Fandom

Psychology Wiki

Zipf–Mandelbrot law

34,203pages on
this wiki
Add New Page
Talk0 Share

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory


Zipf–Mandelbrot
Probability mass function
Cumulative distribution function
Parameters N \in \{1,2,3\ldots\} (integer)
q \in [0;\infty) (real)
s>0\, (real)
Support k \in \{1,2,\ldots,N\}
Template:Probability distribution/link mass \frac{1/(k+q)^s}{H_{N,q,s}}
cdf \frac{H_{k,q,s}}{H_{N,q,s}}
Mean \frac{H_{N,q,s-1}}{H_{N,q,s}}-q
Median
Mode 1\,
Variance
Skewness
Kurtosis
Entropy
mgf
Char. func.

In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot, who subsequently generalized it.

The probability mass function is given by:

f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}

where H_{N,q,s} is given by:

H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}

which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as N approaches infinity, this becomes the Hurwitz zeta function \zeta(q,s). For finite N and q=0 the Zipf–Mandelbrot law becomes Zipf's law. For infinite N and q=0 it becomes a Zeta distribution.

ApplicationsEdit

The distribution of words ranked by their frequency in a random text corpus is generally a power-law distribution, known as Zipf's law.

If one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Gelbukh & Sidorov, 2001).

In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.[1]

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.[2]

NotesEdit

  1. Mouillot, D, Lepretre, A (2000). Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity. Environmental Monitoring and Assessment 63 (2): 279–295.
  2. Manris, B, Vaughan, D, Wagner, CS, Romero, J, Davis, RB. Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music. Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003) 611.

ReferencesEdit

  • Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics" B.B. Wolman and E. Nagel Scientific psychology, Basic Books. Reprinted as
    • Mandelbrot, Benoît [1965] (1968). "Information Theory and Psycholinguistics" R.C. Oldfield and J.C. Marchall Language, Penguin Books.
  • Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language, Cambridge, MA: Harvard University Press.

External linksEdit

Bvn-small Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound PoissondegeneratedegreeGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normalvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropy phase-typeposterior priorquasisampling
</center>


This page uses Creative Commons Licensed content from Wikipedia (view authors).

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki