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Long Memory
Long-range dependence (LRD), also called long memory or long-range persistence, is a phenomenon that may arise in the analysis of Spatial analysis, spatial or time series analysis, time series data. It relates to the rate of decay of statistical dependence of two points with increasing time interval or spatial distance between the points. A phenomenon is usually considered to have long-range dependence if the dependence decays more slowly than an exponential decay, typically a power-like decay. LRD is often related to self-similar processes or fields. LRD has been used in various fields such as internet traffic modelling, econometrics, hydrology, linguistics and the earth sciences. Different mathematical definitions of LRD are used for different contexts and purposes. Short-range dependence versus long-range dependence One way of characterising long-range and short-range dependent stationary process is in terms of their autocovariance functions. For a short-range dependent process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Analysis
Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early development, using different analytic approaches and applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is the technique applied to structures at the human scale, most notably in the analysis of geographic data or transcriptomics data. Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research. The most fundamental of these is the problem of defining the spatial location of the entities being studied. Classification of the techniques of spatial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Increments
In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. Lévy processes) or by construction (e.g. random walks) Definition A stochastic process X=(X_t)_ has stationary increments if for all t \geq 0 and h > 0 , the distribution of the random variables : Y_:=X_ -X_t depends only on h and not on t . Examples Having stationary increments is a defining property for many large families of stochastic processes such as the Lévy processes. Being special Lévy processes, both the Wiener process and the Poisson processes have stationary increments. Other families of stochastic processes such as random walks have stationary increments by construction. An example of a stochastic process with stationary increments that is not a Lévy process is gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurst Exponent
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880–1978), who was the lead researcher in these studies; the use of the standard notation ''H'' for the coefficient also relates to his name. In fractal geometry, the generalized Hurst exponent has been denoted by ''H'' or ''Hq'' in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). ''H'' is directly related to fractal dimension, ''D'', and is a measure of a data series' "mild" or "wild" ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tweedie Distributions
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models. The Tweedie distributions were named by Bent Jørgensen after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984. Definitions The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship. A random variable ''Y'' is Tweedie distributed ''Twp(μ, σ2)'', if Y \sim \mathrm(\mu, \sigma^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Detrended Fluctuation Analysis
In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise. The obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform. Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities. Calculation Consider a bounded time series x_t of length N, where t \in \mathbb, and let its mean value be denoted \langle x\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traffic Generation Model
A traffic generation model is a stochastic model of the traffic flows or data sources in a communication network, for example a cellular network or a computer network. A packet generation model is a traffic generation model of the packet flows or data sources in a packet-switched network. For example, a web traffic model is a model of the data that is sent or received by a user's web-browser. These models are useful during the development of telecommunication technologies, in view to analyse the performance and capacity of various protocols, algorithms and network topologies . Application The network performance can be analyzed by network traffic measurement in a testbed network, using a network traffic generator such as iperf, bwping and Mausezahn. The traffic generator sends dummy packets, often with a unique packet identifier, making it possible to keep track of the packet delivery in the network. Numerical analysis using network simulation is often a less expensive approach. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long-tail Traffic
A long-tailed or heavy-tailed probability distribution is one that assigns relatively high probabilities to regions far from the mean or median. A more formal mathematical definition is given below. In the context of teletraffic engineering a number of quantities of interest have been shown to have a long-tailed distribution. For example, if we consider the sizes of files transferred from a web-server, then, to a good degree of accuracy, the distribution is heavy-tailed, that is, there are a large number of small files transferred but, crucially, the number of very large files transferred remains a major component of the volume downloaded. Many processes are technically long-range dependent but not self-similar. The differences between these two phenomena are subtle. Heavy-tailed refers to a probability distribution, and long-range dependent refers to a property of a time series and so these should be used with care and a distinction should be made. The terms are distinct although s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autoregressive Fractionally Integrated Moving Average
In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA (''autoregressive integrated moving average'') models by allowing non-integer values of the differencing parameter. These models are useful in modeling time series with long memory—that is, in which deviations from the long-run mean decay more slowly than an exponential decay. The acronyms "ARFIMA" or "FARIMA" are often used, although it is also conventional to simply extend the "ARIMA(''p'', ''d'', ''q'')" notation for models, by simply allowing the order of differencing, ''d'', to take fractional values. Basics In an ARIMA model, the ''integrated'' part of the model includes the differencing operator (1 − ''B'') (where ''B'' is the backshift operator) raised to an integer power. For example, :(1-B)^2=1-2B+B^2 \,, where :B^2X_t=X_ \, , so that :(1-B)^2X_t = X_t -2X_ + X_. In a ''fractional'' model, the power is allowed to be fractional, with the meaning of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Model
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Brownian Motion
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process ''BH''(''t'') on , ''T'' that starts at zero, has expectation zero for all ''t'' in , ''T'' and has the following covariance function: :E _H(t) B_H (s)\tfrac (, t, ^+, s, ^-, t-s, ^), where ''H'' is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by . The value of ''H'' determines what kind of process the ''fBm'' is: * if ''H'' = 1/2 then the process is in fact a Brownian motion or Wiener process; * if ''H'' > 1/2 then the increments of the process are positively correlated; * if ''H'' < 1/2 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking throu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
