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Correlation Sum
In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close: :C(\varepsilon) = \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m, where N is the number of considered states \vec(i), \varepsilon is a threshold distance, \, \cdot \, a norm (e.g. Euclidean norm) and \Theta( \cdot ) the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem): :\vec(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)), where u(i) is the time series, m the embedding dimension and \tau the time delay. The correlation sum is used to estimate the correlation dimension. See also *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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Integral
In chaos theory, the correlation integral is the mean probability that the states at two different times are close: :C(\varepsilon) = \lim_ \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m, where N is the number of considered states \vec(i), \varepsilon is a threshold distance, \, \cdot \, a norm (e.g. Euclidean norm) and \Theta( \cdot ) the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem): :\vec(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1))), where u(i) is the time series, m the embedding dimension and \tau the time delay. The correlation integral is used to estimate the correlation dimension. An estimator of the correlation integral is the correlation sum: :C(\varepsilon) = \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m. See also *Recurrence quantification analysis Recurrence quantificati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := 0.html" ;"title=">0">>0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''forecasting' ... [...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 a dynamical 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. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a \nu-dimensional manifold M. The dynamics is given by a smooth map :f: M \to ... [...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 ... [...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 lines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. 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 by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
