macroeco.models.logser_uptrunc = <macroeco.models._distributions.logser_uptrunc_gen object at 0x1086537d0>

Upper truncated logseries random variable.

This distribution was described by Harte (2011) [1]

\[p(x) = \frac{1}{Z} \frac{p^n}{n}\]

where Z is the normalizing factor


x : array_like


q : array_like

lower or upper tail probability

p, b : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)

Alternatively, the object may be called (as a function) to fix the shape and :

location parameters returning a “frozen” discrete RV object: :

rv = logser_uptrunc(p, b, loc=0) :

  • Frozen RV object with the same methods but holding the given shape and location fixed.

p : float

p parameter of the logseries distribution

b : float

Upper bound of the distribution


Code adapted from Ethan White’s macroecology_tools and version 0.1 of macroeco


[1]Harte, J. (2011). Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford, United Kingdom: Oxford University Press.


>>> import macroeco.models as md
>>> # Define a logseries distribution by specifying the necessary parameters
>>> logser_dist = md.logser_uptrunc(p=0.9, b=1000)
>>> # Get the pmf
>>> logser_dist.pmf(1)
>>> # Get the cdf
>>> logser_dist.cdf(10)
>>> # You can also use the following notation
>>> md.logser_uptrunc.pmf(1, 0.9, 1000)
>>> md.logser_uptrunc.cdf(10, 0.9, 1000)
>>> # Get a rank abundance distribution for 30 species
>>> rad = md.logser_uptrunc.rank(30, 0.9, 1000)
>>> rad
array([  1.,   1.,   1.,   1.,   1.,   1.,   1.,   1.,   1.,   1.,   1.,
     1.,   2.,   2.,   2.,   2.,   2.,   3.,   3.,   3.,   4.,   4.,
     5.,   5.,   6.,   7.,   8.,  10.,  13.,  21.])
>>> # Fit the logser_uptrunc to data and estimate the parameters
>>> md.logser_uptrunc.fit_mle(rad)
(0.8957385644316679, 114.0)


rvs(p, b, loc=0, size=1) Random variates.
pmf(x, p, b, loc=0) Probability mass function.
logpmf(x, p, b, loc=0) Log of the probability mass function.
cdf(x, p, b, loc=0) Cumulative density function.
logcdf(x, p, b, loc=0) Log of the cumulative density function.
sf(x, p, b, loc=0) Survival function (1-cdf — sometimes more accurate).
logsf(x, p, b, loc=0) Log of the survival function.
ppf(q, p, b, loc=0) Percent point function (inverse of cdf — percentiles).
isf(q, p, b, loc=0) Inverse survival function (inverse of sf).
stats(p, b, loc=0, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(p, b, loc=0) (Differential) entropy of the RV.
expect(func, p, b, loc=0, lb=None, ub=None, conditional=False) Expected value of a function (of one argument) with respect to the distribution.
median(p, b, loc=0) Median of the distribution.
mean(p, b, loc=0) Mean of the distribution.
var(p, b, loc=0) Variance of the distribution.
std(p, b, loc=0) Standard deviation of the distribution.
interval(alpha, p, b, loc=0) Endpoints of the range that contains alpha percent of the distribution