manyi/venv/Lib/site-packages/scipy/stats/tests/test_morestats.py

# Author:  Travis Oliphant, 2002
#
# Further enhancements and tests added by numerous SciPy developers.
#
from __future__ import division, print_function, absolute_import

import warnings

import numpy as np
from numpy.random import RandomState
from numpy.testing import (assert_array_equal,
    assert_almost_equal, assert_array_less, assert_array_almost_equal,
    assert_, assert_allclose, assert_equal, assert_warns)
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings

from scipy import stats
from .common_tests import check_named_results

# Matplotlib is not a scipy dependency but is optionally used in probplot, so
# check if it's available
try:
    import matplotlib.pyplot as plt
    have_matplotlib = True
except Exception:
    have_matplotlib = False


# test data gear.dat from NIST for Levene and Bartlett test
# https://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm
g1 = [1.006, 0.996, 0.998, 1.000, 0.992, 0.993, 1.002, 0.999, 0.994, 1.000]
g2 = [0.998, 1.006, 1.000, 1.002, 0.997, 0.998, 0.996, 1.000, 1.006, 0.988]
g3 = [0.991, 0.987, 0.997, 0.999, 0.995, 0.994, 1.000, 0.999, 0.996, 0.996]
g4 = [1.005, 1.002, 0.994, 1.000, 0.995, 0.994, 0.998, 0.996, 1.002, 0.996]
g5 = [0.998, 0.998, 0.982, 0.990, 1.002, 0.984, 0.996, 0.993, 0.980, 0.996]
g6 = [1.009, 1.013, 1.009, 0.997, 0.988, 1.002, 0.995, 0.998, 0.981, 0.996]
g7 = [0.990, 1.004, 0.996, 1.001, 0.998, 1.000, 1.018, 1.010, 0.996, 1.002]
g8 = [0.998, 1.000, 1.006, 1.000, 1.002, 0.996, 0.998, 0.996, 1.002, 1.006]
g9 = [1.002, 0.998, 0.996, 0.995, 0.996, 1.004, 1.004, 0.998, 0.999, 0.991]
g10 = [0.991, 0.995, 0.984, 0.994, 0.997, 0.997, 0.991, 0.998, 1.004, 0.997]


class TestBayes_mvs(object):
    def test_basic(self):
        # Expected values in this test simply taken from the function.  For
        # some checks regarding correctness of implementation, see review in
        # gh-674
        data = [6, 9, 12, 7, 8, 8, 13]
        mean, var, std = stats.bayes_mvs(data)
        assert_almost_equal(mean.statistic, 9.0)
        assert_allclose(mean.minmax, (7.1036502226125329, 10.896349777387467),
                        rtol=1e-14)

        assert_almost_equal(var.statistic, 10.0)
        assert_allclose(var.minmax, (3.1767242068607087, 24.45910381334018),
                        rtol=1e-09)

        assert_almost_equal(std.statistic, 2.9724954732045084, decimal=14)
        assert_allclose(std.minmax, (1.7823367265645145, 4.9456146050146312),
                        rtol=1e-14)

    def test_empty_input(self):
        assert_raises(ValueError, stats.bayes_mvs, [])

    def test_result_attributes(self):
        x = np.arange(15)
        attributes = ('statistic', 'minmax')
        res = stats.bayes_mvs(x)

        for i in res:
            check_named_results(i, attributes)


class TestMvsdist(object):
    def test_basic(self):
        data = [6, 9, 12, 7, 8, 8, 13]
        mean, var, std = stats.mvsdist(data)
        assert_almost_equal(mean.mean(), 9.0)
        assert_allclose(mean.interval(0.9), (7.1036502226125329,
                                             10.896349777387467), rtol=1e-14)

        assert_almost_equal(var.mean(), 10.0)
        assert_allclose(var.interval(0.9), (3.1767242068607087,
                                            24.45910381334018), rtol=1e-09)

        assert_almost_equal(std.mean(), 2.9724954732045084, decimal=14)
        assert_allclose(std.interval(0.9), (1.7823367265645145,
                                            4.9456146050146312), rtol=1e-14)

    def test_empty_input(self):
        assert_raises(ValueError, stats.mvsdist, [])

    def test_bad_arg(self):
        # Raise ValueError if fewer than two data points are given.
        data = [1]
        assert_raises(ValueError, stats.mvsdist, data)

    def test_warns(self):
        # regression test for gh-5270
        # make sure there are no spurious divide-by-zero warnings
        with warnings.catch_warnings():
            warnings.simplefilter('error', RuntimeWarning)
            [x.mean() for x in stats.mvsdist([1, 2, 3])]
            [x.mean() for x in stats.mvsdist([1, 2, 3, 4, 5])]


class TestShapiro(object):
    def test_basic(self):
        x1 = [0.11, 7.87, 4.61, 10.14, 7.95, 3.14, 0.46,
              4.43, 0.21, 4.75, 0.71, 1.52, 3.24,
              0.93, 0.42, 4.97, 9.53, 4.55, 0.47, 6.66]
        w, pw = stats.shapiro(x1)
        assert_almost_equal(w, 0.90047299861907959, 6)
        assert_almost_equal(pw, 0.042089745402336121, 6)
        x2 = [1.36, 1.14, 2.92, 2.55, 1.46, 1.06, 5.27, -1.11,
              3.48, 1.10, 0.88, -0.51, 1.46, 0.52, 6.20, 1.69,
              0.08, 3.67, 2.81, 3.49]
        w, pw = stats.shapiro(x2)
        assert_almost_equal(w, 0.9590270, 6)
        assert_almost_equal(pw, 0.52460, 3)

        # Verified against R
        np.random.seed(12345678)
        x3 = stats.norm.rvs(loc=5, scale=3, size=100)
        w, pw = stats.shapiro(x3)
        assert_almost_equal(w, 0.9772805571556091, decimal=6)
        assert_almost_equal(pw, 0.08144091814756393, decimal=3)

        # Extracted from original paper
        x4 = [0.139, 0.157, 0.175, 0.256, 0.344, 0.413, 0.503, 0.577, 0.614,
              0.655, 0.954, 1.392, 1.557, 1.648, 1.690, 1.994, 2.174, 2.206,
              3.245, 3.510, 3.571, 4.354, 4.980, 6.084, 8.351]
        W_expected = 0.83467
        p_expected = 0.000914
        w, pw = stats.shapiro(x4)
        assert_almost_equal(w, W_expected, decimal=4)
        assert_almost_equal(pw, p_expected, decimal=5)

    def test_2d(self):
        x1 = [[0.11, 7.87, 4.61, 10.14, 7.95, 3.14, 0.46,
              4.43, 0.21, 4.75], [0.71, 1.52, 3.24,
              0.93, 0.42, 4.97, 9.53, 4.55, 0.47, 6.66]]
        w, pw = stats.shapiro(x1)
        assert_almost_equal(w, 0.90047299861907959, 6)
        assert_almost_equal(pw, 0.042089745402336121, 6)
        x2 = [[1.36, 1.14, 2.92, 2.55, 1.46, 1.06, 5.27, -1.11,
              3.48, 1.10], [0.88, -0.51, 1.46, 0.52, 6.20, 1.69,
              0.08, 3.67, 2.81, 3.49]]
        w, pw = stats.shapiro(x2)
        assert_almost_equal(w, 0.9590270, 6)
        assert_almost_equal(pw, 0.52460, 3)

    def test_empty_input(self):
        assert_raises(ValueError, stats.shapiro, [])
        assert_raises(ValueError, stats.shapiro, [[], [], []])

    def test_not_enough_values(self):
        assert_raises(ValueError, stats.shapiro, [1, 2])
        assert_raises(ValueError, stats.shapiro, [[], [2]])

    def test_bad_arg(self):
        # Length of x is less than 3.
        x = [1]
        assert_raises(ValueError, stats.shapiro, x)

    def test_nan_input(self):
        x = np.arange(10.)
        x[9] = np.nan

        w, pw = stats.shapiro(x)
        assert_equal(w, np.nan)
        assert_almost_equal(pw, 1.0)


class TestAnderson(object):
    def test_normal(self):
        rs = RandomState(1234567890)
        x1 = rs.standard_exponential(size=50)
        x2 = rs.standard_normal(size=50)
        A, crit, sig = stats.anderson(x1)
        assert_array_less(crit[:-1], A)
        A, crit, sig = stats.anderson(x2)
        assert_array_less(A, crit[-2:])

        v = np.ones(10)
        v[0] = 0
        A, crit, sig = stats.anderson(v)
        # The expected statistic 3.208057 was computed independently of scipy.
        # For example, in R:
        #   > library(nortest)
        #   > v <- rep(1, 10)
        #   > v[1] <- 0
        #   > result <- ad.test(v)
        #   > result$statistic
        #          A
        #   3.208057
        assert_allclose(A, 3.208057)

    def test_expon(self):
        rs = RandomState(1234567890)
        x1 = rs.standard_exponential(size=50)
        x2 = rs.standard_normal(size=50)
        A, crit, sig = stats.anderson(x1, 'expon')
        assert_array_less(A, crit[-2:])
        olderr = np.seterr(all='ignore')
        try:
            A, crit, sig = stats.anderson(x2, 'expon')
        finally:
            np.seterr(**olderr)
        assert_(A > crit[-1])

    def test_gumbel(self):
        # Regression test for gh-6306.  Before that issue was fixed,
        # this case would return a2=inf.
        v = np.ones(100)
        v[0] = 0.0
        a2, crit, sig = stats.anderson(v, 'gumbel')
        # A brief reimplementation of the calculation of the statistic.
        n = len(v)
        xbar, s = stats.gumbel_l.fit(v)
        logcdf = stats.gumbel_l.logcdf(v, xbar, s)
        logsf = stats.gumbel_l.logsf(v, xbar, s)
        i = np.arange(1, n+1)
        expected_a2 = -n - np.mean((2*i - 1) * (logcdf + logsf[::-1]))

        assert_allclose(a2, expected_a2)

    def test_bad_arg(self):
        assert_raises(ValueError, stats.anderson, [1], dist='plate_of_shrimp')

    def test_result_attributes(self):
        rs = RandomState(1234567890)
        x = rs.standard_exponential(size=50)
        res = stats.anderson(x)
        attributes = ('statistic', 'critical_values', 'significance_level')
        check_named_results(res, attributes)

    def test_gumbel_l(self):
        # gh-2592, gh-6337
        # Adds support to 'gumbel_r' and 'gumbel_l' as valid inputs for dist.
        rs = RandomState(1234567890)
        x = rs.gumbel(size=100)
        A1, crit1, sig1 = stats.anderson(x, 'gumbel')
        A2, crit2, sig2 = stats.anderson(x, 'gumbel_l')

        assert_allclose(A2, A1)

    def test_gumbel_r(self):
        # gh-2592, gh-6337
        # Adds support to 'gumbel_r' and 'gumbel_l' as valid inputs for dist.
        rs = RandomState(1234567890)
        x1 = rs.gumbel(size=100)
        x2 = np.ones(100)
        A1, crit1, sig1 = stats.anderson(x1, 'gumbel_r')
        A2, crit2, sig2 = stats.anderson(x2, 'gumbel_r')

        assert_array_less(A1, crit1[-2:])
        assert_(A2 > crit2[-1])


class TestAndersonKSamp(object):
    def test_example1a(self):
        # Example data from Scholz & Stephens (1987), originally
        # published in Lehmann (1995, Nonparametrics, Statistical
        # Methods Based on Ranks, p. 309)
        # Pass a mixture of lists and arrays
        t1 = [38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0]
        t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
        t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
        t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])

        Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=False)

        assert_almost_equal(Tk, 4.449, 3)
        assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
                                  tm[0:5], 4)
        assert_allclose(p, 0.0021, atol=0.00025)

    def test_example1b(self):
        # Example data from Scholz & Stephens (1987), originally
        # published in Lehmann (1995, Nonparametrics, Statistical
        # Methods Based on Ranks, p. 309)
        # Pass arrays
        t1 = np.array([38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0])
        t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
        t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
        t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
        Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=True)

        assert_almost_equal(Tk, 4.480, 3)
        assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
                                  tm[0:5], 4)
        assert_allclose(p, 0.0020, atol=0.00025)

    def test_example2a(self):
        # Example data taken from an earlier technical report of
        # Scholz and Stephens
        # Pass lists instead of arrays
        t1 = [194, 15, 41, 29, 33, 181]
        t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
        t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
        t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
              118, 25, 156, 310, 76, 26, 44, 23, 62]
        t5 = [130, 208, 70, 101, 208]
        t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
        t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
        t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
              12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
        t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
              54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
        t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
               22, 139, 210, 97, 30, 23, 13, 14]
        t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
        t12 = [50, 254, 5, 283, 35, 12]
        t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
        t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
               61, 34]

        Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
                                          t9, t10, t11, t12, t13, t14),
                                         midrank=False)
        assert_almost_equal(Tk, 3.288, 3)
        assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
                                  tm[0:5], 4)
        assert_allclose(p, 0.0041, atol=0.00025)

    def test_example2b(self):
        # Example data taken from an earlier technical report of
        # Scholz and Stephens
        t1 = [194, 15, 41, 29, 33, 181]
        t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
        t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
        t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
              118, 25, 156, 310, 76, 26, 44, 23, 62]
        t5 = [130, 208, 70, 101, 208]
        t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
        t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
        t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
              12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
        t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
              54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
        t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
               22, 139, 210, 97, 30, 23, 13, 14]
        t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
        t12 = [50, 254, 5, 283, 35, 12]
        t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
        t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
               61, 34]

        Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
                                          t9, t10, t11, t12, t13, t14),
                                         midrank=True)

        assert_almost_equal(Tk, 3.294, 3)
        assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
                                  tm[0:5], 4)
        assert_allclose(p, 0.0041, atol=0.00025)

    def test_R_kSamples(self):
        # test values generates with R package kSamples
        # package version 1.2-6 (2017-06-14)
        # r1 = 1:100
        # continuous case (no ties) --> version  1
        # res <- kSamples::ad.test(r1, r1 + 40.5)
        # res$ad[1, "T.AD"] #  41.105
        # res$ad[1, " asympt. P-value"] #  5.8399e-18
        #
        # discrete case (ties allowed) --> version  2 (here: midrank=True)
        # res$ad[2, "T.AD"] #  41.235
        #
        # res <- kSamples::ad.test(r1, r1 + .5)
        # res$ad[1, "T.AD"] #  -1.2824
        # res$ad[1, " asympt. P-value"] #  1
        # res$ad[2, "T.AD"] #  -1.2944
        #
        # res <- kSamples::ad.test(r1, r1 + 7.5)
        # res$ad[1, "T.AD"] # 1.4923
        # res$ad[1, " asympt. P-value"] # 0.077501
        #
        # res <- kSamples::ad.test(r1, r1 + 6)
        # res$ad[2, "T.AD"] # 0.63892
        # res$ad[2, " asympt. P-value"] # 0.17981
        #
        # res <- kSamples::ad.test(r1, r1 + 11.5)
        # res$ad[1, "T.AD"] # 4.5042
        # res$ad[1, " asympt. P-value"] # 0.00545
        #
        # res <- kSamples::ad.test(r1, r1 + 13.5)
        # res$ad[1, "T.AD"] # 6.2982
        # res$ad[1, " asympt. P-value"] # 0.00118

        x1 = np.linspace(1, 100, 100)
        # test case: different distributions;p-value floored at 0.001
        # test case for issue #5493 / #8536
        with suppress_warnings() as sup:
            sup.filter(UserWarning, message='p-value floored')
            s, _, p = stats.anderson_ksamp([x1, x1 + 40.5], midrank=False)
        assert_almost_equal(s, 41.105, 3)
        assert_equal(p, 0.001)

        with suppress_warnings() as sup:
            sup.filter(UserWarning, message='p-value floored')
            s, _, p = stats.anderson_ksamp([x1, x1 + 40.5])
        assert_almost_equal(s, 41.235, 3)
        assert_equal(p, 0.001)

        # test case: similar distributions --> p-value capped at 0.25
        with suppress_warnings() as sup:
            sup.filter(UserWarning, message='p-value capped')
            s, _, p = stats.anderson_ksamp([x1, x1 + .5], midrank=False)
        assert_almost_equal(s, -1.2824, 4)
        assert_equal(p, 0.25)

        with suppress_warnings() as sup:
            sup.filter(UserWarning, message='p-value capped')
            s, _, p = stats.anderson_ksamp([x1, x1 + .5])
        assert_almost_equal(s, -1.2944, 4)
        assert_equal(p, 0.25)

        # test case: check interpolated p-value in [0.01, 0.25] (no ties)
        s, _, p = stats.anderson_ksamp([x1, x1 + 7.5], midrank=False)
        assert_almost_equal(s, 1.4923, 4)
        assert_allclose(p, 0.0775, atol=0.005, rtol=0)

        # test case: check interpolated p-value in [0.01, 0.25] (w/ ties)
        s, _, p = stats.anderson_ksamp([x1, x1 + 6])
        assert_almost_equal(s, 0.6389, 4)
        assert_allclose(p, 0.1798, atol=0.005, rtol=0)

        # test extended critical values for p=0.001 and p=0.005
        s, _, p = stats.anderson_ksamp([x1, x1 + 11.5], midrank=False)
        assert_almost_equal(s, 4.5042, 4)
        assert_allclose(p, 0.00545, atol=0.0005, rtol=0)

        s, _, p = stats.anderson_ksamp([x1, x1 + 13.5], midrank=False)
        assert_almost_equal(s, 6.2982, 4)
        assert_allclose(p, 0.00118, atol=0.0001, rtol=0)

    def test_not_enough_samples(self):
        assert_raises(ValueError, stats.anderson_ksamp, np.ones(5))

    def test_no_distinct_observations(self):
        assert_raises(ValueError, stats.anderson_ksamp,
                      (np.ones(5), np.ones(5)))

    def test_empty_sample(self):
        assert_raises(ValueError, stats.anderson_ksamp, (np.ones(5), []))

    def test_result_attributes(self):
        # Pass a mixture of lists and arrays
        t1 = [38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0]
        t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
        res = stats.anderson_ksamp((t1, t2), midrank=False)

        attributes = ('statistic', 'critical_values', 'significance_level')
        check_named_results(res, attributes)


class TestAnsari(object):

    def test_small(self):
        x = [1, 2, 3, 3, 4]
        y = [3, 2, 6, 1, 6, 1, 4, 1]
        with suppress_warnings() as sup:
            sup.filter(UserWarning, "Ties preclude use of exact statistic.")
            W, pval = stats.ansari(x, y)
        assert_almost_equal(W, 23.5, 11)
        assert_almost_equal(pval, 0.13499256881897437, 11)

    def test_approx(self):
        ramsay = np.array((111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
                           101, 96, 97, 102, 107, 113, 116, 113, 110, 98))
        parekh = np.array((107, 108, 106, 98, 105, 103, 110, 105, 104,
                           100, 96, 108, 103, 104, 114, 114, 113, 108,
                           106, 99))

        with suppress_warnings() as sup:
            sup.filter(UserWarning, "Ties preclude use of exact statistic.")
            W, pval = stats.ansari(ramsay, parekh)

        assert_almost_equal(W, 185.5, 11)
        assert_almost_equal(pval, 0.18145819972867083, 11)

    def test_exact(self):
        W, pval = stats.ansari([1, 2, 3, 4], [15, 5, 20, 8, 10, 12])
        assert_almost_equal(W, 10.0, 11)
        assert_almost_equal(pval, 0.533333333333333333, 7)

    def test_bad_arg(self):
        assert_raises(ValueError, stats.ansari, [], [1])
        assert_raises(ValueError, stats.ansari, [1], [])

    def test_result_attributes(self):
        x = [1, 2, 3, 3, 4]
        y = [3, 2, 6, 1, 6, 1, 4, 1]
        with suppress_warnings() as sup:
            sup.filter(UserWarning, "Ties preclude use of exact statistic.")
            res = stats.ansari(x, y)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)


class TestBartlett(object):

    def test_data(self):
        # https://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm
        args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
        T, pval = stats.bartlett(*args)
        assert_almost_equal(T, 20.78587342806484, 7)
        assert_almost_equal(pval, 0.0136358632781, 7)

    def test_bad_arg(self):
        # Too few args raises ValueError.
        assert_raises(ValueError, stats.bartlett, [1])

    def test_result_attributes(self):
        args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
        res = stats.bartlett(*args)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)

    def test_empty_arg(self):
        args = (g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, [])
        assert_equal((np.nan, np.nan), stats.bartlett(*args))

    # temporary fix for issue #9252: only accept 1d input
    def test_1d_input(self):
        x = np.array([[1, 2], [3, 4]])
        assert_raises(ValueError, stats.bartlett, g1, x)


class TestLevene(object):

    def test_data(self):
        # https://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm
        args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
        W, pval = stats.levene(*args)
        assert_almost_equal(W, 1.7059176930008939, 7)
        assert_almost_equal(pval, 0.0990829755522, 7)

    def test_trimmed1(self):
        # Test that center='trimmed' gives the same result as center='mean'
        # when proportiontocut=0.
        W1, pval1 = stats.levene(g1, g2, g3, center='mean')
        W2, pval2 = stats.levene(g1, g2, g3, center='trimmed',
                                 proportiontocut=0.0)
        assert_almost_equal(W1, W2)
        assert_almost_equal(pval1, pval2)

    def test_trimmed2(self):
        x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
        y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
        np.random.seed(1234)
        x2 = np.random.permutation(x)

        # Use center='trimmed'
        W0, pval0 = stats.levene(x, y, center='trimmed',
                                 proportiontocut=0.125)
        W1, pval1 = stats.levene(x2, y, center='trimmed',
                                 proportiontocut=0.125)
        # Trim the data here, and use center='mean'
        W2, pval2 = stats.levene(x[1:-1], y[1:-1], center='mean')
        # Result should be the same.
        assert_almost_equal(W0, W2)
        assert_almost_equal(W1, W2)
        assert_almost_equal(pval1, pval2)

    def test_equal_mean_median(self):
        x = np.linspace(-1, 1, 21)
        np.random.seed(1234)
        x2 = np.random.permutation(x)
        y = x**3
        W1, pval1 = stats.levene(x, y, center='mean')
        W2, pval2 = stats.levene(x2, y, center='median')
        assert_almost_equal(W1, W2)
        assert_almost_equal(pval1, pval2)

    def test_bad_keyword(self):
        x = np.linspace(-1, 1, 21)
        assert_raises(TypeError, stats.levene, x, x, portiontocut=0.1)

    def test_bad_center_value(self):
        x = np.linspace(-1, 1, 21)
        assert_raises(ValueError, stats.levene, x, x, center='trim')

    def test_too_few_args(self):
        assert_raises(ValueError, stats.levene, [1])

    def test_result_attributes(self):
        args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
        res = stats.levene(*args)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)

    # temporary fix for issue #9252: only accept 1d input
    def test_1d_input(self):
        x = np.array([[1, 2], [3, 4]])
        assert_raises(ValueError, stats.levene, g1, x)


class TestBinomP(object):

    def test_data(self):
        pval = stats.binom_test(100, 250)
        assert_almost_equal(pval, 0.0018833009350757682, 11)
        pval = stats.binom_test(201, 405)
        assert_almost_equal(pval, 0.92085205962670713, 11)
        pval = stats.binom_test([682, 243], p=3.0/4)
        assert_almost_equal(pval, 0.38249155957481695, 11)

    def test_bad_len_x(self):
        # Length of x must be 1 or 2.
        assert_raises(ValueError, stats.binom_test, [1, 2, 3])

    def test_bad_n(self):
        # len(x) is 1, but n is invalid.
        # Missing n
        assert_raises(ValueError, stats.binom_test, [100])
        # n less than x[0]
        assert_raises(ValueError, stats.binom_test, [100], n=50)

    def test_bad_p(self):
        assert_raises(ValueError, stats.binom_test, [50, 50], p=2.0)

    def test_alternatives(self):
        res = stats.binom_test(51, 235, p=1./6, alternative='less')
        assert_almost_equal(res, 0.982022657605858)

        res = stats.binom_test(51, 235, p=1./6, alternative='greater')
        assert_almost_equal(res, 0.02654424571169085)

        res = stats.binom_test(51, 235, p=1./6, alternative='two-sided')
        assert_almost_equal(res, 0.0437479701823997)


class TestFligner(object):

    def test_data(self):
        # numbers from R: fligner.test in package stats
        x1 = np.arange(5)
        assert_array_almost_equal(stats.fligner(x1, x1**2),
                                  (3.2282229927203536, 0.072379187848207877),
                                  11)

    def test_trimmed1(self):
        # Perturb input to break ties in the transformed data
        # See https://github.com/scipy/scipy/pull/8042 for more details
        rs = np.random.RandomState(123)
        _perturb = lambda g: (np.asarray(g) + 1e-10*rs.randn(len(g))).tolist()
        g1_ = _perturb(g1)
        g2_ = _perturb(g2)
        g3_ = _perturb(g3)
        # Test that center='trimmed' gives the same result as center='mean'
        # when proportiontocut=0.
        Xsq1, pval1 = stats.fligner(g1_, g2_, g3_, center='mean')
        Xsq2, pval2 = stats.fligner(g1_, g2_, g3_, center='trimmed',
                                    proportiontocut=0.0)
        assert_almost_equal(Xsq1, Xsq2)
        assert_almost_equal(pval1, pval2)

    def test_trimmed2(self):
        x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
        y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
        # Use center='trimmed'
        Xsq1, pval1 = stats.fligner(x, y, center='trimmed',
                                    proportiontocut=0.125)
        # Trim the data here, and use center='mean'
        Xsq2, pval2 = stats.fligner(x[1:-1], y[1:-1], center='mean')
        # Result should be the same.
        assert_almost_equal(Xsq1, Xsq2)
        assert_almost_equal(pval1, pval2)

    # The following test looks reasonable at first, but fligner() uses the
    # function stats.rankdata(), and in one of the cases in this test,
    # there are ties, while in the other (because of normal rounding
    # errors) there are not.  This difference leads to differences in the
    # third significant digit of W.
    #
    #def test_equal_mean_median(self):
    #    x = np.linspace(-1,1,21)
    #    y = x**3
    #    W1, pval1 = stats.fligner(x, y, center='mean')
    #    W2, pval2 = stats.fligner(x, y, center='median')
    #    assert_almost_equal(W1, W2)
    #    assert_almost_equal(pval1, pval2)

    def test_bad_keyword(self):
        x = np.linspace(-1, 1, 21)
        assert_raises(TypeError, stats.fligner, x, x, portiontocut=0.1)

    def test_bad_center_value(self):
        x = np.linspace(-1, 1, 21)
        assert_raises(ValueError, stats.fligner, x, x, center='trim')

    def test_bad_num_args(self):
        # Too few args raises ValueError.
        assert_raises(ValueError, stats.fligner, [1])

    def test_empty_arg(self):
        x = np.arange(5)
        assert_equal((np.nan, np.nan), stats.fligner(x, x**2, []))


class TestMood(object):
    def test_mood(self):
        # numbers from R: mood.test in package stats
        x1 = np.arange(5)
        assert_array_almost_equal(stats.mood(x1, x1**2),
                                  (-1.3830857299399906, 0.16663858066771478),
                                  11)

    def test_mood_order_of_args(self):
        # z should change sign when the order of arguments changes, pvalue
        # should not change
        np.random.seed(1234)
        x1 = np.random.randn(10, 1)
        x2 = np.random.randn(15, 1)
        z1, p1 = stats.mood(x1, x2)
        z2, p2 = stats.mood(x2, x1)
        assert_array_almost_equal([z1, p1], [-z2, p2])

    def test_mood_with_axis_none(self):
        # Test with axis = None, compare with results from R
        x1 = [-0.626453810742332, 0.183643324222082, -0.835628612410047,
               1.59528080213779, 0.329507771815361, -0.820468384118015,
               0.487429052428485, 0.738324705129217, 0.575781351653492,
              -0.305388387156356, 1.51178116845085, 0.389843236411431,
              -0.621240580541804, -2.2146998871775, 1.12493091814311,
              -0.0449336090152309, -0.0161902630989461, 0.943836210685299,
               0.821221195098089, 0.593901321217509]

        x2 = [-0.896914546624981, 0.184849184646742, 1.58784533120882,
              -1.13037567424629, -0.0802517565509893, 0.132420284381094,
               0.707954729271733, -0.23969802417184, 1.98447393665293,
              -0.138787012119665, 0.417650750792556, 0.981752777463662,
              -0.392695355503813, -1.03966897694891, 1.78222896030858,
              -2.31106908460517, 0.878604580921265, 0.035806718015226,
               1.01282869212708, 0.432265154539617, 2.09081920524915,
              -1.19992581964387, 1.58963820029007, 1.95465164222325,
               0.00493777682814261, -2.45170638784613, 0.477237302613617,
              -0.596558168631403, 0.792203270299649, 0.289636710177348]

        x1 = np.array(x1)
        x2 = np.array(x2)
        x1.shape = (10, 2)
        x2.shape = (15, 2)
        assert_array_almost_equal(stats.mood(x1, x2, axis=None),
                                  [-1.31716607555, 0.18778296257])

    def test_mood_2d(self):
        # Test if the results of mood test in 2-D case are consistent with the
        # R result for the same inputs.  Numbers from R mood.test().
        ny = 5
        np.random.seed(1234)
        x1 = np.random.randn(10, ny)
        x2 = np.random.randn(15, ny)
        z_vectest, pval_vectest = stats.mood(x1, x2)

        for j in range(ny):
            assert_array_almost_equal([z_vectest[j], pval_vectest[j]],
                                      stats.mood(x1[:, j], x2[:, j]))

        # inverse order of dimensions
        x1 = x1.transpose()
        x2 = x2.transpose()
        z_vectest, pval_vectest = stats.mood(x1, x2, axis=1)

        for i in range(ny):
            # check axis handling is self consistent
            assert_array_almost_equal([z_vectest[i], pval_vectest[i]],
                                      stats.mood(x1[i, :], x2[i, :]))

    def test_mood_3d(self):
        shape = (10, 5, 6)
        np.random.seed(1234)
        x1 = np.random.randn(*shape)
        x2 = np.random.randn(*shape)

        for axis in range(3):
            z_vectest, pval_vectest = stats.mood(x1, x2, axis=axis)
            # Tests that result for 3-D arrays is equal to that for the
            # same calculation on a set of 1-D arrays taken from the
            # 3-D array
            axes_idx = ([1, 2], [0, 2], [0, 1])  # the two axes != axis
            for i in range(shape[axes_idx[axis][0]]):
                for j in range(shape[axes_idx[axis][1]]):
                    if axis == 0:
                        slice1 = x1[:, i, j]
                        slice2 = x2[:, i, j]
                    elif axis == 1:
                        slice1 = x1[i, :, j]
                        slice2 = x2[i, :, j]
                    else:
                        slice1 = x1[i, j, :]
                        slice2 = x2[i, j, :]

                    assert_array_almost_equal([z_vectest[i, j],
                                               pval_vectest[i, j]],
                                              stats.mood(slice1, slice2))

    def test_mood_bad_arg(self):
        # Raise ValueError when the sum of the lengths of the args is
        # less than 3
        assert_raises(ValueError, stats.mood, [1], [])


class TestProbplot(object):

    def test_basic(self):
        np.random.seed(12345)
        x = stats.norm.rvs(size=20)
        osm, osr = stats.probplot(x, fit=False)
        osm_expected = [-1.8241636, -1.38768012, -1.11829229, -0.91222575,
                        -0.73908135, -0.5857176, -0.44506467, -0.31273668,
                        -0.18568928, -0.06158146, 0.06158146, 0.18568928,
                        0.31273668, 0.44506467, 0.5857176, 0.73908135,
                        0.91222575, 1.11829229, 1.38768012, 1.8241636]
        assert_allclose(osr, np.sort(x))
        assert_allclose(osm, osm_expected)

        res, res_fit = stats.probplot(x, fit=True)
        res_fit_expected = [1.05361841, 0.31297795, 0.98741609]
        assert_allclose(res_fit, res_fit_expected)

    def test_sparams_keyword(self):
        np.random.seed(123456)
        x = stats.norm.rvs(size=100)
        # Check that None, () and 0 (loc=0, for normal distribution) all work
        # and give the same results
        osm1, osr1 = stats.probplot(x, sparams=None, fit=False)
        osm2, osr2 = stats.probplot(x, sparams=0, fit=False)
        osm3, osr3 = stats.probplot(x, sparams=(), fit=False)
        assert_allclose(osm1, osm2)
        assert_allclose(osm1, osm3)
        assert_allclose(osr1, osr2)
        assert_allclose(osr1, osr3)
        # Check giving (loc, scale) params for normal distribution
        osm, osr = stats.probplot(x, sparams=(), fit=False)

    def test_dist_keyword(self):
        np.random.seed(12345)
        x = stats.norm.rvs(size=20)
        osm1, osr1 = stats.probplot(x, fit=False, dist='t', sparams=(3,))
        osm2, osr2 = stats.probplot(x, fit=False, dist=stats.t, sparams=(3,))
        assert_allclose(osm1, osm2)
        assert_allclose(osr1, osr2)

        assert_raises(ValueError, stats.probplot, x, dist='wrong-dist-name')
        assert_raises(AttributeError, stats.probplot, x, dist=[])

        class custom_dist(object):
            """Some class that looks just enough like a distribution."""
            def ppf(self, q):
                return stats.norm.ppf(q, loc=2)

        osm1, osr1 = stats.probplot(x, sparams=(2,), fit=False)
        osm2, osr2 = stats.probplot(x, dist=custom_dist(), fit=False)
        assert_allclose(osm1, osm2)
        assert_allclose(osr1, osr2)

    @pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
    def test_plot_kwarg(self):
        np.random.seed(7654321)
        fig = plt.figure()
        fig.add_subplot(111)
        x = stats.t.rvs(3, size=100)
        res1, fitres1 = stats.probplot(x, plot=plt)
        plt.close()
        res2, fitres2 = stats.probplot(x, plot=None)
        res3 = stats.probplot(x, fit=False, plot=plt)
        plt.close()
        res4 = stats.probplot(x, fit=False, plot=None)
        # Check that results are consistent between combinations of `fit` and
        # `plot` keywords.
        assert_(len(res1) == len(res2) == len(res3) == len(res4) == 2)
        assert_allclose(res1, res2)
        assert_allclose(res1, res3)
        assert_allclose(res1, res4)
        assert_allclose(fitres1, fitres2)

        # Check that a Matplotlib Axes object is accepted
        fig = plt.figure()
        ax = fig.add_subplot(111)
        stats.probplot(x, fit=False, plot=ax)
        plt.close()

    def test_probplot_bad_args(self):
        # Raise ValueError when given an invalid distribution.
        assert_raises(ValueError, stats.probplot, [1], dist="plate_of_shrimp")

    def test_empty(self):
        assert_equal(stats.probplot([], fit=False),
                     (np.array([]), np.array([])))
        assert_equal(stats.probplot([], fit=True),
                     ((np.array([]), np.array([])),
                      (np.nan, np.nan, 0.0)))

    def test_array_of_size_one(self):
        with np.errstate(invalid='ignore'):
            assert_equal(stats.probplot([1], fit=True),
                         ((np.array([0.]), np.array([1])),
                          (np.nan, np.nan, 0.0)))


def test_wilcoxon_bad_arg():
    # Raise ValueError when two args of different lengths are given or
    # zero_method is unknown.
    assert_raises(ValueError, stats.wilcoxon, [1], [1, 2])
    assert_raises(ValueError, stats.wilcoxon, [1, 2], [1, 2], "dummy")


def test_wilcoxon_arg_type():
    # Should be able to accept list as arguments.
    # Address issue 6070.
    arr = [1, 2, 3, 0, -1, 3, 1, 2, 1, 1, 2]

    _ = stats.wilcoxon(arr, zero_method="pratt")
    _ = stats.wilcoxon(arr, zero_method="zsplit")
    _ = stats.wilcoxon(arr, zero_method="wilcox")


class TestKstat(object):
    def test_moments_normal_distribution(self):
        np.random.seed(32149)
        data = np.random.randn(12345)
        moments = []
        for n in [1, 2, 3, 4]:
            moments.append(stats.kstat(data, n))

        expected = [0.011315, 1.017931, 0.05811052, 0.0754134]
        assert_allclose(moments, expected, rtol=1e-4)

        # test equivalence with `stats.moment`
        m1 = stats.moment(data, moment=1)
        m2 = stats.moment(data, moment=2)
        m3 = stats.moment(data, moment=3)
        assert_allclose((m1, m2, m3), expected[:-1], atol=0.02, rtol=1e-2)

    def test_empty_input(self):
        assert_raises(ValueError, stats.kstat, [])

    def test_nan_input(self):
        data = np.arange(10.)
        data[6] = np.nan

        assert_equal(stats.kstat(data), np.nan)

    def test_kstat_bad_arg(self):
        # Raise ValueError if n > 4 or n < 1.
        data = np.arange(10)
        for n in [0, 4.001]:
            assert_raises(ValueError, stats.kstat, data, n=n)


class TestKstatVar(object):
    def test_empty_input(self):
        assert_raises(ValueError, stats.kstatvar, [])

    def test_nan_input(self):
        data = np.arange(10.)
        data[6] = np.nan

        assert_equal(stats.kstat(data), np.nan)

    def test_bad_arg(self):
        # Raise ValueError is n is not 1 or 2.
        data = [1]
        n = 10
        assert_raises(ValueError, stats.kstatvar, data, n=n)


class TestPpccPlot(object):
    def setup_method(self):
        np.random.seed(7654321)
        self.x = stats.loggamma.rvs(5, size=500) + 5

    def test_basic(self):
        N = 5
        svals, ppcc = stats.ppcc_plot(self.x, -10, 10, N=N)
        ppcc_expected = [0.21139644, 0.21384059, 0.98766719, 0.97980182,
                         0.93519298]
        assert_allclose(svals, np.linspace(-10, 10, num=N))
        assert_allclose(ppcc, ppcc_expected)

    def test_dist(self):
        # Test that we can specify distributions both by name and as objects.
        svals1, ppcc1 = stats.ppcc_plot(self.x, -10, 10, dist='tukeylambda')
        svals2, ppcc2 = stats.ppcc_plot(self.x, -10, 10,
                                        dist=stats.tukeylambda)
        assert_allclose(svals1, svals2, rtol=1e-20)
        assert_allclose(ppcc1, ppcc2, rtol=1e-20)
        # Test that 'tukeylambda' is the default dist
        svals3, ppcc3 = stats.ppcc_plot(self.x, -10, 10)
        assert_allclose(svals1, svals3, rtol=1e-20)
        assert_allclose(ppcc1, ppcc3, rtol=1e-20)

    @pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
    def test_plot_kwarg(self):
        # Check with the matplotlib.pyplot module
        fig = plt.figure()
        ax = fig.add_subplot(111)
        stats.ppcc_plot(self.x, -20, 20, plot=plt)
        fig.delaxes(ax)

        # Check that a Matplotlib Axes object is accepted
        ax = fig.add_subplot(111)
        stats.ppcc_plot(self.x, -20, 20, plot=ax)
        plt.close()

    def test_invalid_inputs(self):
        # `b` has to be larger than `a`
        assert_raises(ValueError, stats.ppcc_plot, self.x, 1, 0)

        # Raise ValueError when given an invalid distribution.
        assert_raises(ValueError, stats.ppcc_plot, [1, 2, 3], 0, 1,
                      dist="plate_of_shrimp")

    def test_empty(self):
        # For consistency with probplot return for one empty array,
        # ppcc contains all zeros and svals is the same as for normal array
        # input.
        svals, ppcc = stats.ppcc_plot([], 0, 1)
        assert_allclose(svals, np.linspace(0, 1, num=80))
        assert_allclose(ppcc, np.zeros(80, dtype=float))


class TestPpccMax(object):
    def test_ppcc_max_bad_arg(self):
        # Raise ValueError when given an invalid distribution.
        data = [1]
        assert_raises(ValueError, stats.ppcc_max, data, dist="plate_of_shrimp")

    def test_ppcc_max_basic(self):
        np.random.seed(1234567)
        x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
        # On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
        # it is accurate up to 16 decimals
        assert_almost_equal(stats.ppcc_max(x), -0.71215366521264145, decimal=5)

    def test_dist(self):
        np.random.seed(1234567)
        x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4

        # Test that we can specify distributions both by name and as objects.
        max1 = stats.ppcc_max(x, dist='tukeylambda')
        max2 = stats.ppcc_max(x, dist=stats.tukeylambda)
        assert_almost_equal(max1, -0.71215366521264145, decimal=5)
        assert_almost_equal(max2, -0.71215366521264145, decimal=5)

        # Test that 'tukeylambda' is the default dist
        max3 = stats.ppcc_max(x)
        assert_almost_equal(max3, -0.71215366521264145, decimal=5)

    def test_brack(self):
        np.random.seed(1234567)
        x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
        assert_raises(ValueError, stats.ppcc_max, x, brack=(0.0, 1.0, 0.5))

        # On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
        # it is accurate up to 16 decimals
        assert_almost_equal(stats.ppcc_max(x, brack=(0, 1)),
                            -0.71215366521264145, decimal=5)

        # On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
        # it is accurate up to 16 decimals
        assert_almost_equal(stats.ppcc_max(x, brack=(-2, 2)),
                            -0.71215366521264145, decimal=5)


class TestBoxcox_llf(object):

    def test_basic(self):
        np.random.seed(54321)
        x = stats.norm.rvs(size=10000, loc=10)
        lmbda = 1
        llf = stats.boxcox_llf(lmbda, x)
        llf_expected = -x.size / 2. * np.log(np.sum(x.std()**2))
        assert_allclose(llf, llf_expected)

    def test_array_like(self):
        np.random.seed(54321)
        x = stats.norm.rvs(size=100, loc=10)
        lmbda = 1
        llf = stats.boxcox_llf(lmbda, x)
        llf2 = stats.boxcox_llf(lmbda, list(x))
        assert_allclose(llf, llf2, rtol=1e-12)

    def test_2d_input(self):
        # Note: boxcox_llf() was already working with 2-D input (sort of), so
        # keep it like that.  boxcox() doesn't work with 2-D input though, due
        # to brent() returning a scalar.
        np.random.seed(54321)
        x = stats.norm.rvs(size=100, loc=10)
        lmbda = 1
        llf = stats.boxcox_llf(lmbda, x)
        llf2 = stats.boxcox_llf(lmbda, np.vstack([x, x]).T)
        assert_allclose([llf, llf], llf2, rtol=1e-12)

    def test_empty(self):
        assert_(np.isnan(stats.boxcox_llf(1, [])))


class TestBoxcox(object):

    def test_fixed_lmbda(self):
        np.random.seed(12345)
        x = stats.loggamma.rvs(5, size=50) + 5
        xt = stats.boxcox(x, lmbda=1)
        assert_allclose(xt, x - 1)
        xt = stats.boxcox(x, lmbda=-1)
        assert_allclose(xt, 1 - 1/x)

        xt = stats.boxcox(x, lmbda=0)
        assert_allclose(xt, np.log(x))

        # Also test that array_like input works
        xt = stats.boxcox(list(x), lmbda=0)
        assert_allclose(xt, np.log(x))

    def test_lmbda_None(self):
        np.random.seed(1234567)
        # Start from normal rv's, do inverse transform to check that
        # optimization function gets close to the right answer.
        np.random.seed(1245)
        lmbda = 2.5
        x = stats.norm.rvs(loc=10, size=50000)
        x_inv = (x * lmbda + 1)**(-lmbda)
        xt, maxlog = stats.boxcox(x_inv)

        assert_almost_equal(maxlog, -1 / lmbda, decimal=2)

    def test_alpha(self):
        np.random.seed(1234)
        x = stats.loggamma.rvs(5, size=50) + 5

        # Some regular values for alpha, on a small sample size
        _, _, interval = stats.boxcox(x, alpha=0.75)
        assert_allclose(interval, [4.004485780226041, 5.138756355035744])
        _, _, interval = stats.boxcox(x, alpha=0.05)
        assert_allclose(interval, [1.2138178554857557, 8.209033272375663])

        # Try some extreme values, see we don't hit the N=500 limit
        x = stats.loggamma.rvs(7, size=500) + 15
        _, _, interval = stats.boxcox(x, alpha=0.001)
        assert_allclose(interval, [0.3988867, 11.40553131])
        _, _, interval = stats.boxcox(x, alpha=0.999)
        assert_allclose(interval, [5.83316246, 5.83735292])

    def test_boxcox_bad_arg(self):
        # Raise ValueError if any data value is negative.
        x = np.array([-1])
        assert_raises(ValueError, stats.boxcox, x)

    def test_empty(self):
        assert_(stats.boxcox([]).shape == (0,))


class TestBoxcoxNormmax(object):
    def setup_method(self):
        np.random.seed(12345)
        self.x = stats.loggamma.rvs(5, size=50) + 5

    def test_pearsonr(self):
        maxlog = stats.boxcox_normmax(self.x)
        assert_allclose(maxlog, 1.804465, rtol=1e-6)

    def test_mle(self):
        maxlog = stats.boxcox_normmax(self.x, method='mle')
        assert_allclose(maxlog, 1.758101, rtol=1e-6)

        # Check that boxcox() uses 'mle'
        _, maxlog_boxcox = stats.boxcox(self.x)
        assert_allclose(maxlog_boxcox, maxlog)

    def test_all(self):
        maxlog_all = stats.boxcox_normmax(self.x, method='all')
        assert_allclose(maxlog_all, [1.804465, 1.758101], rtol=1e-6)


class TestBoxcoxNormplot(object):
    def setup_method(self):
        np.random.seed(7654321)
        self.x = stats.loggamma.rvs(5, size=500) + 5

    def test_basic(self):
        N = 5
        lmbdas, ppcc = stats.boxcox_normplot(self.x, -10, 10, N=N)
        ppcc_expected = [0.57783375, 0.83610988, 0.97524311, 0.99756057,
                         0.95843297]
        assert_allclose(lmbdas, np.linspace(-10, 10, num=N))
        assert_allclose(ppcc, ppcc_expected)

    @pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
    def test_plot_kwarg(self):
        # Check with the matplotlib.pyplot module
        fig = plt.figure()
        ax = fig.add_subplot(111)
        stats.boxcox_normplot(self.x, -20, 20, plot=plt)
        fig.delaxes(ax)

        # Check that a Matplotlib Axes object is accepted
        ax = fig.add_subplot(111)
        stats.boxcox_normplot(self.x, -20, 20, plot=ax)
        plt.close()

    def test_invalid_inputs(self):
        # `lb` has to be larger than `la`
        assert_raises(ValueError, stats.boxcox_normplot, self.x, 1, 0)
        # `x` can not contain negative values
        assert_raises(ValueError, stats.boxcox_normplot, [-1, 1], 0, 1)

    def test_empty(self):
        assert_(stats.boxcox_normplot([], 0, 1).size == 0)


class TestYeojohnson_llf(object):

    def test_array_like(self):
        np.random.seed(54321)
        x = stats.norm.rvs(size=100, loc=0)
        lmbda = 1
        llf = stats.yeojohnson_llf(lmbda, x)
        llf2 = stats.yeojohnson_llf(lmbda, list(x))
        assert_allclose(llf, llf2, rtol=1e-12)

    def test_2d_input(self):
        np.random.seed(54321)
        x = stats.norm.rvs(size=100, loc=10)
        lmbda = 1
        llf = stats.yeojohnson_llf(lmbda, x)
        llf2 = stats.yeojohnson_llf(lmbda, np.vstack([x, x]).T)
        assert_allclose([llf, llf], llf2, rtol=1e-12)

    def test_empty(self):
        assert_(np.isnan(stats.yeojohnson_llf(1, [])))


class TestYeojohnson(object):

    def test_fixed_lmbda(self):
        np.random.seed(12345)

        # Test positive input
        x = stats.loggamma.rvs(5, size=50) + 5
        assert np.all(x > 0)
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt, x)
        xt = stats.yeojohnson(x, lmbda=-1)
        assert_allclose(xt, 1 - 1 / (x + 1))
        xt = stats.yeojohnson(x, lmbda=0)
        assert_allclose(xt, np.log(x + 1))
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt, x)

        # Test negative input
        x = stats.loggamma.rvs(5, size=50) - 5
        assert np.all(x < 0)
        xt = stats.yeojohnson(x, lmbda=2)
        assert_allclose(xt, -np.log(-x + 1))
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt, x)
        xt = stats.yeojohnson(x, lmbda=3)
        assert_allclose(xt, 1 / (-x + 1) - 1)

        # test both positive and negative input
        x = stats.loggamma.rvs(5, size=50) - 2
        assert not np.all(x < 0)
        assert not np.all(x >= 0)
        pos = x >= 0
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt[pos], x[pos])
        xt = stats.yeojohnson(x, lmbda=-1)
        assert_allclose(xt[pos], 1 - 1 / (x[pos] + 1))
        xt = stats.yeojohnson(x, lmbda=0)
        assert_allclose(xt[pos], np.log(x[pos] + 1))
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt[pos], x[pos])

        neg = ~pos
        xt = stats.yeojohnson(x, lmbda=2)
        assert_allclose(xt[neg], -np.log(-x[neg] + 1))
        xt = stats.yeojohnson(x, lmbda=1)
        assert_allclose(xt[neg], x[neg])
        xt = stats.yeojohnson(x, lmbda=3)
        assert_allclose(xt[neg], 1 / (-x[neg] + 1) - 1)

    @pytest.mark.parametrize('lmbda', [0, .1, .5, 2])
    def test_lmbda_None(self, lmbda):
        # Start from normal rv's, do inverse transform to check that
        # optimization function gets close to the right answer.

        def _inverse_transform(x, lmbda):
            x_inv = np.zeros(x.shape, dtype=x.dtype)
            pos = x >= 0

            # when x >= 0
            if abs(lmbda) < np.spacing(1.):
                x_inv[pos] = np.exp(x[pos]) - 1
            else:  # lmbda != 0
                x_inv[pos] = np.power(x[pos] * lmbda + 1, 1 / lmbda) - 1

            # when x < 0
            if abs(lmbda - 2) > np.spacing(1.):
                x_inv[~pos] = 1 - np.power(-(2 - lmbda) * x[~pos] + 1,
                                           1 / (2 - lmbda))
            else:  # lmbda == 2
                x_inv[~pos] = 1 - np.exp(-x[~pos])

            return x_inv

        np.random.seed(1234567)
        n_samples = 20000
        x = np.random.normal(loc=0, scale=1, size=(n_samples))

        x_inv = _inverse_transform(x, lmbda)
        xt, maxlog = stats.yeojohnson(x_inv)

        assert_allclose(maxlog, lmbda, atol=1e-2)

        assert_almost_equal(0, np.linalg.norm(x - xt) / n_samples, decimal=2)
        assert_almost_equal(0, xt.mean(), decimal=1)
        assert_almost_equal(1, xt.std(), decimal=1)

    def test_empty(self):
        assert_(stats.yeojohnson([]).shape == (0,))

    def test_array_like(self):
        np.random.seed(54321)
        x = stats.norm.rvs(size=100, loc=0)
        lmbda = 1.5
        xt1, _ = stats.yeojohnson(x)
        xt2, _ = stats.yeojohnson(list(x))
        assert_allclose(xt1, xt2, rtol=1e-12)


class TestYeojohnsonNormmax(object):
    def setup_method(self):
        np.random.seed(12345)
        self.x = stats.loggamma.rvs(5, size=50) + 5

    def test_mle(self):
        maxlog = stats.yeojohnson_normmax(self.x)
        assert_allclose(maxlog, 1.876393, rtol=1e-6)

    def test_darwin_example(self):
        # test from original paper "A new family of power transformations to
        # improve normality or symmetry" by Yeo and Johnson.
        x = [6.1, -8.4, 1.0, 2.0, 0.7, 2.9, 3.5, 5.1, 1.8, 3.6, 7.0, 3.0, 9.3,
             7.5, -6.0]
        lmbda = stats.yeojohnson_normmax(x)
        assert np.allclose(lmbda, 1.305, atol=1e-3)


class TestCircFuncs(object):
    def test_circfuncs(self):
        x = np.array([355, 5, 2, 359, 10, 350])
        M = stats.circmean(x, high=360)
        Mval = 0.167690146
        assert_allclose(M, Mval, rtol=1e-7)

        V = stats.circvar(x, high=360)
        Vval = 42.51955609
        assert_allclose(V, Vval, rtol=1e-7)

        S = stats.circstd(x, high=360)
        Sval = 6.520702116
        assert_allclose(S, Sval, rtol=1e-7)

    def test_circfuncs_small(self):
        x = np.array([20, 21, 22, 18, 19, 20.5, 19.2])
        M1 = x.mean()
        M2 = stats.circmean(x, high=360)
        assert_allclose(M2, M1, rtol=1e-5)

        V1 = x.var()
        V2 = stats.circvar(x, high=360)
        assert_allclose(V2, V1, rtol=1e-4)

        S1 = x.std()
        S2 = stats.circstd(x, high=360)
        assert_allclose(S2, S1, rtol=1e-4)

    def test_circmean_axis(self):
        x = np.array([[355, 5, 2, 359, 10, 350],
                      [351, 7, 4, 352, 9, 349],
                      [357, 9, 8, 358, 4, 356]])
        M1 = stats.circmean(x, high=360)
        M2 = stats.circmean(x.ravel(), high=360)
        assert_allclose(M1, M2, rtol=1e-14)

        M1 = stats.circmean(x, high=360, axis=1)
        M2 = [stats.circmean(x[i], high=360) for i in range(x.shape[0])]
        assert_allclose(M1, M2, rtol=1e-14)

        M1 = stats.circmean(x, high=360, axis=0)
        M2 = [stats.circmean(x[:, i], high=360) for i in range(x.shape[1])]
        assert_allclose(M1, M2, rtol=1e-14)

    def test_circvar_axis(self):
        x = np.array([[355, 5, 2, 359, 10, 350],
                      [351, 7, 4, 352, 9, 349],
                      [357, 9, 8, 358, 4, 356]])

        V1 = stats.circvar(x, high=360)
        V2 = stats.circvar(x.ravel(), high=360)
        assert_allclose(V1, V2, rtol=1e-11)

        V1 = stats.circvar(x, high=360, axis=1)
        V2 = [stats.circvar(x[i], high=360) for i in range(x.shape[0])]
        assert_allclose(V1, V2, rtol=1e-11)

        V1 = stats.circvar(x, high=360, axis=0)
        V2 = [stats.circvar(x[:, i], high=360) for i in range(x.shape[1])]
        assert_allclose(V1, V2, rtol=1e-11)

    def test_circstd_axis(self):
        x = np.array([[355, 5, 2, 359, 10, 350],
                      [351, 7, 4, 352, 9, 349],
                      [357, 9, 8, 358, 4, 356]])

        S1 = stats.circstd(x, high=360)
        S2 = stats.circstd(x.ravel(), high=360)
        assert_allclose(S1, S2, rtol=1e-11)

        S1 = stats.circstd(x, high=360, axis=1)
        S2 = [stats.circstd(x[i], high=360) for i in range(x.shape[0])]
        assert_allclose(S1, S2, rtol=1e-11)

        S1 = stats.circstd(x, high=360, axis=0)
        S2 = [stats.circstd(x[:, i], high=360) for i in range(x.shape[1])]
        assert_allclose(S1, S2, rtol=1e-11)

    def test_circfuncs_array_like(self):
        x = [355, 5, 2, 359, 10, 350]
        assert_allclose(stats.circmean(x, high=360), 0.167690146, rtol=1e-7)
        assert_allclose(stats.circvar(x, high=360), 42.51955609, rtol=1e-7)
        assert_allclose(stats.circstd(x, high=360), 6.520702116, rtol=1e-7)

    def test_empty(self):
        assert_(np.isnan(stats.circmean([])))
        assert_(np.isnan(stats.circstd([])))
        assert_(np.isnan(stats.circvar([])))

    def test_circmean_scalar(self):
        x = 1.
        M1 = x
        M2 = stats.circmean(x)
        assert_allclose(M2, M1, rtol=1e-5)

    def test_circmean_range(self):
        # regression test for gh-6420: circmean(..., high, low) must be
        # between `high` and `low`
        m = stats.circmean(np.arange(0, 2, 0.1), np.pi, -np.pi)
        assert_(m < np.pi)
        assert_(m > -np.pi)

    def test_circfuncs_unit8(self):
        # regression test for gh-7255: overflow when working with
        # numpy uint8 data type
        x = np.array([150, 10], dtype='uint8')
        assert_equal(stats.circmean(x, high=180), 170.0)
        assert_allclose(stats.circvar(x, high=180), 437.45871686, rtol=1e-7)
        assert_allclose(stats.circstd(x, high=180), 20.91551378, rtol=1e-7)

def test_accuracy_wilcoxon():
    freq = [1, 4, 16, 15, 8, 4, 5, 1, 2]
    nums = range(-4, 5)
    x = np.concatenate([[u] * v for u, v in zip(nums, freq)])
    y = np.zeros(x.size)

    T, p = stats.wilcoxon(x, y, "pratt")
    assert_allclose(T, 423)
    assert_allclose(p, 0.00197547303533107)

    T, p = stats.wilcoxon(x, y, "zsplit")
    assert_allclose(T, 441)
    assert_allclose(p, 0.0032145343172473055)

    T, p = stats.wilcoxon(x, y, "wilcox")
    assert_allclose(T, 327)
    assert_allclose(p, 0.00641346115861)

    # Test the 'correction' option, using values computed in R with:
    # > wilcox.test(x, y, paired=TRUE, exact=FALSE, correct={FALSE,TRUE})
    x = np.array([120, 114, 181, 188, 180, 146, 121, 191, 132, 113, 127, 112])
    y = np.array([133, 143, 119, 189, 112, 199, 198, 113, 115, 121, 142, 187])
    T, p = stats.wilcoxon(x, y, correction=False)
    assert_equal(T, 34)
    assert_allclose(p, 0.6948866, rtol=1e-6)
    T, p = stats.wilcoxon(x, y, correction=True)
    assert_equal(T, 34)
    assert_allclose(p, 0.7240817, rtol=1e-6)


def test_wilcoxon_result_attributes():
    x = np.array([120, 114, 181, 188, 180, 146, 121, 191, 132, 113, 127, 112])
    y = np.array([133, 143, 119, 189, 112, 199, 198, 113, 115, 121, 142, 187])
    res = stats.wilcoxon(x, y, correction=False)
    attributes = ('statistic', 'pvalue')
    check_named_results(res, attributes)


def test_wilcoxon_tie():
    # Regression test for gh-2391.
    # Corresponding R code is:
    #   > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=FALSE)
    #   > result$p.value
    #   [1] 0.001565402
    #   > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=TRUE)
    #   > result$p.value
    #   [1] 0.001904195
    stat, p = stats.wilcoxon([0.1] * 10)
    expected_p = 0.001565402
    assert_equal(stat, 0)
    assert_allclose(p, expected_p, rtol=1e-6)

    stat, p = stats.wilcoxon([0.1] * 10, correction=True)
    expected_p = 0.001904195
    assert_equal(stat, 0)
    assert_allclose(p, expected_p, rtol=1e-6)


class TestMedianTest(object):

    def test_bad_n_samples(self):
        # median_test requires at least two samples.
        assert_raises(ValueError, stats.median_test, [1, 2, 3])

    def test_empty_sample(self):
        # Each sample must contain at least one value.
        assert_raises(ValueError, stats.median_test, [], [1, 2, 3])

    def test_empty_when_ties_ignored(self):
        # The grand median is 1, and all values in the first argument are
        # equal to the grand median.  With ties="ignore", those values are
        # ignored, which results in the first sample being (in effect) empty.
        # This should raise a ValueError.
        assert_raises(ValueError, stats.median_test,
                      [1, 1, 1, 1], [2, 0, 1], [2, 0], ties="ignore")

    def test_empty_contingency_row(self):
        # The grand median is 1, and with the default ties="below", all the
        # values in the samples are counted as being below the grand median.
        # This would result a row of zeros in the contingency table, which is
        # an error.
        assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1])

        # With ties="above", all the values are counted as above the
        # grand median.
        assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1],
                      ties="above")

    def test_bad_ties(self):
        assert_raises(ValueError, stats.median_test, [1, 2, 3], [4, 5],
                      ties="foo")

    def test_bad_nan_policy(self):
        assert_raises(ValueError, stats.median_test, [1, 2, 3], [4, 5], nan_policy='foobar')

    def test_bad_keyword(self):
        assert_raises(TypeError, stats.median_test, [1, 2, 3], [4, 5],
                      foo="foo")

    def test_simple(self):
        x = [1, 2, 3]
        y = [1, 2, 3]
        stat, p, med, tbl = stats.median_test(x, y)

        # The median is floating point, but this equality test should be safe.
        assert_equal(med, 2.0)

        assert_array_equal(tbl, [[1, 1], [2, 2]])

        # The expected values of the contingency table equal the contingency
        # table, so the statistic should be 0 and the p-value should be 1.
        assert_equal(stat, 0)
        assert_equal(p, 1)

    def test_ties_options(self):
        # Test the contingency table calculation.
        x = [1, 2, 3, 4]
        y = [5, 6]
        z = [7, 8, 9]
        # grand median is 5.

        # Default 'ties' option is "below".
        stat, p, m, tbl = stats.median_test(x, y, z)
        assert_equal(m, 5)
        assert_equal(tbl, [[0, 1, 3], [4, 1, 0]])

        stat, p, m, tbl = stats.median_test(x, y, z, ties="ignore")
        assert_equal(m, 5)
        assert_equal(tbl, [[0, 1, 3], [4, 0, 0]])

        stat, p, m, tbl = stats.median_test(x, y, z, ties="above")
        assert_equal(m, 5)
        assert_equal(tbl, [[0, 2, 3], [4, 0, 0]])

    def test_nan_policy_options(self):
        x = [1, 2, np.nan]
        y = [4, 5, 6]
        mt1 = stats.median_test(x, y, nan_policy='propagate')
        s, p, m, t = stats.median_test(x, y, nan_policy='omit')

        assert_equal(mt1, (np.nan, np.nan, np.nan, None))
        assert_allclose(s, 0.31250000000000006)
        assert_allclose(p, 0.57615012203057869)
        assert_equal(m, 4.0)
        assert_equal(t, np.array([[0, 2],[2, 1]]))
        assert_raises(ValueError, stats.median_test, x, y, nan_policy='raise')

    def test_basic(self):
        # median_test calls chi2_contingency to compute the test statistic
        # and p-value.  Make sure it hasn't screwed up the call...

        x = [1, 2, 3, 4, 5]
        y = [2, 4, 6, 8]

        stat, p, m, tbl = stats.median_test(x, y)
        assert_equal(m, 4)
        assert_equal(tbl, [[1, 2], [4, 2]])

        exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl)
        assert_allclose(stat, exp_stat)
        assert_allclose(p, exp_p)

        stat, p, m, tbl = stats.median_test(x, y, lambda_=0)
        assert_equal(m, 4)
        assert_equal(tbl, [[1, 2], [4, 2]])

        exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, lambda_=0)
        assert_allclose(stat, exp_stat)
        assert_allclose(p, exp_p)

        stat, p, m, tbl = stats.median_test(x, y, correction=False)
        assert_equal(m, 4)
        assert_equal(tbl, [[1, 2], [4, 2]])

        exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, correction=False)
        assert_allclose(stat, exp_stat)
        assert_allclose(p, exp_p)