The Zipf-Mandelbrot Distribution
Density, distribution function, quantile function and random generation for the Mandelbrot distribution.
dzipfmb(x, shape, start = 1, log = FALSE) pzipfmb(q, shape, start = 1, lower.tail = TRUE, log.p = FALSE) qzipfmb(p, shape, start = 1) rzipfmb(n, shape, start = 1)
x |
vector of (non-negative integer) quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of random values to return. |
shape |
vector of positive shape parameter. |
start |
integer, the minimum value of the support of the distribution. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p) |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]. |
The probability mass function of the zipf-mandelbrot distribution is given by
P(Y=y) = ((b)Gamma(a))/(Gamma(a-b)) * Gamma(y+-b)/Gamma(y+1)
where 0<=b<1 and the starting value start being by default 1.
dzipfmb gives the density,
pzipfmb gives the distribution function,
qzipfmb gives the quantile function, and
rzipfmb generates random deviates.
M. Chou, with edits by T. W. Yee.
Mandelbrot, B. (1961). On the theory of word frequencies and on related Markovian models of discourse. In R. Jakobson, Structure of Language and its Mathematical Aspects, pp. 190–219, Providence, RI, USA. American Mathematical Society.
Moreno-Sanchez, I. and Font-Clos, F. and Corral, A. (2016). Large-Scale Analysis of Zipf's Law in English Texts. PLos ONE, 11(1), 1–19.
Zipf.
aa <- 1:10
(pp <- pzipfmb(aa, shape = 0.5, start = 1))
cumsum(dzipfmb(aa, shape = 0.5, start = 1)) # Should be same
qzipfmb(pp, shape = 0.5, start = 1) - aa # Should be all 0s
rdiffzeta(30, 0.5)
## Not run:
x <- 1:10
plot(x, dzipfmb(x, shape = 0.5), type = "h", ylim = 0:1,
sub = "shape=0.5", las = 1, col = "blue", ylab = "Probability",
main = "Zipf-Mandelbrot distribution: blue=PMF; orange=CDF")
lines(x + 0.1, pzipfmb(x, shape = 0.8), col = "orange", lty = 3, type = "h")
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.