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Mean Of Circular Quantities
In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle.Christopher M. Bishop: ''Pattern Recognition and Machine Learning (Information Science and Statistics)'', As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of circular statistics and of statistics of non-Euclidean spaces. This computation produces a different result than the arithmetic mean, with the difference being greater when the angles are widely distributed. For example, the arithmetic mean of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arctangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of ''θ'' radians will correspond to an arc whose length is ''rθ'', where ''r'' is the radius of the circle. Thus in the unit cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center Of Mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion. In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in ''n''-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the ''barycenter'' or ''center of mass'' coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin. In physics, if variations in gravity are considered, then a ''center of gravity'' can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. History The term "centroid" is of recent coinage (1814). It is used as a substitute for the older te ... [...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] |
Circular Standard Deviation
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] |
Circular Distribution
In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range A circular distribution is often a continuous probability distribution, and hence has a probability density, but such distributions can also be discrete, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the lower or upper end of the range, and the division of the range could notionally be made at any point. Graphical representation If a circular distribution has a density :p(\phi) \qquad \qquad (0\le\phi<2\pi),\, it can be graphically represented as a closed : |
Spherical Linear Interpolation
In computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius great circle arc, given the ends and an interpolation parameter between 0 and 1. Geometric Slerp Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends. Let ''p''0 and ''p''1 be the first and last points of the arc, and let ''t'' be the parameter, 0 ≤ ''t'' ≤ 1. Compute Ω as the angle subtended by the arc, so that , the ''n''-dimensional dot product of the unit vectors from the origin to the ends. The geometric formula is then : \operatorname(p_0,p_1; t) = \frac p_0 + \frac p_1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Mises Distribution
In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle \theta on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the ''N''-dimensional sphere. Definition The von Mises probability density funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more ''likely'' than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as \mathcal(\theta\mid X) instead of P(X \mid \theta), to emphasize that it is to be understood as a function of the parameters \theta instead of the random variable X. In maximum likelihood estimation, the arg max of the likelihood function serves as a point estimate for \theta, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' r ... [...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] |
