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
q : array_like
skew : array_like
loc : array_like, optional
scale : array_like, optional
size : int or tuple of ints, optional
moments : str, optional
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)
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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()
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 |