[Download 48 Laws Of Power 57
Rancul Ratha
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, 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.[1] The rate of change exhibited in these relationships is said to be multiplicative.
The distributions of a wide variety of physical, biological, and human-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,[2] cloud sizes,[3] the foraging pattern of various species,[4] the sizes of activity patterns of neuronal populations,[5] the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms,[6] the sizes of power outages, volcanic eruptions,[7] human judgments of stimulus intensity[8][9] and many other quantities.[10] Empirical distributions can only fit a power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values.Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.
On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis).[12] On the other hand, this also allows for cost-efficient interventions.[11] For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially.[13]
Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them.[14] The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems;[15] see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, neuroscience, systematics, sociology, economics and more.
However, much of the recent interest in power laws comes from the study of probability distributions: The distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory), which considers the frequency of extremely rare events like stock market crashes and large natural disasters. It is primarily in the study of statistical distributions that the name "power law" is used.
In empirical contexts, an approximation to a power-law o ( x k ) \displaystyle o(x^k) often includes a deviation term ε \displaystyle \varepsilon , which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from the power-law function (perhaps for stochastic reasons):
More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income).[16] Among them are:
The Tweedie distributions are a family of statistical models characterized by closure under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have a fundamental role as foci of mathematical convergence similar to the role that the normal distribution has as a focus in the central limit theorem. This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with Taylor's law in ecology and with fluctuation scaling[50] in physics. It can also be shown that this variance-to-mean power law, when demonstrated by the method of expanding bins, implies the presence of 1/f noise and that 1/f noise can arise as a consequence of this Tweedie convergence effect.[51]
On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i-th order statistic versus the i-th order statistic, for i = 1, ..., n, where n is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to "stabilize" about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots [55]
Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided.[10][65] The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.
The maximum of this likelihood is found by differentiating with respect to parameter α \displaystyle \alpha , setting the result equal to zero. Upon rearrangement, this yields the estimator equation:
where ζ ( α , x m i n ) \displaystyle \zeta (\alpha ,x_\mathrm min ) is the incomplete zeta function. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for α ^ \displaystyle \hat \alpha are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.
Further, both of these estimators require the choice of x min \displaystyle x_\min . For functions with a non-trivial L ( x ) \displaystyle L(x) function, choosing x min \displaystyle x_\min too small produces a significant bias in α ^ \displaystyle \hat \alpha , while choosing it too large increases the uncertainty in α ^ \displaystyle \hat \alpha , and reduces the statistical power of our model. In general, the best choice of x min \displaystyle x_\min depends strongly on the particular form of the lower tail, represented by L ( x ) \displaystyle L(x) above.
More about these methods, and the conditions under which they can be used, can be found in .[10] Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions.
This criterion[67] can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides a more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the cumulative distribution function, by the cumulative frequency of a property X, defined as the number of elements per meter (or area unit, second etc.) for which X > x applies, where x is a variable real number. As an example,[citation needed] the cumulative distribution of the fracture aperture, X, for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope).
Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data.[25] This is important for understanding the mechanism that gives rise to the distribution: superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation.
For example, log-normal distributions are often mistaken for power-law distributions:[68] a data set drawn from a lognormal distribution will be approximately linear for large values (corresponding to the upper tail of the lognormal being close to a power law)[clarification needed], but for small values the lognormal will drop off significantly (bowing down), corresponding to the lower tail of the lognormal being small (there are very few small values, rather than many small values in a power law).[citation needed]
In general, many alternative functional forms can appear to follow a power-law form for some extent.[69] Stumpf & Porter (2012) proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude.[70] Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a power law. As a solution to this problem, Diaz[54] proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails. However, Stumpf & Porter (2012) claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process.[70]
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