scipy.stats.fisk#
- scipy.stats.fisk = <scipy.stats._continuous_distns.fisk_gen object>[source]#
A Fisk continuous random variable.
The Fisk distribution is also known as the log-logistic distribution.
As an instance of the
rv_continuous
class,fisk
object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.See also
Notes
The probability density function for
fisk
is:\[f(x, c) = \frac{c x^{c-1}} {(1 + x^c)^2}\]for \(x >= 0\) and \(c > 0\).
Please note that the above expression can be transformed into the following one, which is also commonly used:
\[f(x, c) = \frac{c x^{-c-1}} {(1 + x^{-c})^2}\]fisk
takesc
as a shape parameter for \(c\).fisk
is a special case ofburr
orburr12
withd=1
.The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the
loc
andscale
parameters. Specifically,fisk.pdf(x, c, loc, scale)
is identically equivalent tofisk.pdf(y, c) / scale
withy = (x - loc) / scale
. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.Examples
>>> from scipy.stats import fisk >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> c = 3.09 >>> mean, var, skew, kurt = fisk.stats(c, moments='mvsk')
Display the probability density function (
pdf
):>>> x = np.linspace(fisk.ppf(0.01, c), ... fisk.ppf(0.99, c), 100) >>> ax.plot(x, fisk.pdf(x, c), ... 'r-', lw=5, alpha=0.6, label='fisk pdf')
Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.
Freeze the distribution and display the frozen
pdf
:>>> rv = fisk(c) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of
cdf
andppf
:>>> vals = fisk.ppf([0.001, 0.5, 0.999], c) >>> np.allclose([0.001, 0.5, 0.999], fisk.cdf(vals, c)) True
Generate random numbers:
>>> r = fisk.rvs(c, size=1000)
And compare the histogram:
>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()
Methods
rvs(c, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, c, loc=0, scale=1)
Probability density function.
logpdf(x, c, loc=0, scale=1)
Log of the probability density function.
cdf(x, c, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, c, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, c, loc=0, scale=1)
Survival function (also defined as
1 - cdf
, but sf is sometimes more accurate).logsf(x, c, loc=0, scale=1)
Log of the survival function.
ppf(q, c, loc=0, scale=1)
Percent point function (inverse of
cdf
— percentiles).isf(q, c, loc=0, scale=1)
Inverse survival function (inverse of
sf
).moment(order, c, loc=0, scale=1)
Non-central moment of the specified order.
stats(c, loc=0, scale=1, moments=’mv’)
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(c, loc=0, scale=1)
(Differential) entropy of the RV.
fit(data)
Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(c,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
Expected value of a function (of one argument) with respect to the distribution.
median(c, loc=0, scale=1)
Median of the distribution.
mean(c, loc=0, scale=1)
Mean of the distribution.
var(c, loc=0, scale=1)
Variance of the distribution.
std(c, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, c, loc=0, scale=1)
Confidence interval with equal areas around the median.