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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] |
Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bingham Distribution
In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the ''n''-sphere. It is a generalization of the Watson distribution and a special case of the Kent and Fisher-Bingham distributions. The Bingham distribution is widely used in paleomagnetic data analysis, and has been reported as being of use in the field of computer vision. Its probability density function is given by : f(\mathbf\,;\,M,Z)\; dS^ \;=\; _F_(;;Z)^\;\cdot\; \exp\left(\right)\; dS^ which may also be written : f(\mathbf\,;\,M,Z)\; dS^ \;=\; _F_(;;Z)^\;\cdot\; \exp\left(\right)\; dS^ where x is an axis (i.e., a unit vector), ''M'' is an orthogonal orientation matrix, ''Z'' is a diagonal concentration matrix, and _F_(\cdot;\cdot,\cdot) is a confluent hypergeometric function of matrix argument. The matrices ''M'' and ''Z'' are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Von Mises–Fisher Distribution
Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchise) * Matrix (mathematics), a rectangular array of numbers, symbols or expressions Matrix (or its plural form matrices) may also refer to: Science and mathematics * Matrix (mathematics), algebraic structure, extension of vector into 2 dimensions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryotic organism's cells * Matrix (chemical analysis), the non-analyte components of a sample * Matrix (geology), the fine-grained material in which larger objects are embedded * Matrix (composite), the constituent of a composite material * Hair matrix, produces hair * Nail matrix, part of the nail in anatomy Arts and entertainment Fictional entities * Matrix (comics), two comic b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivariate Von Mises Distribution
In probability theory and statistics, the bivariate von Mises distribution is a probability distribution describing values on a torus. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail, such as backbone-dependent rotamer libraries. Definition The bivariate von Mises distribution is a probability distribution defined on the torus, S^1 \times S^1 in \mathbb^3. The probability density function of the general bivariate von Mises distribution for the angles \phi, \psi \in , 2\pi/math> is given by : f(\phi, \psi) \propto \exp \kappa_1 \cos(\phi - \mu) + \kappa_2 \cos(\psi - \nu) + (\cos(\phi-\mu), \sin(\phi-\mu)) \mathbf (\cos(\psi - \nu), \sin(\psi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kent 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\ where \mathbf\, is a three-dimensional unit vector, (\cdot)^ 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(\kappa^)\sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
