zm {UCS}R Documentation

The Zipf-Mandelbrot LNRE Model (zm)


Object constructor for a Zipf-Mandelbrot (ZM) LNRE model with parameters α and C (Evert, 2004a). Either the parameters are specified explicitly, or one or both of them can be estimated from an observed frequency spectrum.


zm(alpha=NULL, C=NULL, N=NULL, V=NULL, spc=NULL, m.max=15, stepmax=10, debug=FALSE)


zm(alpha, C)

zm(alpha, N, V)

zm(N, V, spc, m.max=15, stepmax=10, debug=FALSE)


alpha a number in the range (0,1), the shape parameter α of the ZM model. alpha can automatically be estimated from N, V, and spc.
C a positive number, the parameter C of the ZM model. C can automatically be estimated from N and V.
N the sample size, i.e. number of observed tokens
V the vocabulary size, i.e. the number of observed types
spc a vector of non-negative integers representing the class sizes V_m of the observed frequency spectrum. The vector is usually read from a file in lexstats format with the read.spectrum function.
m.max the number of ranks from spc that will be used to estimate the α parameter
stepmax maximal step size of the nlm function used for parameter estimation. It should not be necessary to change the default value.
debug if TRUE, print debugging information during the parameter estimation process. This feature can be useful to find out why parameter estimation fails.


The ZM model with parameters α \in (0,1) and C > 0 is defined by the type density function

g(p) := C * p^(-alpha - 1)

for 0 <= p <= B, where the upper bound B is determined from C by the normalisation condition

integral_0^Inf p * g(p) dp = 1

The parameter α is estimated by nonlinear minimisation (nlm) of a multinomial chi-squared statistic for the observed against the expected frequency spectrum. Note that this is different from the multivariate chi-squared test used to measure the goodness-of-fit of the final model (Baayen, 2001, Sec. 3.3).

See Evert (2004, Ch. 4) for further mathematical details, especially concerning the expected vocabulary size, frequency spectrum and conditional parameter distribution, as well as their variances.


An object of class "zm" with the following components:
alpha value of the α parameter
B value of the upper bound B (a normalisation device)
C value of the C parameter
N number of observed tokens (if specified)
V number of observed types (if specified)
spc observed frequency spectrum (if specified)
This object prints a short summary, including a comparison of the first ranks of the observed and expected frequency spectrum (if available).


Baayen, R. Harald (2001). Word Frequency Distributions. Kluwer, Dordrecht.

Evert, Stefan (2004). The Statistics of Word Cooccurrences: Word Pairs and Collocations. PhD Thesis, IMS, University of Stuttgart.

Evert, Stefan (2004a). A simple LNRE model for random character sequences. In Proceedings of JADT 2004, Louvain-la-Neuve, Belgium, pages 411–422.

See Also

fzm, EV, EVm, VV, VVm, write.lexstats,, read.spectrum, and spectrum.plot

[Package UCS version 0.5 Index]