Power law

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In statistics, the attorney is a functional relationship between two quantities, where the relative change in one quantity yields a relative change proportionately in another quantity, independent of the initial size of the quantity: one quantity varies as the power of another. For example, consider the square area in terms of the length of the sides, if the length is multiplied, the area multiplied by a factor of four.

Video Power law

Contoh empiris

The distribution of various physical, biological, and man-made phenomena roughly follows the law of forces over a wide range of magnitudes: these include the size of the moon crater and the sun beacon, the pattern of foraging of various species, the size of the patterns of neuronal population activity, the frequency of words in most languages, frequency of surnames, species richness in clades of organisms, size of power outages, criminal charges per inmate, volcanic eruptions, human judgment on stimulus intensity and many others. Few empirical distributions are in accordance with the law of force for all their values, but it is better to follow the law of force in the tail. Acoustic damping follows the law of frequency forces in wide bands for many complex media. The allometric scale law for the relationship between biological variables is one of the most recognized functions of legal force in nature.

Maps Power law

Properties

Invariant scale

Satu atribut hukum kekuasaan adalah invarian skala mereka. Diberikan relasi ${\ displaystyle f (x) = ax ^ {- k}}  ÂÂ$ , penskalasan argumen ${\ displaystyle x}  ÂÂ$ oleh faktor konstan ${\ displaystyle c}  ÂÂ$ hanya menyebabkan penskalaan proporsional dari fungsi itu sendiri. Itu adalah,

${\ displaystyle f (cx) = a (cx) ^ {- k} = c ^ {- k} f (x) \ propto f (x), \!}  ÂÂ$

di mana ${\ displaystyle \ propto}  ÂÂ$ menunjukkan proporsionalitas langsung. Yaitu, penskalaan dengan konstanta ${\ displaystyle c}  ÂÂ$ hanya mengalikan relasi kuasa-hukum asli dengan konstanta ${\ displaystyle c ^ {- k}}  ÂÂ$ . Jadi, maka semua hukum kekuatan dengan eksponen penskalaan tertentu setara dengan faktor konstan, karena masing-masing hanyalah versi skala dari yang lain. Perilaku ini adalah apa yang menghasilkan hubungan linear ketika logaritma diambil dari kedua ${\ displaystyle f (x)}  ÂÂ$ dan ${\ displaystyle x}  ÂÂ$ , dan garis lurus pada plot log-log sering disebut tanda tangan dari hukum kekuatan. Dengan data nyata, kelurusan seperti itu adalah kondisi yang diperlukan, tetapi tidak cukup, untuk data yang mengikuti hubungan kekuasaan hukum. Bahkan, ada banyak cara untuk menghasilkan jumlah data yang terbatas yang meniru perilaku tanda tangan ini, tetapi, dalam batas asimtotiknya, bukan hukum kekuatan yang sebenarnya (misalnya, jika proses pembangkit beberapa data mengikuti distribusi Log-normal). Dengan demikian, model hukum kekuatan yang akurat dan memvalidasi adalah bidang aktif penelitian dalam statistik; Lihat di bawah.

Kurangnya nilai rata-rata yang terdefinisi dengan baik

Kekuatan-hukum ${\ displaystyle x ^ {- k}}  ÂÂ$ memiliki mean yang terdefinisi dengan baik di atas ${\ displaystyle x \ dalam [1, \ infty]}  ÂÂ$ hanya jika ${\ displaystyle k & gt; 2}  ÂÂ$ , dan ia memiliki varian terbatas hanya jika ${\ displaystyle k & gt; 3}  ÂÂ$ ; sebagian besar hukum kekuasaan yang teridentifikasi di alam memiliki eksponen sedemikian rupa sehingga mean didefinisikan dengan baik tetapi varians tidak, menyiratkan mereka mampu perilaku angsa hitam. Ini dapat dilihat dalam eksperimen pemikiran berikut: bayangkan ruangan dengan teman-teman Anda dan perkirakan rata-rata penghasilan bulanan di dalam ruangan. Sekarang bayangkan orang terkaya di dunia memasuki ruangan, dengan penghasilan bulanan sekitar 1 miliar US $. Apa yang terjadi dengan pendapatan rata-rata di dalam ruangan? Penghasilan didistribusikan menurut kuasa hukum yang dikenal sebagai distribusi Pareto (misalnya, kekayaan bersih orang Amerika didistribusikan menurut hukum kekuatan dengan eksponen 2).

On the one hand, this makes the mistake of applying traditional statistics based on variance and standard deviation (such as regression analysis). On the other hand, it also allows cost-effective interventions. For example, since car exhausts are distributed according to the legal force among cars (very few cars contribute to contamination) will be enough to remove very few cars from the road to reduce exhaust substantially.

The median exists, however: for the power law x ^{- k , with an exponent ${\ displaystyle k & gt; 1}$ , it takes a value of 2 1/( k - 1) x _min , where x _min is the minimum value applicable to legal authorities}

Universalization

The equality of power law with a certain scale exponent can have a deeper origin in the dynamic process that results in a power-law relationship. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of a certain quantity of legal force distribution, whose exponents are referred to as critical exponents of the system. Multiple systems with the same critical exponent - that is, displaying identical scaling behaviors as they approach criticality - can be shown, through group theory of renormalization, to share the same fundamental dynamics. For example, water behavior and CO ₂ at their boiling point fall in the same class of universality because they have identical critical exponents. In fact, almost all material phase transitions are represented by a small set of classes of universality. Similar observations have been made, though not comprehensively, to self-regulated critical systems, where the critical point of the system is the puller. Formally, this division of dynamics is called universality, and the system with the exact same critical exponent is said to be from the same class of universality.

src: diplateevo.com

Power-law function

Scientific interest in power-law relations comes in part from the ease with which a particular class mechanism generates it. Demonstration of power law relations in some data can point to certain types of mechanisms that may underlie natural phenomena in question, and can show deep relationships with other seemingly unrelated systems; see also the universality above. The existence of power-law relationships in physics is in part due to dimensional limitations, whereas in complex systems power laws are often regarded as signatures of specific hierarchies or stochastic processes. Some examples of prominent power laws are the law of Pareto income distribution, structural self-similarity fractals, and the scale of law in biological systems. Research on the origins of power-law relationships, and attempts 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, sociology, economics and more.

However, much of the new interest in power law comes from the study of probability distributions: The distribution of a wide variety of numbers seems to follow a law-power form, at least in the upper tail (major events). The behavior of this great event connects this amount to the study of the theory of major deviations (also called extreme value theory), which considers the frequency of extremely rare events such as stock market collapse and major natural disasters. This is primarily in the study of statistical distribution that the name "legal counsel" is used.

Dalam konteks empiris, perkiraan untuk kekuatan-hukum ${\ displaystyle o (x ^ {k})}  ÂÂ$ sering menyertakan istilah penyimpangan ${\ displaystyle \ varepsilon}  ÂÂ$ , yang dapat mewakili ketidakpastian dalam nilai-nilai yang diamati (mungkin kesalahan pengukuran atau sampling) atau memberikan cara sederhana untuk mengamati menyimpang dari fungsi kekuasaan-hukum (mungkin untuk alasan stokastik):

${\ displaystyle y = axe ^ {k} \ varepsilon. \!}  ÂÂ$

Secara matematis, hukum kekuatan yang ketat tidak dapat menjadi distribusi probabilitas, tetapi distribusi yang merupakan fungsi kekuasaan terpotong adalah mungkin: ${\ displaystyle p (x) = Cx ^ {- \ alpha}}  ÂÂ$ untuk ${\ displaystyle x & gt; x _ {\ text {min}}}  ÂÂ$ di mana eksponen ${\ displaystyle \ alpha}  ÂÂ$ (huruf Yunani alfa, jangan dikelirukan dengan faktor skala ${\ displaystyle a}  ÂÂ$ yang digunakan di atas) lebih besar dari 1 (jika tidak, ekor memiliki area tak terbatas), nilai minimum ${\ displaystyle x _ {\ text {min}}}  ÂÂ$ diperlukan jika tidak distribusi memiliki area tak terbatas karena x mendekati 0, dan konstanta C adalah penskalaan faktor untuk memastikan bahwa total area adalah 1, seperti yang dipersyaratkan oleh distribusi probabilitas. Lebih sering seseorang menggunakan hukum kekuatan asimtotik - yang hanya benar dalam batasan; lihat distribusi probabilitas power-law di bawah ini untuk detailnya. Biasanya eksponen jatuh dalam rentang ${\ displaystyle 2 & lt; \ alpha & lt; 3}  ÂÂ$ , meskipun tidak selalu.

Contoh

Over a hundred distribution of power law has been identified in physics (eg avalanches), biology (eg species extinction and body mass), and social sciences (eg city size and income). Among others are:

• Angstrom exponents in aerosol optics
• The frequency dependence of acoustic attenuation in complex media
• Stevens' psychophysical power law "
• Stefan-Boltzmann's Law
• The current-voltage-current-output curve of the field-effect transistor and the vacuum tube approaches the squared law relation, a factor in "tube sound".
• Legal square cube (ratio of surface area to volume)
• Kleiber's law connects animal metabolism with size, and general allometric law
• The 3/2 power law can be found on the characteristic curve of the plates on the triode.
• The inverse square law of Newton and electrostatic gravity, as evidenced by the potential of gravity and electrostatic potential, respectively.
• Self-criticality with tipping point as puller
• Rain-shower cell size, energy dissipation in cyclones and diameter of dust devils on Earth and Mars
• Exponential growth and random observation (or killing)
• Progress through exponential growth and exponential diffusion of innovation
• Highly optimized tolerance
• Van der Waals style model
• Force and potential in simple harmonic motion
• Kepler's third law
• The star's initial mass function
• The M-sigma Relation
• Gamma correction attributes the intensity of light to the voltage
• The law of two-thirds power, connecting speed to curvature in the human motor system.
• Taylor's law attributes the average population size and population size variance in ecology
• Behavior near second-order phase transitions involving critical exponents
• Proposed curve effect effect form
• The cosmic-ray core differential energy spectrum
• Fractal
• Pareto distribution and Pareto principles are also called "80-20 rules"
• Zipf laws in corpus and population distribution analysis, among others, where the frequency of a good or event is inversely proportional to the frequency rating (ie the second most frequent item/event occurs half as often as the most frequent, the third most frequent). items/events that often occur one-third as often as the most frequent items, and so on).
• A secure operating area is associated with the maximum simultaneous current and voltage in the power semiconductor.
• Supercritical state of matter and supercritical fluids, such as supercritical expansion of heat capacity and viscosity.
• Zeta Distribution (discrete)
• Yule-Simon Distribution (discrete)
• Student t distribution (continuous), where Cauchy distribution is a special case
• Lotka's Law
• Network model without scaling
• Pink sound
• Neuronal avalanches
• Law of the amount of current, and the law of long flow (Horton's law describing the river system)
• Urban population (Gibrat law)
• Bibliogram, and the frequency of words in text (Zipf law)
• 90-9-1 principle on the wiki (also referred to as 1% Rule)
• Artist distribution with an average price of their artwork.
• Richardson's law for violent conflict (war and terrorism) {Lewis Fry Richardson, Statistics from Deadly Quarrels, 1950}
• The relationship between CPU cache size and cache misses counts the law of Power of cache misses.
• The Law of Curie-von Schweidler in the dielectric response to input DC voltage input.
• Species of wealth (number of species) in freshwater fish clades (Albert, JS, HJ Bart, & RE Reis.. 2011. Species richness & amp; cladal diversity pp. Ã, 89-104 in Historical Biogeography of Water Bargaining Neotropic Fish (Albert, JS, & RE Reis, Eds.) University of California Press, Berkeley.)
• Damping force over speed relation in antiseismic damper calculus

Variant

Damaged power laws

Kekuasaan yang rusak hukum adalah fungsi piecewise, yang terdiri dari dua atau lebih hukum kekuatan, dikombinasikan dengan ambang batas. Misalnya, dengan dua undang-undang kekuatan:

${\ displaystyle f (x) \ propto x ^ {\ alpha _ {1}}}  ÂÂ$ untuk ${\ displaystyle x & lt; x_ {\ text {th}},}  ÂÂ$

${\ displaystyle f (x) \ propto x _ {\ text {th}} ^ {\ alpha_ {1} - \ alpha _ {2}} x ^ {\ alpha _ {2}} {\ text {for}} x & gt; x_ {\ text {th}}}  ÂÂ$ .

Hukum kuasa dengan pemutusan eksponensial

Kekuasaan hukum dengan cutoff eksponensial hanyalah hukum kekuatan dikalikan dengan fungsi eksponensial:

${\ displaystyle f (x) \ propto x ^ {\ alpha} e ^ {\ beta x}.}  ÂÂ$

Hukum kekuatan melengkung

${\ displaystyle f (x) \ propto x ^ {\ alpha \ beta x}}  ÂÂ$

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Distribusi probabilitas hukum-daya

Dalam arti yang lebih longgar, distribusi probabilitas hukum kekuatan adalah distribusi yang fungsi densitasnya (atau fungsi massa dalam kasus diskrit) memiliki bentuk, untuk nilai-nilai besar ${\ displaystyle x}  ÂÂ$ ,

${\ displaystyle P (X & gt; x) \ sim L (x) x ^ {- (\ alpha 1)}}  ÂÂ$

di mana pra-faktor untuk ${\ displaystyle {\ frac {\ alpha -1} {x _ {\ min}}}}  ÂÂ$ adalah konstanta normalisasi. Kami sekarang dapat mempertimbangkan beberapa properti dari distribusi ini. Misalnya, momennya diberikan oleh

${\ displaystyle \ langle x ^ {m} \ rangle = \ int _ {x _ {\ min}} ^ {\ infty} x ^ {m} p (x) \ , \ mathrm {d} x = {\ frac {\ alpha -1} {\ alpha -1-m}} x _ {\ min} ^ {m}}  ÂÂ$

yang hanya didefinisikan dengan baik untuk ${\ displaystyle m & lt; \ alpha -1}  ÂÂ$ . Yaitu, semua momen ${\ displaystyle m \ geq \ alpha -1}  ÂÂ$ menyimpang: ketika ${\ displaystyle \ alpha \ leq 2}  ÂÂ$ , rata-rata dan semua momen tingkat tinggi tidak terbatas; ketika ${\ displaystyle 2 & lt; \ alpha & lt; 3}  ÂÂ$ , artinya ada, tetapi varians dan momen tingkat tinggi tidak terbatas, dll. Untuk sampel ukuran terbatas yang diambil dari distribusi tersebut, perilaku ini menyiratkan bahwa pusat estimator momen (seperti mean dan varians) untuk saat-saat menyimpang tidak akan pernah menyatu - karena semakin banyak data yang terakumulasi, mereka akan terus tumbuh. Distribusi probabilitas power-law ini juga disebut distribusi tipe Pareto, distribusi dengan ekor Pareto, atau distribusi dengan ekor yang bervariasi secara teratur.

Modifikasi, yang tidak memenuhi bentuk umum di atas, dengan cutoff eksponensial, adalah

${\ displaystyle p (x) \ propto L (x) x ^ {- \ alpha} \ mathrm {e} ^ {- \ lambda x}.}  ÂÂ$

Dalam distribusi ini, istilah peluruhan eksponensial ${\ displaystyle \ mathrm {e} ^ {- \ lambda x}}  ÂÂ$ akhirnya menguasai perilaku hukum kekuatan pada nilai yang sangat besar ${\ displaystyle x}  ÂÂ$ . Distribusi ini tidak skala dan dengan demikian tidak asimtotik sebagai kekuatan hukum; Namun, itu kira-kira skala atas wilayah yang terbatas sebelum cutoff. (Perhatikan bahwa bentuk murni di atas adalah bagian dari keluarga ini, dengan ${\ displaystyle \ lambda = 0}  ÂÂ$ .) Distribusi ini adalah alternatif umum untuk distribusi kekuasaan-hukum asimtotik karena secara alami menangkap efek ukuran terbatas.

The Tweedie distribution is a family of statistical models characterized by closure under additive and reproductive conventions as well as lower-scale transformations. Consequently, all of these models express the legal relationship of forces between variance and mean. These models have a fundamental role as the focus of mathematical convergence is similar to the role that normal distribution has as a focus in the central limit theorem. This convergence effect explains why the law of variance-to-mean power manifests so widely in natural processes, such as Taylor's law in ecology and with the scale of fluctuations in physics. It can also be shown that this variance-to-mean law of power, when demonstrated by the method of expanding the waste, implies the presence of noise and that 1/ f noise can arise as a consequence of this Tweedie convergence effect.

Graphical methods for identification

Although more sophisticated and powerful methods have been proposed, the most commonly used graphical methods for identifying probability-power probabilities distributions using random samples are Quantitative-Quantile Pareto plots (or Pareto QQ plots), average plots of residual life and log-log plots. Another stronger graphical method uses a collection of leftover quartile functions. (Please note that the distribution of force law is also called Pareto-type distribution). Here it is assumed that a random sample is obtained from the probability distribution, and that we want to know whether the distribution tails follow the law of force (in other words, we want to know if the distribution has "Pareto tail"). Here, a random sample is called "data".

The pareto plot Q-Q compares the quintiles of the log-transformed data to the corresponding quantile of the exponential distribution with mean 1 (or to the quantile of the standard Pareto distribution) by plotting the previous versus the latter. If the resulting scatterplot indicates that points plotted "asymptotically merged" in a straight line, then the distribution of legal force should be suspected. The limitations of the Pareto QQ plot are that they behave badly when the tail index ${\ displaystyle \ alpha}$ (also called the Pareto index) approaches 0, because the Pareto Q-Q plot is not designed to identify distributions with slowly changing tails.

On the other hand, in its version to identify the probability-power distribution law, the residual residual life plot consists of first log-transforming data, and then plotting the mean of the log-transformed data higher than i - statistical sequence versus i -the statistical action, to i Ã, = Ã, 1, Ã,..., Ã, n , where n is a random sample size. If the resulting scatterplot shows that the plotted points tend to be "stable" about horizontal straight lines, then the distribution of legal force should be suspected. Since the average residual plot is very sensitive to emission (not strong), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots

Plot logs are an alternative way of graphically checking the tail of a distribution using a random sample. Attention should be done however as plot logs are required but not enough evidence for a power relations law, since many non-legal distributions of law will appear as a straight line on the log-log plot. This method consists of plotting the logarithm of the probability estimator that a certain number of distributions occur versus the logarithm of a particular number. Typically, this estimator is the proportion of times that number occurs in the data set. If the points in the plot tend to "converge" into straight lines for large quantities in the x-axis, then the researcher concludes that the distribution has a tail-law power. An example of applying this type of plot has been published. The disadvantage of these plots is that, in order for them to deliver reliable results, they require large amounts of data. In addition, they are only appropriate for discrete (or grouped) data.

Other graphical methods for identification of power-law probability distributions using random samples have been proposed. This methodology consists of plotting the bundle for the log-transformation sample . Originally proposed as a tool for exploring the existence of moments and functions of generations when using random samples, the bundle methodology was based on a residual quintile function (RQFs), also called the residual function of percentiles, which gives full characterization of tail behavior to many well known probability distributions, including distribution of legal powers, with other types of heavy tail, and even non-heavy-tailed distributions. The plot bundle has no disadvantages of the Pareto QQ plot, meaning the remaining plots and plots of the logs mentioned above (they are strong for slicing, allowing the identification of the law of visual forces with small values ​​ ${\ displaystyle \ alpha}$ , and does not require the collection of a lot of data). In addition, other types of tail behavior can be identified using bundles.

Designing power distribution laws

Secara umum, distribusi kekuasaan hukum diplot pada sumbu logaritmik ganda, yang menekankan wilayah ekor bagian atas. Cara paling mudah untuk melakukan ini adalah melalui distribusi kumulatif (komplementer) (cdf), ${\ displaystyle P (x) = \ mathrm {Pr} (X & gt; x)}  ÂÂ$ ,

Source of the article : Wikipedia