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580 lines
19 KiB
Python
580 lines
19 KiB
Python
from __future__ import absolute_import
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from scipy.stats._distn_infrastructure import * # @UnusedWildImport
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from scipy.stats._distn_infrastructure import (_skew, # @UnusedImport
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_kurtosis, _lazywhere, _ncx2_log_pdf, # @IgnorePep8 @UnusedImport
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_ncx2_pdf, _ncx2_cdf) # @UnusedImport @IgnorePep8
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from .estimation import FitDistribution
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from ._constants import _XMAX
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_doc_default_example = """\
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Examples
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--------
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>>> from wafo.stats import %(name)s
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>>> import matplotlib.pyplot as plt
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>>> fig, ax = plt.subplots(1, 1)
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Calculate a few first moments:
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%(set_vals_stmt)s
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>>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
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Display the probability density function (``pdf``):
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>>> x = np.linspace(%(name)s.ppf(0.01, %(shapes)s),
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... %(name)s.ppf(0.99, %(shapes)s), 100)
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>>> ax.plot(x, %(name)s.pdf(x, %(shapes)s),
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... 'r-', lw=5, alpha=0.6, label='%(name)s pdf')
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Alternatively, the distribution object can be called (as a function)
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to fix the shape, location and scale parameters. This returns a "frozen"
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RV object holding the given parameters fixed.
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Freeze the distribution and display the frozen ``pdf``:
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>>> rv = %(name)s(%(shapes)s)
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>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
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Check accuracy of ``cdf`` and ``ppf``:
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>>> vals = %(name)s.ppf([0.001, 0.5, 0.999], %(shapes)s)
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>>> np.allclose([0.001, 0.5, 0.999], %(name)s.cdf(vals, %(shapes)s))
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True
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Generate random numbers:
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>>> r = %(name)s.rvs(%(shapes)s, size=1000)
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And compare the histogram:
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>>> ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2)
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>>> ax.legend(loc='best', frameon=False)
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>>> plt.show()
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Compare ML and MPS method
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>>> phat = %(name)s.fit2(R, method='ml');>>> phat.plotfitsummary()
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>>> plt.figure(plt.gcf().number+1)
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>>> phat2 = %(name)s.fit2(R, method='mps')
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>>> phat2.plotfitsummary(); plt.figure(plt.gcf().number+1)
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Fix loc=0 and estimate shapes and scale
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>>> phat3 = %(name)s.fit2(R, scale=1, floc=0, method='mps')
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>>> phat3.plotfitsummary(); plt.figure(plt.gcf().number+1)
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Accurate confidence interval with profile loglikelihood
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>>> lp = phat3.profile()
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>>> lp.plot()
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>>> pci = lp.get_bounds()
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"""
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# Frozen RV class
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class rv_frozen(object):
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''' Frozen continous or discrete 1D Random Variable object (RV)
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Methods
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-------
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rvs(size=1)
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Random variates.
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pdf(x)
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Probability density function.
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cdf(x)
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Cumulative density function.
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sf(x)
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Survival function (1-cdf --- sometimes more accurate).
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ppf(q)
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Percent point function (inverse of cdf --- percentiles).
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isf(q)
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Inverse survival function (inverse of sf).
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stats(moments='mv')
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Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
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moment(n)
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n-th order non-central moment of distribution.
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entropy()
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(Differential) entropy of the RV.
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interval(alpha)
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Confidence interval with equal areas around the median.
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expect(func, lb, ub, conditional=False)
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Calculate expected value of a function with respect to the
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distribution.
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'''
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def __init__(self, dist, *args, **kwds):
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# create a new instance
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self.dist = dist # .__class__(**dist._ctor_param)
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shapes, loc, scale = self.dist._parse_args(*args, **kwds)
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if isinstance(dist, rv_continuous):
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self.par = shapes + (loc, scale)
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else: # rv_discrete
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self.par = shapes + (loc,)
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self.a = self.dist.a
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self.b = self.dist.b
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self.shapes = self.dist.shapes
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# @property
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# def shapes(self):
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# return self.dist.shapes
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@property
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def random_state(self):
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return self.dist._random_state
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@random_state.setter
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def random_state(self, seed):
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self.dist._random_state = check_random_state(seed)
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def pdf(self, x):
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''' Probability density function at x of the given RV.'''
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return self.dist.pdf(x, *self.par)
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def logpdf(self, x):
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return self.dist.logpdf(x, *self.par)
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def cdf(self, x):
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'''Cumulative distribution function at x of the given RV.'''
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return self.dist.cdf(x, *self.par)
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def logcdf(self, x):
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return self.dist.logcdf(x, *self.par)
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def ppf(self, q):
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'''Percent point function (inverse of cdf) at q of the given RV.'''
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return self.dist.ppf(q, *self.par)
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def isf(self, q):
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'''Inverse survival function at q of the given RV.'''
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return self.dist.isf(q, *self.par)
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def rvs(self, size=None, random_state=None):
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kwds = {'size': size, 'random_state': random_state}
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return self.dist.rvs(*self.par, **kwds)
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def sf(self, x):
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'''Survival function (1-cdf) at x of the given RV.'''
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return self.dist.sf(x, *self.par)
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def logsf(self, x):
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return self.dist.logsf(x, *self.par)
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def stats(self, moments='mv'):
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''' Some statistics of the given RV'''
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kwds = dict(moments=moments)
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return self.dist.stats(*self.par, **kwds)
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def median(self):
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return self.dist.median(*self.par)
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def mean(self):
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return self.dist.mean(*self.par)
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def var(self):
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return self.dist.var(*self.par)
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def std(self):
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return self.dist.std(*self.par)
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def moment(self, n):
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return self.dist.moment(n, *self.par)
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def entropy(self):
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return self.dist.entropy(*self.par)
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def pmf(self, k):
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'''Probability mass function at k of the given RV'''
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return self.dist.pmf(k, *self.par)
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def logpmf(self, k):
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return self.dist.logpmf(k, *self.par)
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def interval(self, alpha):
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return self.dist.interval(alpha, *self.par)
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def expect(self, func=None, lb=None, ub=None, conditional=False, **kwds):
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if isinstance(self.dist, rv_continuous):
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a, loc, scale = self.par[:-2], self.par[:-2], self.par[-1]
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return self.dist.expect(func, a, loc, scale, lb, ub, conditional,
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**kwds)
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a, loc = self.par[:-1], self.par[-1]
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if kwds:
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raise ValueError("Discrete expect does not accept **kwds.")
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return self.dist.expect(func, a, loc, lb, ub, conditional)
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def freeze(self, *args, **kwds):
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"""Freeze the distribution for the given arguments.
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Parameters
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----------
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arg1, arg2, arg3,... : array_like
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The shape parameter(s) for the distribution. Should include all
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the non-optional arguments, may include ``loc`` and ``scale``.
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Returns
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-------
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rv_frozen : rv_frozen instance
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The frozen distribution.
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"""
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return rv_frozen(self, *args, **kwds)
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def link(self, x, logSF, theta, i):
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'''
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Return theta[i] as function of quantile, survival probability and
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theta[j] for j!=i.
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Parameters
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----------
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x : quantile
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logSF : logarithm of the survival probability
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theta : list
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all distribution parameters including location and scale.
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Returns
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-------
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theta[i] : real scalar
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fixed distribution parameter theta[i] as function of x, logSF and
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theta[j] where j != i.
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LINK is a function connecting the fixed distribution parameter theta[i]
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with the quantile (x) and the survival probability (SF) and the
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remaining free distribution parameters theta[j] for j!=i, i.e.:
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theta[i] = link(x, logSF, theta, i),
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where logSF = log(Prob(X>x; theta)).
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See also
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estimation.Profile
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'''
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return self._link(x, logSF, theta, i)
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def _link(self, x, logSF, theta, i):
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msg = ('Link function not implemented for the %s distribution' %
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self.name)
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raise NotImplementedError(msg)
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def nlogps(self, theta, x):
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""" Moran's negative log Product Spacings statistic
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where theta are the parameters (including loc and scale)
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Note the data in x must be sorted
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References
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-----------
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R. C. H. Cheng; N. A. K. Amin (1983)
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"Estimating Parameters in Continuous Univariate Distributions with a
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Shifted Origin.",
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Journal of the Royal Statistical Society. Series B (Methodological),
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Vol. 45, No. 3. (1983), pp. 394-403.
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R. C. H. Cheng; M. A. Stephens (1989)
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"A Goodness-Of-Fit Test Using Moran's Statistic with Estimated
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Parameters", Biometrika, 76, 2, pp 385-392
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Wong, T.S.T. and Li, W.K. (2006)
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"A note on the estimation of extreme value distributions using maximum
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product of spacings.",
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IMS Lecture Notes Monograph Series 2006, Vol. 52, pp. 272-283
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"""
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try:
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loc = theta[-2]
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scale = theta[-1]
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args = tuple(theta[:-2])
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except IndexError:
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raise ValueError("Not enough input arguments.")
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if not self._argcheck(*args) or scale <= 0:
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return inf
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x = asarray((x - loc) / scale)
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cond0 = (x <= self.a) | (self.b <= x)
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Nbad = np.sum(cond0)
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if Nbad > 0:
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x = argsreduce(~cond0, x)[0]
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lowertail = True
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if lowertail:
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prb = np.hstack((0.0, self.cdf(x, *args), 1.0))
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dprb = np.diff(prb)
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else:
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prb = np.hstack((1.0, self.sf(x, *args), 0.0))
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dprb = -np.diff(prb)
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logD = log(dprb)
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dx = np.diff(x, axis=0)
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tie = (dx == 0)
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if any(tie):
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# TODO : implement this method for treating ties in data:
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# Assume measuring error is delta. Then compute
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# yL = F(xi-delta,theta)
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# yU = F(xi+delta,theta)
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# and replace
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# logDj = log((yU-yL)/(r-1)) for j = i+1,i+2,...i+r-1
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# The following is OK when only minimization of T is wanted
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i_tie, = np.nonzero(tie)
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tiedata = x[i_tie]
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logD[i_tie + 1] = log(self._pdf(tiedata, *args)) - log(scale)
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finiteD = np.isfinite(logD)
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nonfiniteD = 1 - finiteD
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Nbad += np.sum(nonfiniteD, axis=0)
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if Nbad > 0:
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T = -np.sum(logD[finiteD], axis=0) + 100.0 * np.log(_XMAX) * Nbad
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else:
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T = -np.sum(logD, axis=0)
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return T
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def _reduce_func(self, args, options):
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# First of all, convert fshapes params to fnum: eg for stats.beta,
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# shapes='a, b'. To fix `a`, can specify either `f1` or `fa`.
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# Convert the latter into the former.
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kwds = options.copy()
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if self.shapes:
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shapes = self.shapes.replace(',', ' ').split()
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for j, s in enumerate(shapes):
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val = kwds.pop('f' + s, None) or kwds.pop('fix_' + s, None)
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if val is not None:
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key = 'f%d' % j
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if key in kwds:
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raise ValueError("Duplicate entry for %s." % key)
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else:
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kwds[key] = val
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args = list(args)
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Nargs = len(args)
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fixedn = []
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names = ['f%d' % n for n in range(Nargs - 2)] + ['floc', 'fscale']
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x0 = []
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for n, key in enumerate(names):
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if key in kwds:
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fixedn.append(n)
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args[n] = kwds.pop(key)
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else:
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x0.append(args[n])
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method = kwds.pop('method', 'ml').lower()
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if method.startswith('mps'):
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fitfun = self.nlogps
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else:
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fitfun = self._penalized_nnlf
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if len(fixedn) == 0:
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func = fitfun
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restore = None
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else:
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if len(fixedn) == Nargs:
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raise ValueError(
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"All parameters fixed. There is nothing to optimize.")
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def restore(args, theta):
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# Replace with theta for all numbers not in fixedn
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# This allows the non-fixed values to vary, but
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# we still call self.nnlf with all parameters.
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i = 0
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for n in range(Nargs):
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if n not in fixedn:
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args[n] = theta[i]
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i += 1
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return args
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def func(theta, x):
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newtheta = restore(args[:], theta)
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return fitfun(newtheta, x)
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return x0, func, restore, args
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def fit(self, data, *args, **kwargs):
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"""
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Return ML/MPS estimate for shape, location, and scale parameters from data.
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ML and MPS stands for Maximum Likelihood and Maximum Product Spacing,
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respectively. Starting estimates for
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the fit are given by input arguments; for any arguments not provided
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with starting estimates, ``self._fitstart(data)`` is called to generate
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such.
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One can hold some parameters fixed to specific values by passing in
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keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)
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and ``floc`` and ``fscale`` (for location and scale parameters,
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respectively).
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Parameters
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----------
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data : array_like
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Data to use in calculating the MLEs.
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args : floats, optional
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Starting value(s) for any shape-characterizing arguments (those not
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provided will be determined by a call to ``_fitstart(data)``).
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No default value.
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kwds : floats, optional
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Starting values for the location and scale parameters; no default.
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Special keyword arguments are recognized as holding certain
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parameters fixed:
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- f0...fn : hold respective shape parameters fixed.
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Alternatively, shape parameters to fix can be specified by name.
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For example, if ``self.shapes == "a, b"``, ``fa``and ``fix_a``
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are equivalent to ``f0``, and ``fb`` and ``fix_b`` are
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equivalent to ``f1``.
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- floc : hold location parameter fixed to specified value.
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- fscale : hold scale parameter fixed to specified value.
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- optimizer : The optimizer to use. The optimizer must take ``func``,
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and starting position as the first two arguments,
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plus ``args`` (for extra arguments to pass to the
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function to be optimized) and ``disp=0`` to suppress
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output as keyword arguments.
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Returns
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-------
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shape, loc, scale : tuple of floats
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MLEs for any shape statistics, followed by those for location and
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scale.
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Notes
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-----
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This fit is computed by maximizing a log-likelihood function, with
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penalty applied for samples outside of range of the distribution. The
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returned answer is not guaranteed to be the globally optimal MLE, it
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may only be locally optimal, or the optimization may fail altogether.
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Examples
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--------
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Generate some data to fit: draw random variates from the `beta`
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distribution
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>>> from wafo.stats import beta
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>>> a, b = 1., 2.
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>>> x = beta.rvs(a, b, size=1000)
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Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):
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>>> a1, b1, loc1, scale1 = beta.fit(x)
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We can also use some prior knowledge about the dataset: let's keep
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``loc`` and ``scale`` fixed:
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>>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)
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>>> loc1, scale1
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(0, 1)
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We can also keep shape parameters fixed by using ``f``-keywords. To
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keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,
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equivalently, ``fa=1``:
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>>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)
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>>> a1
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1
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"""
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Narg = len(args)
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if Narg > self.numargs:
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raise TypeError("Too many input arguments.")
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kwds = kwargs.copy()
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start = [None]*2
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if (Narg < self.numargs) or not ('loc' in kwds and
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'scale' in kwds):
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# get distribution specific starting locations
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start = self._fitstart(data)
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args += start[Narg:-2]
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loc = kwds.pop('loc', start[-2])
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scale = kwds.pop('scale', start[-1])
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args += (loc, scale)
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x0, func, restore, args = self._reduce_func(args, kwds)
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optimizer = kwds.pop('optimizer', optimize.fmin)
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# convert string to function in scipy.optimize
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if not callable(optimizer) and isinstance(optimizer, string_types):
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if not optimizer.startswith('fmin_'):
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optimizer = "fmin_"+optimizer
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if optimizer == 'fmin_':
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optimizer = 'fmin'
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try:
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optimizer = getattr(optimize, optimizer)
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except AttributeError:
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raise ValueError("%s is not a valid optimizer" % optimizer)
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# by now kwds must be empty, since everybody took what they needed
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|
if kwds:
|
|
raise TypeError("Unknown arguments: %s." % kwds)
|
|
|
|
vals = optimizer(func, x0, args=(ravel(data),), disp=0)
|
|
if restore is not None:
|
|
vals = restore(args, vals)
|
|
vals = tuple(vals)
|
|
return vals
|
|
|
|
|
|
def fit2(self, data, *args, **kwds):
|
|
''' Return Maximum Likelihood or Maximum Product Spacing estimator object
|
|
|
|
Parameters
|
|
----------
|
|
data : array-like
|
|
Data to use in calculating the ML or MPS estimators
|
|
args : optional
|
|
Starting values for any shape arguments (those not specified
|
|
will be determined by dist._fitstart(data))
|
|
kwds : loc, scale
|
|
Starting values for the location and scale parameters
|
|
Special keyword arguments are recognized as holding certain
|
|
parameters fixed:
|
|
f0..fn : hold respective shape paramters fixed
|
|
floc : hold location parameter fixed to specified value
|
|
fscale : hold scale parameter fixed to specified value
|
|
method : of estimation. Options are
|
|
'ml' : Maximum Likelihood method (default)
|
|
'mps': Maximum Product Spacing method
|
|
alpha : scalar, optional
|
|
Confidence coefficent (default=0.05)
|
|
search : bool
|
|
If true search for best estimator (default),
|
|
otherwise return object with initial distribution parameters
|
|
copydata : bool
|
|
If true copydata (default)
|
|
optimizer : The optimizer to use. The optimizer must take func,
|
|
and starting position as the first two arguments,
|
|
plus args (for extra arguments to pass to the
|
|
function to be optimized) and disp=0 to suppress
|
|
output as keyword arguments.
|
|
|
|
Return
|
|
------
|
|
phat : FitDistribution object
|
|
Fitted distribution object with following member variables:
|
|
LLmax : loglikelihood function evaluated using par
|
|
LPSmax : log product spacing function evaluated using par
|
|
pvalue : p-value for the fit
|
|
par : distribution parameters (fixed and fitted)
|
|
par_cov : covariance of distribution parameters
|
|
par_fix : fixed distribution parameters
|
|
par_lower : lower (1-alpha)% confidence bound for the parameters
|
|
par_upper : upper (1-alpha)% confidence bound for the parameters
|
|
|
|
Note
|
|
----
|
|
`data` is sorted using this function, so if `copydata`==False the data
|
|
in your namespace will be sorted as well.
|
|
'''
|
|
return FitDistribution(self, data, args, **kwds)
|
|
|
|
|
|
rv_generic.freeze = freeze
|
|
rv_discrete.freeze = freeze
|
|
rv_continuous.freeze = freeze
|
|
rv_continuous.link = link
|
|
rv_continuous._link = _link
|
|
rv_continuous.nlogps = nlogps
|
|
rv_continuous._reduce_func = _reduce_func
|
|
rv_continuous.fit = fit
|
|
rv_continuous.fit2 = fit2
|