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Wrapped Lévy Distribution
In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle. Description The pdf of the wrapped Lévy distribution is : f_(\theta;\mu,c)=\sum_^\infty \sqrt\,\frac where the value of the summand is taken to be zero when \theta+2\pi n-\mu \le 0, c is the scale factor and \mu is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields: : f_(\theta;\mu,c)=\frac\sum_^\infty e^=\frac\left(1 + 2\sum_^\infty e^\cos\left(n(\theta-\mu) - \sqrt\,\right)\right) In terms of the circular variable z=e^ the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments: :\langle z^n\rangle=\int_\Gamma e^\,f_(\theta;\mu,c)\,d\theta = e^. where \Gamma\, is some interval of length 2\pi. The first moment is th ... [...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] |
Directional Statistics
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes (lines through the origin in R''n'') or rotations in R''n''. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold. The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, dihedral angles in molecules, orientations, rotations and so on. Circular distributions Any probability density function (pdf) \ p(x) on the line can be "wrappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wrapped Distribution
In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit ''n''-sphere. In one dimension, a wrapped distribution consists of points on the unit circle. If \phi is a random variate in the interval (-\infty,\infty) with probability density function (PDF) p(\phi), then z = e^ is a circular variable distributed according to the wrapped distribution p_(\theta) and \theta = \arg(z) is an angular variable in the interval (-\pi,\pi] distributed according to the wrapped distribution p_w(\theta). Any probability density function p(\phi) on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable :\theta=\phi \mod 2\pi in some interval of length 2\pi is : p_w(\theta)=\sum_^\infty which is a periodic summation, periodic sum of period 2\pi. The preferred interval is generally (-\pi<\theta\le\pi) for which [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile."van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, , and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995/ref> It is a special case of the inverse-gamma distribution. It is a stable distribution. Definition The probability density function of the Lévy distribution over the domain x\ge \mu is :f(x;\mu,c)=\sqrt~~\frac where \mu is the location parameter and c is the scale parameter. The cumulative distribution function is :F(x;\mu,c)=1 - \textrm\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wrapped Distribution
In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit ''n''-sphere. In one dimension, a wrapped distribution consists of points on the unit circle. If \phi is a random variate in the interval (-\infty,\infty) with probability density function (PDF) p(\phi), then z = e^ is a circular variable distributed according to the wrapped distribution p_(\theta) and \theta = \arg(z) is an angular variable in the interval (-\pi,\pi] distributed according to the wrapped distribution p_w(\theta). Any probability density function p(\phi) on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable :\theta=\phi \mod 2\pi in some interval of length 2\pi is : p_w(\theta)=\sum_^\infty which is a periodic summation, periodic sum of period 2\pi. The preferred interval is generally (-\pi<\theta\le\pi) for which [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directional Statistics
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes (lines through the origin in R''n'') or rotations in R''n''. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold. The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, dihedral angles in molecules, orientations, rotations and so on. Circular distributions Any probability density function (pdf) \ p(x) on the line can be "wrappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Distributions
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic secti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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