macroeco.models.dgamma

macroeco.models.dgamma = <macroeco.models._distributions.dgamma_gen object at 0x108653850>

A discrete gamma random variable.

\[P(x) = k * x^{(\alpha - 1)} * e^{(-1 / \theta)*x}\]

for x >= 1, \theta > 0. k is the normalizing constant. \alpha is analogous to the k_{agg} parameter in the zero-truncated negative binomial.

Parameters:

x : array_like

quantiles

q : array_like

lower or upper tail probability

alpha, theta : 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 = dgamma(alpha, theta, loc=0) :

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

alpha : float

distribution parameter

theta : float

distribution parameter

Notes

This parameterization of the discrete gamma was taken from [1].

References

[1]	Frank, F. (2011). Measurement scale in maximum entropy models of species abundance. Journal of Evolutionary Biology, 24(3), 485-496

Examples

>>> import macroeco.models as md

>>> # dgamma takes two parameters
>>> dgamma_dist = md.dgamma(alpha=1, theta=2)

>>> # When alpha = 1 and theta = 2, should be similar to a to
>>> # zero_truncated NBD with mu = 2.541.
>>> dgamma_dist.pmf(np.arange(1, 10))
array([ 0.39346934,  0.23865122,  0.14474928,  0.08779488,  0.05325028,
    0.03229793,  0.01958968,  0.01188174,  0.00720664])

>>> md.nbinom_ztrunc.pmf(np.arange(1, 10), mu=2.541, k_agg=1)
array([ 0.39354585,  0.23866751,  0.1447409 ,  0.08777872,  0.05323377,
    0.03228384,  0.01957867,  0.01187357,  0.00720077])

>>> # Get the approximate mean and variance
>>> dgamma_dist.stats()
(array(2.541494082536799), array(3.917698089032762))

>>> # Draw random numbers from the discrete gamma distribution
>>> dgamma_dist.rvs(size=20)
array([5, 4, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 1, 3])

Methods

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