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Recurrence Plots
In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment j in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits roughly the same area in the phase space as at time j. In other words, it is a plot of :\vec(i)\approx \vec(j), showing i on a horizontal axis and j on a vertical axis, where \vec is the state of the system (or its phase space trajectory). Background Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles), but also irregular cyclicities (as El Niño Southern Oscillation, heart beat intervals). Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf. Poincaré recurrence theorem). The recurrence of states i ... [...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] |
Angular Distance
Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtended by the radius, radii through two points on a sphere. When the rays are Line of sight, lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation. Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque. Use The term ''angular distance'' (or ''separation'') is technically synonymous with ''angle'' itself, but is meant to suggest the linear distance between objects (for instance, a pair of stars observed from Earth). Measurement Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Synchronization
{{no footnotes, date=June 2017 Phase synchronization is the process by which two or more cyclic signals tend to oscillate with a repeating sequence of relative phase angles. Phase synchronisation is usually applied to two waveforms of the same frequency with identical phase angles with each cycle. However it can be applied if there is an integer relationship of frequency, such that the cyclic signals share a repeating sequence of phase angles over consecutive cycles. These integer relationships are called Arnold tongues which follow from bifurcation of the circle map. One example of phase synchronization of multiple oscillators can be seen in the behavior of Southeast Asian fireflies. At dusk, the flies begin to flash periodically with random phases and a gaussian distribution of native frequencies. As night falls, the flies, sensitive to one another's behavior, begin to synchronize their flashing. After some time all the fireflies within a given tree (or even larger area) will ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Product
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Notation This article will use the following notat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual Statistical dependence, dependence between the two variables. More specifically, it quantifies the "Information content, amount of information" (in Units of information, units such as shannon (unit), shannons (bits), Nat (unit), nats or Hartley (unit), hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of Entropy (information theory), entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the Pearson correlation coefficient, correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is from the product of the marginal distributions of X and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K2 Entropy
K, or k, is the eleventh letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''kay'' (pronounced ), plural ''kays''. The letter K usually represents the voiceless velar plosive. History The letter K comes from the Greek letter Κ (kappa), which was taken from the Semitic kaph, the symbol for an open hand. This, in turn, was likely adapted by Semitic tribes who had lived in Egypt from the hieroglyph for "hand" representing /ḏ/ in the Egyptian word for hand, ⟨ ḏ-r-t⟩ (likely pronounced in Old Egyptian). The Semites evidently assigned it the sound value instead, because their word for hand started with that sound. K was brought into the Latin alphabet with the name ''ka'' /kaː/ to differentiate it from C, named ''ce'' (pronounced /keː/) and Q, named ''qu'' and pronounced /kuː/. In the earliest Latin inscriptions, the letters C, K and Q were all u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Dimension
In chaos theory, the correlation dimension (denoted by ''ν'') is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ''ν'' = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or ''m''-dimensional space), the correlation dimension will be ''ν'' = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Invariant
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Quantification Analysis
Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory. Background The recurrence quantification analysis (RQA) was developed in order to quantify differently appearing recurrence plots (RPs), based on the small-scale structures therein. Recurrence plots are tools which visualise the recurrence behaviour of the phase space trajectory \vec(i) of dynamical systems: :(i,j) = \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), where \Theta: \mathbf \rightarrow \ is the Heaviside function and \varepsilon a predefined tolerance. Recurrence plots mostly contain single dots and lines which are parallel to the mean diagonal (''line of identity'', LOI) or which are vertical/horizontal. Lines parallel to the LOI are referred to as ''diagonal lines'' and the vertical structures as ''vertical line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rp Examples740
RP, R-P, Rp, R-p, or rp may refer to: Businesses and organizations * Rainforest Partnership, an environmental organization based in Austin, Texas * RallyPoint, a social network for the US military * Reform Party (Singapore), an opposition party in Singapore led by Kenneth Jeyaretnam * Republic Polytechnic, a polytechnic in Singapore * ''Rheinische Post'', a German newspaper * Rhône-Poulenc, a former French chemical company * Royal Society of Portrait Painters (London), with membership indicated RP * Roma Party (''Romska partija''), a political party in Serbia * Welfare Party, or ''Refah Partisi'', in Turkey * Chautauqua Airlines, the IATA airline designator RP Economics and finance *Repurchase agreement, the sale of securities together with an agreement for the seller to buy back the securities at a later date *Reservation price, the highest price a buyer is willing to pay for goods or a service *Rupee, common name for the currencies of several countries *Rupiah, the offic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorenz System
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. The term "butterfly effect" in popular media may stem from the real-world implications of the Lorenz attractor, namely that tiny changes in initial conditions evolve to completely different trajectories. This underscores that chaotic systems can be completely deterministic and yet still be inherently impractical or even impossible to predict over longer periods of time. For example, even the small flap of a butterfly's wings could set the earth's atmosphere on a vastly different trajectory, in which for example a hurricane occurs where it otherwise would have not (see Saddle points). The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takens' Theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of that system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. It is the most commonly used method for attractor reconstruction. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a -dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
