133 lines
5.2 KiB
Python
133 lines
5.2 KiB
Python
from __future__ import division, print_function, absolute_import
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from collections import namedtuple
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import numpy as np
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import warnings
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from ._continuous_distns import chi2
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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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