean spaces over the field of complex numbers, so facilitating applications in time-series analysis. (wikipedia.org)
- Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example. (wikipedia.org)
- There exist elliptical distributions that have undefined mean, such as the Cauchy distribution (even in the univariate case). (wikipedia.org)
- Thus any linear combination of the components of X is elliptical (though not necessarily with the same elliptical distribution), and any subset of X is elliptical. (wikipedia.org)
- Elliptical distributions are used in statistics and in economics. (wikipedia.org)
- In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance. (wikipedia.org)

**Approximation**- So the representation of heavy-tailed or leptokurtic distribution by phase type is an approximation, even if the precision of the approximation can be as good as we want. (wikipedia.org)

**multivariate normal distr**- Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent. (wikipedia.org)
- In statistics, the multivariate normal distribution (of Gauss) is used in classical multivariate analysis, in which most methods for estimation and hypothesis-testing are motivated for the normal distribution. (wikipedia.org)

**symmetric**- The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. (wikipedia.org)

**satisfies**- and, conversely, satisfies boundedness properties unavailable to classical distributions. (wikipedia.org)

**particles**- Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. (wikipedia.org)
- The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. (wikipedia.org)
- Significant energy is usually required to disintegrate soil, etc. particles into the PSD that is then called a grain size distribution. (wikipedia.org)
- The wave function of physical particles is a typical example of a probability amplitude distribution. (science20.com)

**empirical**- However, bed elevation time series collected using sonar transducers and lidar can be translated into empirical residence time distributions at each elevation in the bed and for the entire bed thickness. (wiley.com)
- The distribution patterns and volumes of coarse-grained channel-fill deposits developed mainly during falling- and lowstand stages can be estimated from outcrop and seismic data on the basis of several empirical relationships among width, depth, and sinuosity of deep-water channels. (go.jp)

**particle**- Particle size distribution can greatly affect the efficiency of any collection device. (wikipedia.org)
- The squared modulus of a CPAD can be interpreted as the distribution of the probability density of the presence of the carrier of the properties of the particle. (science20.com)

**uncertainty**- QMU focuses on the identification, characterization, and analysis of performance thresholds and their associated margins for engineering systems that are evaluated under conditions of uncertainty, particularly when portions of those results are generated using computational modeling and simulation. (wikipedia.org)
- The characterization of uncertainty supports comparisons of design margins for key system performance metrics to the uncertainty associated with their calculation by the model. (wikipedia.org)
- The identification and characterization of these values allows the ratios of margin-to-uncertainty (M/U) to be calculated for the system. (wikipedia.org)

**simulation**- Examples outside of the nuclear weapons field include applications at NASA for interplanetary spacecraft and rover development, missile six-degree-of-freedom (6DOF) simulation results, and characterization of material properties in terminal ballistic encounters. (wikipedia.org)
- He/she will learn several methodologies of simulation and experimental characterization. (universite-paris-saclay.fr)

**exponential distribution**- Exponential distribution - 1 phase. (wikipedia.org)
- Hyper-exponential distribution (also called a mixture of exponential) - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. (wikipedia.org)
- Note: The exponential distribution is the degenerate situation when all the parallel phases are identical. (wikipedia.org)
- The simplest non-trivial example of a phase-type distribution is the exponential distribution of parameter λ. (wikipedia.org)

**thus**- In the Hilbert Book Model this argument is used in order to explain why physics contains fields, thus it has far reaching consequences. (science20.com)
- Thus quaternionic distributions exist in eight different sign flavors. (science20.com)

**Statistics**- The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback-Leibler divergence. (wikipedia.org)

**random**- We introduce an automated segmentation method, extended Markov Random Field ( eMRF ) to classify 21 neuroanatomical structures of mouse brain based on three dimensional (3D) magnetic resonance imaging (MRI). (pubmedcentralcanada.ca)
- The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. (wikipedia.org)

**quantum**- For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model-and is a convenient indicator of quantum mechanical interference. (wikipedia.org)
- In quantum physics this inaccuracy is represented by the wave function, which is a probability amplitude distribution. (science20.com)

**Examples**- In all the following examples it is assumed that there is no probability mass at zero, that is α0 = 0. (wikipedia.org)

**functions**- See below for a characterization of pure states whose Wigner functions are non-negative. (wikipedia.org)
- In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(x,p) uniquely associated to suitably ordered operators Ĝ through Weyl's transform (cf. (wikipedia.org)

**spaces**- One could apply non-separable Hilbert spaces, such as a rigged Hilbert space, but that takes the argument away why physics must contain fields. (science20.com)

**fundamental**- Photorefractive materials: fundamental concepts, holographic recording and materials characterization. (wikipedia.org)

**laboratory**- Residence time for individual grains is difficult to measure, and residence time distributions appropriate to field and laboratory settings are typically inferred theoretically or from overall transport characteristics. (wiley.com)

**disambiguation**- See also Wigner distribution (disambiguation). (wikipedia.org)

**complex**- Most physicists use a complex probability amplitude distribution (CPAD) for that purpose. (science20.com)
- Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. (wikipedia.org)

**values**- Quaternionic distributions keep the same sign selection through all of their values. (science20.com)

**frequency**- The Wigner distribution P(x,p) of a pure state is defined as: where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair (i.e. real and imaginary parts of the electric field or frequency and time of a signal). (wikipedia.org)
- These moments can then be used to determine an appropriate frequency distribution, which can then be used as a probability model. (wikipedia.org)
- The frequency of extremal events, such as severe droughts and storms, often requires the use of distributions that focus on the tail of the distribution, rather than the data nearest the mean. (wikipedia.org)

**Normal Distribution**- The log-normal distribution mass median diameter. (wikipedia.org)

**imaginary**- A QPAD can be considered as the combination of a real charge density distribution and an imaginary current density distribution. (science20.com)
- The parameter space can be seen as the imaginary part of a quaternionic distribution. (science20.com)

**infinite**- In those set-ups Jean-François Mertens provided an extension of the characterization of the minmax and maxmin value for the infinite game in the dependent case with state independent signals. (wikipedia.org)
- Deterministic distribution (or constant) - The limiting case of an Erlang distribution, as the number of phases become infinite, while the time in each state becomes zero. (wikipedia.org)

**identical**- Erlang distribution - 2 or more identical phases in sequence. (wikipedia.org)
- Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase. (wikipedia.org)
- Hypoexponential distribution - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang. (wikipedia.org)

**results**- Smoothing the Wigner distribution through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one. (wikipedia.org)
- From the results of the 2D and 3D seismic survey acquired in the Nankai Trough area by METI in 2001 and 2002, widely-distributed BSR was recognized in the area, which was expected the wide distribution of methane hydrate. (go.jp)

**parameter**- The parameter of the phase-type distribution are : S = -λ and α = 1. (wikipedia.org)

**Class**- In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). (wikipedia.org)

**material**- Curiosity and taste for learning new material from new fields is essential. (universite-paris-saclay.fr)
- With the net redistribution of electrons into the dark regions of the material, leaving holes in the bright areas, the resulting charge distribution causes an electric field, known as a space charge field to be set up in the crystal. (wikipedia.org)

**possible**- Verification of conditions of this or that characterization theorem in practice is possible only with some error, i.e., only to a certain degree of accuracy. (wikipedia.org)
- However, it is equally well possible to use a quaternionic probability amplitude distribution (QPAD) instead of a CPAD. (science20.com)

**Science**- Rao was awarded the United States National Medal of Science, that nation's highest award for lifetime achievement in fields of scientific research, in June 2002. (wikipedia.org)

**neural**- Neural decoding is a neuroscience-related field concerned with the reconstruction of sensory and other stimuli from information that has already been encoded and represented in the brain by networks of neurons. (wikipedia.org)

**known**- Mertens and Zamir implemented John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types. (wikipedia.org)