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Von Mises–Fisher Distribution
In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the (p-1)-sphere in \mathbb^. If p=2 the distribution reduces to the von Mises distribution on the circle. Definition The probability density function of the von Mises–Fisher distribution for the random ''p''-dimensional unit vector \mathbf is given by: :f_(\mathbf; \boldsymbol, \kappa) = C_(\kappa) \exp \left( \right), where \kappa \ge 0, \left \Vert \boldsymbol \right \Vert = 1 and the normalization constant C_(\kappa) is equal to : C_(\kappa)=\frac , where I_ denotes the modified Bessel function of the first kind at order v. If p = 3, the normalization constant reduces to : C_(\kappa) = \frac = \frac . The parameters \boldsymbol and \kappa are called the ''mean direction'' and '' concentration parameter'', respectively. The greater the value of \kappa, the higher the concentration of the distribution around the mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projected Normal Distribution
In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution) is a probability distribution over directions that describes the radial projection of a random variable with n-variate normal distribution over the unit (n-1)-sphere. Definition and properties Given a random variable \boldsymbol X \in \R^n that follows a multivariate normal distribution \mathcal_n(\boldsymbol\mu,\, \boldsymbol\Sigma), the projected normal distribution \mathcal_n(\boldsymbol\mu, \boldsymbol\Sigma) represents the distribution of the random variable \boldsymbol Y = \frac obtained projecting \boldsymbol X over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case \boldsymbol \mu is parallel to an eigenvector of \boldsymbol \Sigma, the distribution is symmetric. The first version of such distribution was introduced in Pukkila and Rao (198 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...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 "wr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold (mathematics)
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rejection Sampling
In numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in \mathbb^m with a density. Rejection sampling is based on the observation that to sample a random variable in one dimension, one can perform a uniformly random sampling of the two-dimensional Cartesian graph, and keep the samples in the region under the graph of its density function. Note that this property can be extended to ''N''-dimension functions. Description To visualize the motivation behind rejection sampling, imagine graphing the probability density function (PDF) of a random variable onto a large rectangular board and throwing darts at it. Assume that the darts are uniformly distributed around the board. Now remove all of the darts that are outside the area und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (mathematics)
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: * The projection from a point onto a plane or central projection: If is a point, called the center of projection, then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher-Bingham Distribution
In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere ''S''2 in 3-space R3). It is the analogue on ''S''2 of the bivariate normal distribution with an unconstrained covariance matrix. The Kent distribution was proposed by John T. Kent in 1982, and is used in geology as well as bioinformatics. Definition The probability density function f(\mathbf)\, of the Kent distribution is given by: : f(\mathbf) = \frac\exp\left\ where \mathbf\, is a three-dimensional unit vector, (\cdot)^T denotes the transpose of (\cdot), and the normalizing constant \textrm(\kappa,\beta)\, is: : c(\kappa,\beta) = 2\pi\sum_^\infty \frac\beta^\left(\frac\kappa\right)^ I_(\kappa) Where I_v(\kappa) is the modified Bessel function and \Gamma(\cdot) is the gamma function. Note that c(0,0) = 4\pi and c(\kappa,0)=4\pi(\kapp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restriction (mathematics)
In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A. Formal definition Let f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the restriction of f to A is the function _A : A \to F given by _A(x) = f(x) for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A. If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by its graph, :G(_A) = \ = G(f)\cap (A\times F), where the pairs (x,f(x)) represent ordered pairs in the graph G. Extensions A function F is said to be an ' of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname f \subseteq \operatorname F and F\big\vert_ = f. ... [...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] |
Isotropic
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables, each of which clusters around a mean value. Definitions Notation and parametrization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero To The Power Of Zero
Zero to the power of zero, denoted as , is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents. For instance, in combinatorics, defining aligns with the interpretation of choosing 0 elements from a set (mathematics), set and simplifies Polynomial expansion, polynomial and binomial expansions. However, in other contexts, particularly in mathematical analysis, is often considered an indeterminate form. This is because the value of as both and approach zero can lead to different results based on the Limit process, limiting process. The expression arises in limit problems and may result in a range of values or diverge to infinity, making it difficult to assign a single consistent value in these cases. The treatment of also varies across different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
