scipy.stats.pearson3#
- scipy.stats.pearson3 = <scipy.stats._continuous_distns.pearson3_gen object>[source]#
- A pearson type III continuous random variable. - As an instance of the - rv_continuousclass,- pearson3object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- 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(order, skew, loc=0, scale=1) - Non-central moment of the specified order. - 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) - Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. - 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(confidence, skew, loc=0, scale=1) - Confidence interval with equal areas around the median. - Notes - The probability density function for - pearson3is:\[f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)} (\beta (x - \zeta))^{\alpha - 1} \exp(-\beta (x - \zeta))\]- where: \[ \begin{align}\begin{aligned}\beta = \frac{2}{\kappa}\\\alpha = \beta^2 = \frac{4}{\kappa^2}\\\zeta = -\frac{\alpha}{\beta} = -\beta\end{aligned}\end{align} \]- \(\Gamma\) is the gamma function ( - scipy.special.gamma). Pass the skew \(\kappa\) into- pearson3as the shape parameter- skew.- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- pearson3.pdf(x, skew, loc, scale)is identically equivalent to- pearson3.pdf(y, skew) / scalewith- y = (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.- 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 - >>> import numpy as np >>> from scipy.stats import pearson3 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> skew = -2 >>> lb, ub = pearson3.support(skew) - Calculate the first four moments: - >>> 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 - cdfand- 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, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show() 