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392 lines
14 KiB
Python
392 lines
14 KiB
Python
from collections import namedtuple
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import numpy as np
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import warnings
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from . import distributions
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from ._continuous_distns import chi2
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from scipy.special import gamma, kv, gammaln
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from . import _wilcoxon_data
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Epps_Singleton_2sampResult = namedtuple('Epps_Singleton_2sampResult',
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('statistic', 'pvalue'))
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def epps_singleton_2samp(x, y, t=(0.4, 0.8)):
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"""
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Compute the Epps-Singleton (ES) test statistic.
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Test the null hypothesis that two samples have the same underlying
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probability distribution.
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Parameters
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----------
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x, y : array-like
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The two samples of observations to be tested. Input must not have more
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than one dimension. Samples can have different lengths.
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t : array-like, optional
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The points (t1, ..., tn) where the empirical characteristic function is
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to be evaluated. It should be positive distinct numbers. The default
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value (0.4, 0.8) is proposed in [1]_. Input must not have more than
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one dimension.
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Returns
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-------
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statistic : float
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The test statistic.
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pvalue : float
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The associated p-value based on the asymptotic chi2-distribution.
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See Also
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--------
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ks_2samp, anderson_ksamp
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Notes
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-----
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Testing whether two samples are generated by the same underlying
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distribution is a classical question in statistics. A widely used test is
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the Kolmogorov-Smirnov (KS) test which relies on the empirical
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distribution function. Epps and Singleton introduce a test based on the
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empirical characteristic function in [1]_.
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One advantage of the ES test compared to the KS test is that is does
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not assume a continuous distribution. In [1]_, the authors conclude
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that the test also has a higher power than the KS test in many
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examples. They recommend the use of the ES test for discrete samples as
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well as continuous samples with at least 25 observations each, whereas
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`anderson_ksamp` is recommended for smaller sample sizes in the
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continuous case.
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The p-value is computed from the asymptotic distribution of the test
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statistic which follows a `chi2` distribution. If the sample size of both
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`x` and `y` is below 25, the small sample correction proposed in [1]_ is
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applied to the test statistic.
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The default values of `t` are determined in [1]_ by considering
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various distributions and finding good values that lead to a high power
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of the test in general. Table III in [1]_ gives the optimal values for
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the distributions tested in that study. The values of `t` are scaled by
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the semi-interquartile range in the implementation, see [1]_.
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References
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----------
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.. [1] T. W. Epps and K. J. Singleton, "An omnibus test for the two-sample
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problem using the empirical characteristic function", Journal of
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Statistical Computation and Simulation 26, p. 177--203, 1986.
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.. [2] S. J. Goerg and J. Kaiser, "Nonparametric testing of distributions
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- the Epps-Singleton two-sample test using the empirical characteristic
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function", The Stata Journal 9(3), p. 454--465, 2009.
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"""
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x, y, t = np.asarray(x), np.asarray(y), np.asarray(t)
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# check if x and y are valid inputs
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if x.ndim > 1:
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raise ValueError('x must be 1d, but x.ndim equals {}.'.format(x.ndim))
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if y.ndim > 1:
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raise ValueError('y must be 1d, but y.ndim equals {}.'.format(y.ndim))
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nx, ny = len(x), len(y)
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if (nx < 5) or (ny < 5):
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raise ValueError('x and y should have at least 5 elements, but len(x) '
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'= {} and len(y) = {}.'.format(nx, ny))
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if not np.isfinite(x).all():
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raise ValueError('x must not contain nonfinite values.')
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if not np.isfinite(y).all():
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raise ValueError('y must not contain nonfinite values.')
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n = nx + ny
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# check if t is valid
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if t.ndim > 1:
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raise ValueError('t must be 1d, but t.ndim equals {}.'.format(t.ndim))
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if np.less_equal(t, 0).any():
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raise ValueError('t must contain positive elements only.')
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# rescale t with semi-iqr as proposed in [1]; import iqr here to avoid
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# circular import
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from scipy.stats import iqr
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sigma = iqr(np.hstack((x, y))) / 2
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ts = np.reshape(t, (-1, 1)) / sigma
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# covariance estimation of ES test
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gx = np.vstack((np.cos(ts*x), np.sin(ts*x))).T # shape = (nx, 2*len(t))
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gy = np.vstack((np.cos(ts*y), np.sin(ts*y))).T
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cov_x = np.cov(gx.T, bias=True) # the test uses biased cov-estimate
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cov_y = np.cov(gy.T, bias=True)
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est_cov = (n/nx)*cov_x + (n/ny)*cov_y
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est_cov_inv = np.linalg.pinv(est_cov)
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r = np.linalg.matrix_rank(est_cov_inv)
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if r < 2*len(t):
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warnings.warn('Estimated covariance matrix does not have full rank. '
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'This indicates a bad choice of the input t and the '
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'test might not be consistent.') # see p. 183 in [1]_
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# compute test statistic w distributed asympt. as chisquare with df=r
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g_diff = np.mean(gx, axis=0) - np.mean(gy, axis=0)
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w = n*np.dot(g_diff.T, np.dot(est_cov_inv, g_diff))
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# apply small-sample correction
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if (max(nx, ny) < 25):
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corr = 1.0/(1.0 + n**(-0.45) + 10.1*(nx**(-1.7) + ny**(-1.7)))
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w = corr * w
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p = chi2.sf(w, r)
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return Epps_Singleton_2sampResult(w, p)
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class CramerVonMisesResult:
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def __init__(self, statistic, pvalue):
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self.statistic = statistic
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self.pvalue = pvalue
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def __repr__(self):
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return (f"{self.__class__.__name__}(statistic={self.statistic}, "
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f"pvalue={self.pvalue})")
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def _psi1_mod(x):
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"""
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psi1 is defined in equation 1.10 in Csorgo, S. and Faraway, J. (1996).
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This implements a modified version by excluding the term V(x) / 12
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(here: _cdf_cvm_inf(x) / 12) to avoid evaluating _cdf_cvm_inf(x)
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twice in _cdf_cvm.
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Implementation based on MAPLE code of Julian Faraway and R code of the
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function pCvM in the package goftest (v1.1.1), permission granted
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by Adrian Baddeley. Main difference in the implementation: the code
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here keeps adding terms of the series until the terms are small enough.
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"""
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def _ed2(y):
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z = y**2 / 4
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b = kv(1/4, z) + kv(3/4, z)
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return np.exp(-z) * (y/2)**(3/2) * b / np.sqrt(np.pi)
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def _ed3(y):
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z = y**2 / 4
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c = np.exp(-z) / np.sqrt(np.pi)
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return c * (y/2)**(5/2) * (2*kv(1/4, z) + 3*kv(3/4, z) - kv(5/4, z))
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def _Ak(k, x):
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m = 2*k + 1
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sx = 2 * np.sqrt(x)
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y1 = x**(3/4)
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y2 = x**(5/4)
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e1 = m * gamma(k + 1/2) * _ed2((4 * k + 3)/sx) / (9 * y1)
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e2 = gamma(k + 1/2) * _ed3((4 * k + 1) / sx) / (72 * y2)
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e3 = 2 * (m + 2) * gamma(k + 3/2) * _ed3((4 * k + 5) / sx) / (12 * y2)
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e4 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 1) / sx) / (144 * y1)
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e5 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 5) / sx) / (144 * y1)
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return e1 + e2 + e3 + e4 + e5
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x = np.asarray(x)
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tot = np.zeros_like(x, dtype='float')
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cond = np.ones_like(x, dtype='bool')
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k = 0
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while np.any(cond):
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z = -_Ak(k, x[cond]) / (np.pi * gamma(k + 1))
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tot[cond] = tot[cond] + z
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cond[cond] = np.abs(z) >= 1e-7
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k += 1
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return tot
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def _cdf_cvm_inf(x):
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"""
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Calculate the cdf of the Cramér-von Mises statistic (infinite sample size).
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See equation 1.2 in Csorgo, S. and Faraway, J. (1996).
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Implementation based on MAPLE code of Julian Faraway and R code of the
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function pCvM in the package goftest (v1.1.1), permission granted
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by Adrian Baddeley. Main difference in the implementation: the code
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here keeps adding terms of the series until the terms are small enough.
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The function is not expected to be accurate for large values of x, say
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x > 4, when the cdf is very close to 1.
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"""
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x = np.asarray(x)
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def term(x, k):
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# this expression can be found in [2], second line of (1.3)
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u = np.exp(gammaln(k + 0.5) - gammaln(k+1)) / (np.pi**1.5 * np.sqrt(x))
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y = 4*k + 1
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q = y**2 / (16*x)
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b = kv(0.25, q)
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return u * np.sqrt(y) * np.exp(-q) * b
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tot = np.zeros_like(x, dtype='float')
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cond = np.ones_like(x, dtype='bool')
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k = 0
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while np.any(cond):
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z = term(x[cond], k)
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tot[cond] = tot[cond] + z
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cond[cond] = np.abs(z) >= 1e-7
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k += 1
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return tot
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def _cdf_cvm(x, n=None):
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"""
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Calculate the cdf of the Cramér-von Mises statistic for a finite sample
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size n. If N is None, use the asymptotic cdf (n=inf)
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See equation 1.8 in Csorgo, S. and Faraway, J. (1996) for finite samples,
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1.2 for the asymptotic cdf.
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The function is not expected to be accurate for large values of x, say
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x > 2, when the cdf is very close to 1 and it might return values > 1
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in that case, e.g. _cdf_cvm(2.0, 12) = 1.0000027556716846.
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"""
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x = np.asarray(x)
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if n is None:
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y = _cdf_cvm_inf(x)
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else:
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# support of the test statistic is [12/n, n/3], see 1.1 in [2]
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y = np.zeros_like(x, dtype='float')
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sup = (1./(12*n) < x) & (x < n/3.)
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# note: _psi1_mod does not include the term _cdf_cvm_inf(x) / 12
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# therefore, we need to add it here
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y[sup] = _cdf_cvm_inf(x[sup]) * (1 + 1./(12*n)) + _psi1_mod(x[sup]) / n
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y[x >= n/3] = 1
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if y.ndim == 0:
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return y[()]
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return y
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def cramervonmises(rvs, cdf, args=()):
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"""
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Perform the Cramér-von Mises test for goodness of fit.
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This performs a test of the goodness of fit of a cumulative distribution
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function (cdf) :math:`F` compared to the empirical distribution function
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:math:`F_n` of observed random variates :math:`X_1, ..., X_n` that are
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assumed to be independent and identically distributed ([1]_).
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The null hypothesis is that the :math:`X_i` have cumulative distribution
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:math:`F`.
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Parameters
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----------
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rvs : array_like
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A 1-D array of observed values of the random variables :math:`X_i`.
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cdf : str or callable
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The cumulative distribution function :math:`F` to test the
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observations against. If a string, it should be the name of a
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distribution in `scipy.stats`. If a callable, that callable is used
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to calculate the cdf: ``cdf(x, *args) -> float``.
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args : tuple, optional
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Distribution parameters. These are assumed to be known; see Notes.
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Returns
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-------
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res : object with attributes
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statistic : float
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Cramér-von Mises statistic.
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pvalue : float
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The p-value.
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See Also
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--------
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kstest
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Notes
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-----
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.. versionadded:: 1.6.0
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The p-value relies on the approximation given by equation 1.8 in [2]_.
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It is important to keep in mind that the p-value is only accurate if
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one tests a simple hypothesis, i.e. the parameters of the reference
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distribution are known. If the parameters are estimated from the data
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(composite hypothesis), the computed p-value is not reliable.
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Cramér-von_Mises_criterion
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.. [2] Csorgo, S. and Faraway, J. (1996). The Exact and Asymptotic
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Distribution of Cramér-von Mises Statistics. Journal of the
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Royal Statistical Society, pp. 221-234.
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Examples
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--------
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Suppose we wish to test whether data generated by ``scipy.stats.norm.rvs``
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were, in fact, drawn from the standard normal distribution. We choose a
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significance level of alpha=0.05.
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>>> import numpy as np
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>>> from scipy import stats
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>>> np.random.seed(626)
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>>> x = stats.norm.rvs(size=500)
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>>> res = stats.cramervonmises(x, 'norm')
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>>> res.statistic, res.pvalue
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(0.06342154705518796, 0.792680516270629)
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The p-value 0.79 exceeds our chosen significance level, so we do not
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reject the null hypothesis that the observed sample is drawn from the
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standard normal distribution.
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Now suppose we wish to check whether the same sampels shifted by 2.1 is
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consistent with being drawn from a normal distribution with a mean of 2.
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>>> y = x + 2.1
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>>> res = stats.cramervonmises(y, 'norm', args=(2,))
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>>> res.statistic, res.pvalue
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(0.4798693195559657, 0.044782228803623814)
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Here we have used the `args` keyword to specify the mean (``loc``)
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of the normal distribution to test the data against. This is equivalent
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to the following, in which we create a frozen normal distribution with
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mean 2.1, then pass its ``cdf`` method as an argument.
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>>> frozen_dist = stats.norm(loc=2)
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>>> res = stats.cramervonmises(y, frozen_dist.cdf)
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>>> res.statistic, res.pvalue
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(0.4798693195559657, 0.044782228803623814)
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In either case, we would reject the null hypothesis that the observed
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sample is drawn from a normal distribution with a mean of 2 (and default
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variance of 1) because the p-value 0.04 is less than our chosen
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significance level.
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"""
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if isinstance(cdf, str):
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cdf = getattr(distributions, cdf).cdf
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vals = np.sort(np.asarray(rvs))
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if vals.size <= 1:
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raise ValueError('The sample must contain at least two observations.')
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if vals.ndim > 1:
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raise ValueError('The sample must be one-dimensional.')
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n = len(vals)
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cdfvals = cdf(vals, *args)
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u = (2*np.arange(1, n+1) - 1)/(2*n)
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w = 1/(12*n) + np.sum((u - cdfvals)**2)
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# avoid small negative values that can occur due to the approximation
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p = max(0, 1. - _cdf_cvm(w, n))
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return CramerVonMisesResult(statistic=w, pvalue=p)
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def _get_wilcoxon_distr(n):
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"""
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Distribution of counts of the Wilcoxon ranksum statistic r_plus (sum of
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ranks of positive differences).
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Returns an array with the counts/frequencies of all the possible ranks
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r = 0, ..., n*(n+1)/2
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"""
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cnt = _wilcoxon_data.COUNTS.get(n)
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if cnt is None:
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raise ValueError("The exact distribution of the Wilcoxon test "
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"statistic is not implemented for n={}".format(n))
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return np.array(cnt, dtype=int)
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