SciPy

scipy.stats.pearson3

scipy.stats.pearson3 = <scipy.stats._continuous_distns.pearson3_gen object at 0x4e07e90>[source]

A pearson type III continuous random variable.

Continuous random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:

Parameters :

x : array_like

quantiles

q : array_like

lower or upper tail probability

skew : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

scale : array_like, optional

scale parameter (default=1)

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, :

location, and scale parameters returning a “frozen” continuous RV object: :

rv = pearson3(skew, loc=0, scale=1) :

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

Notes

The probability density function for pearson3 is:

pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
    (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))

where:

beta = 2 / (skew * stddev)
alpha = (stddev * beta)**2
zeta = loc - alpha / beta

References

R.W. Vogel and D.E. McMartin, “Probability Plot Goodness-of-Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution”, Water Resources Research, Vol.27, 3149-3158 (1991).

L.R. Salvosa, “Tables of Pearson’s Type III Function”, Ann. Math. Statist., Vol.1, 191-198 (1930).

“Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data”, Office of Aviation Research (2003).

Examples

>>> from scipy.stats import pearson3
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> skew = 0.1
>>> mean, var, skew, kurt = pearson3.stats(skew, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(pearson3.ppf(0.01, skew),
...               pearson3.ppf(0.99, skew), 100)
>>> ax.plot(x, pearson3.pdf(x, skew),
...          'r-', lw=5, alpha=0.6, label='pearson3 pdf')

Alternatively, freeze the distribution and display the frozen pdf:

>>> rv = pearson3(skew)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = pearson3.ppf([0.001, 0.5, 0.999], skew)
>>> np.allclose([0.001, 0.5, 0.999], pearson3.cdf(vals, skew))
True

Generate random numbers:

>>> r = pearson3.rvs(skew, size=1000)

And compare the histogram:

>>> ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

(Source code)

../_images/scipy-stats-pearson3-1.png

Methods

rvs(skew, loc=0, scale=1, size=1) Random variates.
pdf(x, skew, loc=0, scale=1) Probability density function.
logpdf(x, skew, loc=0, scale=1) Log of the probability density function.
cdf(x, skew, loc=0, scale=1) Cumulative density function.
logcdf(x, skew, loc=0, scale=1) Log of the cumulative density function.
sf(x, skew, loc=0, scale=1) Survival function (1-cdf — sometimes more accurate).
logsf(x, skew, loc=0, scale=1) Log of the survival function.
ppf(q, skew, loc=0, scale=1) Percent point function (inverse of cdf — percentiles).
isf(q, skew, loc=0, scale=1) Inverse survival function (inverse of sf).
moment(n, skew, loc=0, scale=1) Non-central moment of order n
stats(skew, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(skew, loc=0, scale=1) (Differential) entropy of the RV.
fit(data, skew, loc=0, scale=1) Parameter estimates for generic data.
expect(func, skew, 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(skew, loc=0, scale=1) Median of the distribution.
mean(skew, loc=0, scale=1) Mean of the distribution.
var(skew, loc=0, scale=1) Variance of the distribution.
std(skew, loc=0, scale=1) Standard deviation of the distribution.
interval(alpha, skew, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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