scipy.stats.pearson3¶
- scipy.stats.pearson3 = <scipy.stats._continuous_distns.pearson3_gen object at 0x2b2318ec4510>[source]¶
A pearson type III continuous random variable.
As an instance of the rv_continuous class, pearson3 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.
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
pearson3 takes skew as a shape parameter.
The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, pearson3.pdf(x, skew, loc, scale) is identically equivalent to pearson3.pdf(y, skew) / scale with y = (x - loc) / scale.
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, 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 = 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()
Methods
rvs(skew, loc=0, scale=1, size=1, random_state=None) 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 distribution function. logcdf(x, skew, loc=0, scale=1) Log of the cumulative distribution function. sf(x, skew, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is 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, args=(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