|
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 used in the field of computer vision. Its probability density function is given by : f(\mathbf\,;\,M,Z) \; dS^ = _1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^ \cdot \exp \left( \operatorname Z M^T \mathbf \mathbf^T M \right)\; dS^ which may also be written : f(\mathbf\,;\,M,Z)\; dS^ \;=\; _1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^ \cdot \exp\left( \mathbf^T M Z M^T \mathbf \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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), 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 statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Bingham
Christopher Bingham is an American statistician who introduced the Bingham distribution. In joint work with C. M. D. Godfrey and John Tukey he introduced complex demodulation into the analysis of time series. The Kent distribution, also known as the Fisher–Bingham distribution, is named after Bingham and the English biologist and statistician Ronald Fisher. Bingham studied mathematics at Yale University, receiving his PhD under the supervision of Alan Treleven James in 1964. His PhD theis was titled ''Distributions on the sphere and on the projective plane''. He subsequently worked as a research associate at Princeton University with John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi .... References External links Christopher Binghamfaculty web page * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antipodal Symmetry
In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation. An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetries is called ''centrosymmetric''. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersphere
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] |
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\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(\kap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paleomagnetic
Paleomagnetism (occasionally palaeomagnetism) is the study of prehistoric Earth's magnetic fields recorded in rocks, sediment, or archeological materials. Geophysicists who specialize in paleomagnetism are called ''paleomagnetists.'' Certain magnetic minerals in rocks can record the direction and intensity of Earth's magnetic field at the time they formed. This record provides information on the past behavior of the geomagnetic field and the past location of tectonic plates. The record of geomagnetic reversals preserved in volcanic and sedimentary rock sequences (magnetostratigraphy) provides a time-scale that is used as a geochronologic tool. Evidence from paleomagnetism led to the revival of the continental drift hypothesis and its transformation into the modern theory of plate tectonics. Apparent polar wander paths provided the first clear geophysical evidence for continental drift, while marine magnetic anomalies did the same for seafloor spreading. Paleomagnetic data ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Vision
Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the form of decisions. "Understanding" in this context signifies the transformation of visual images (the input to the retina) into descriptions of the world that make sense to thought processes and can elicit appropriate action. This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory. The scientific discipline of computer vision is concerned with the theory behind artificial systems that extract information from images. Image data can take many forms, such as video sequences, views from multiple cameras, multi-dimensional data from a 3D scanning, 3D scanner, 3D point clouds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint ( conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergeometric Function Of A Matrix Argument
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals. Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument. Definition Let p\ge 0 and q\ge 0 be integers, and let X be an m\times m complex symmetric matrix. Then the hypergeometric function of a matrix argument X and parameter \alpha>0 is defined as : _pF_q^(a_1,\ldots,a_p; b_1,\ldots,b_q;X) = \sum_^\infty\sum_ \frac\cdot \frac \cdot C_\kappa^(X), where \kappa\vdash k means \kappa is a partition of k, (a_i)^_ is the generalized Pochhammer symbol, and C_\kappa^(X) is the "C" normalization of the Jack function. Two matrix arguments If X and Y are two m\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonalizable Matrix
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is matrix similarity, similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not unique.) This property exists for any linear map: for a dimension (vector space), finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an Basis (linear algebra)#Ordered bases and coordinates, ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix (mathematics), matrix representation A = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, T is represented by Diagonalization is the process of finding the above P and and makes many subsequent computations easier. One can raise a diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
