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Rank–size Distribution
Rank–size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as the rank–frequency distribution, when the source data are from a frequency distribution. These are particularly of interest when the data vary significantly in scales, such as city size or word frequency. These distributions frequently follow a power law distribution, or less well-known ones such as a stretched exponential function or parabolic fractal distribution, at least approximately for certain ranges of ranks; see below. A rank-size distribution is not a probability distribution or cumulative distribution function. Rather, it is a discrete form of a quantile function (inverse cumulative distribution) in reverse order, giving the size of the element at a given rank. Simple rank–size distributions In the case of city popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Order Countries
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * Hierarchy of the Catholic Church * Military rank * Police ranks of the United States * Ranking member, S politicsthe most senior member of a committee from the minority party, and thus second-most senior member of a committee * Imperial, royal and noble ranks Level or position in society * Social class * Social position * Social status Places * Rank, Iran, a village * Rank, Nepal, a village development committee People * Rank (surname), a list of people with the name Arts, entertainment, and media Music * ''Rank'' (album), a live album by the Smiths * "Rank", a song by Artwork from '' A Bugged Out Mix'' Other arts, entertainment, and media * Rank (chess), a row of the chessboard * ''Rank'' (film), a short film directed by David Y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes a particular aspect of a probability distribution. There are different ways to quantify kurtosis for a theoretical distribution, and there are corresponding ways of estimating it using a sample from a population. Different measures of kurtosis may have different interpretations. The standard measure of a distribution's kurtosis, originating with Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean. It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then b ... [...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, 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 () of log45 ≈ 1.16 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yule Distribution
Yule, actually Yuletide ("Yule time") is a festival observed by the historical Germanic peoples, later undergoing Christianised reformulation resulting in the now better-known Christmastide. The earliest references to Yule are by way of indigenous Germanic month names ' (Before Yule) or ' and ' (After Yule). Scholars have connected the celebration to the Wild Hunt, the god Odin and the pagan Anglo-Saxon Mōdraniht. Terms with an etymological equivalent to ''Yule'' are used in the Nordic countries for Christmas with its religious rites, but also for the holidays of this season. ''Yule'' is also used to a lesser extent in English-speaking countries to refer to Christmas. Customs such as the Yule log, Yule goat, Yule boar, Yule singing, and others stem from Yule. A number of Neopagans have introduced their own rites. Etymology ''Yule'' is the modern English representation of the Old English words ' or ' and ' or ''ġéoli'', with the former indicating the 12-day festival of " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preferential Attachment
A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not. "Preferential attachment" is only the most recent of many names that have been given to such processes. They are also referred to under the names Yule process, cumulative advantage, the rich get richer, and the Matthew effect. They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions. If preferential attachment is non-linear, measured distributions may deviate from a power law. These mechanisms may generate distributions which are approximately power law over transient periods. Definition A preferential attachment process is a stochastic urn process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yule–Simon Distribution
In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the ''Yule distribution''. The probability mass function (pmf) of the Yule–Simon (''ρ'') distribution is :f(k;\rho) = \rho\operatorname(k, \rho+1), for integer k \geq 1 and real \rho > 0, where \operatorname is the beta function. Equivalently the pmf can be written in terms of the rising factorial as : f(k;\rho) = \frac, where \Gamma is the gamma function. Thus, if \rho is an integer, : f(k;\rho) = \frac. The parameter \rho can be estimated using a fixed point algorithm. The probability mass function ''f'' has the property that for sufficiently large ''k'' we have : f(k;\rho) \approx \frac \propto \frac 1 . This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: f(k;\rho) can be used to model, for example, the relative frequency of the kth most frequent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broken Power Law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Empirical examples The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, volcanic eruptions, human judgments of stimulus intensity and many other quantit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Function
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example. Notation and interpretation Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King Effect
In statistics, economics, and econophysics, the king effect is the phenomenon in which the top one or two members of a ranked set show up as clear outliers. These top one or two members are unexpectedly large because they do not conform to the statistical distribution or rank-distribution which the remainder of the set obeys. Distributions typically followed include the power-law distribution, that is a basis for the stretched exponential function,"The individual success of musicians, like that of physicists, follows a stretch exponential" J.A. Davies and |
Pareto Index
Pareto may refer to: People * Vilfredo Pareto (1848–1923), Italian economist, political scientist, and philosopher, works named for him include: ** Pareto analysis, a statistical analysis tool in problem solving **Pareto distribution, a power-law probability distribution **Pareto efficiency **Pareto front, the set of all Pareto efficient solutions **Pareto principle, or the 80-20 rule * Bartolomeo Pareto, medieval priest and cartographer from Genoa * Graziella Pareto (1889–1973), Catalan soprano * Lorenzo Pareto (1800–1865), Italian geologist and statesman * Paula Pareto (born 1986), Argentine judoka * Benedetto Pareto, builder of the Shrine of Nostra Signora della Guardia in Liguria, Italy Other uses * Pareto, Piedmont, a town in Italy * Pareto Group, a Norwegian finance company See also * Paret, a surname * Pereto, a town in Italy * Perito Perito ( Cilentan: ''Prito'') is a town and ''comune'' in the province of Salerno in the Campania region of south-western Ita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
