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Self-similar Process
Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property. Several related properties have this name, and some are defined here. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension. Because stochastic processes are random variables with a time and a space component, their self-similarity properties are defined in terms of how a scaling in time relates to a scaling in space. Distributional self-similarity Definition A continuous-time stochastic process (X_t)_ is called ''self-similar'' with parameter H>0 if for all a>0, the processes (X_)_ and (a^HX_t)_ have the same law. Examples *The Wiener process (or Brownian motion) is self-similar with H=1/2. *The fractional Brownian motion is a generalisation of Brownian motion that preserves self-similarity; it can be self-similar for any H\in(0,1). *The class of self-similar Lévy processes are call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. 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 Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial science, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The ''Pareto principle'' or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () precisely reflect it. Empirical observation has shown that this 80:20 distribution fits a wide range of cases, including natural phenomena and human activities. Definitions If ''X'' is a random variable with a Pareto (Type I) distribution, then the probability that ''X'' is greater than some nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. 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, 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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steven E
Stephen or Steven is an English first name. It is particularly significant to Christians, as it belonged to Saint Stephen ( ), an early disciple and deacon who, according to the Book of Acts, was stoned to death; he is widely regarded as the first martyr (or " protomartyr") of the Christian Church. The name, in both the forms Stephen and Steven, is often shortened to Steve or Stevie. In English, the female version of the name is Stephanie. Many surnames are derived from the first name, including Stephens, Stevens, Stephenson, and Stevenson, all of which mean "Stephen's (son)". In modern times the name has sometimes been given with intentionally non-standard spelling, such as Stevan or Stevon. A common variant of the name used in English is Stephan ( ); related names that have found some currency or significance in English include Stefan (pronounced or in English), Esteban (often pronounced ), and the Shakespearean Stephano ( ). Origins The name "Stephen" (and its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE
The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) organization, 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE has a corporate office in New York City and an operations center in Piscataway, New Jersey. The IEEE was formed in 1963 as an amalgamation of the American Institute of Electrical Engineers and the Institute of Radio Engineers. History The IEEE traces its founding to 1884 and the American Institute of Electrical Engineers. In 1912, the rival Institute of Radio Engineers was formed. Although the AIEE was initially larger, the IRE attracted more students and was larger by the mid-1950s. The AIEE and IRE merged in 1963. The IEEE is headquartered in New York City, but most business is done at the IEEE Operations Center in Piscataway, New Jersey, opened in 1975. The Australian Section of the IEEE existed between 1972 and 1985, after which it s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murad Taqqu
Murad Salman Taqqu (Arabic language, Arabic: مراد طقو) is an Iraqi probabilist and statistician specializing in time series and stochastic processes. His research areas have included Long-range dependency, long-range dependence, self-similar processes, and Heavy-tailed distribution, heavy tails. Taqqu is a professor emeritus at Boston University Department of Mathematics and Statistics, and a fellow of the American Mathematical Society. Education and career Taqqu was born in Baghdad but grew up in Switzerland. As an undergraduate, he studied physics and mathematics at the École Polytechnique Fédérale de Lausanne. He obtained his Ph.D. at Columbia University in 1972, with his dissertation ''Limit Theorems for Sums of Strongly Dependent Random Variables'' supervised by Benoit Mandelbrot. Between 1972 and 1974 Taqqu lectured at the Hebrew University of Jerusalem and worked as a post-doctoral fellow at the Weizmann Institute in Rehovot. From 1974 to 1985, he was a faculty memb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ethernet
Ethernet ( ) is a family of wired computer networking technologies commonly used in local area networks (LAN), metropolitan area networks (MAN) and wide area networks (WAN). It was commercially introduced in 1980 and first standardized in 1983 as IEEE 802.3. Ethernet has since been refined to support higher bit rates, a greater number of nodes, and longer link distances, but retains much backward compatibility. Over time, Ethernet has largely replaced competing wired LAN technologies such as Token Ring, FDDI and ARCNET. The original 10BASE5 Ethernet uses a thick coaxial cable as a shared medium. This was largely superseded by 10BASE2, which used a thinner and more flexible cable that was both less expensive and easier to use. More modern Ethernet variants use Ethernet over twisted pair, twisted pair and fiber optic links in conjunction with Network switch, switches. Over the course of its history, Ethernet data transfer rates have been increased from the original to the lates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multifractal System
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pink Noise
Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each Octave (electronics), octave interval (halving or doubling in frequency) carries an equal amount of noise energy. Pink noise sounds like a waterfall. It is often used to tune loudspeaker systems in professional audio. Pink noise is one of the most commonly observed signals in biological systems. The name arises from the pink appearance of visible light with this power spectrum. This is in contrast with white noise which has equal intensity per frequency interval. Definition Within the scientific literature, the term "1/f noise" is sometimes used loosely to refer to any noise with a power spectral density of the form S(f) \propto \frac, where is frequency, and , with exponent usually close to 1. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor's Law
Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007). Taylor's original name for this relationship was the law of the mean. The name ''Taylor's law'' was coined by Southwood in 1966. Definition This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms. For a population count Y with mean \mu and variance \operatorname (Y), Taylor's law is written : \operatorname (Y) = a\mu^b, where ''a'' and ''b'' are both positive constants. Taylor proposed this relationship in 1961, suggesting that the exponent ''b'' be considered a species specific index of aggregation. This power law has subsequently been confirmed for many hundreds of species. Taylor's law has also been applied to assess the time d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
