test_histograms.py

import numpy as np

from numpy.lib.histograms import histogram, histogramdd, histogram_bin_edges
from numpy.testing import (
    assert_, assert_equal, assert_array_equal, assert_almost_equal,
    assert_array_almost_equal, assert_raises, assert_allclose,
    assert_array_max_ulp, assert_raises_regex, suppress_warnings,
    )
from numpy.testing._private.utils import requires_memory
import pytest


class TestHistogram:

    def setup_method(self):
        pass

    def teardown_method(self):
        pass

    def test_simple(self):
        n = 100
        v = np.random.rand(n)
        (a, b) = histogram(v)
        # check if the sum of the bins equals the number of samples
        assert_equal(np.sum(a, axis=0), n)
        # check that the bin counts are evenly spaced when the data is from
        # a linear function
        (a, b) = histogram(np.linspace(0, 10, 100))
        assert_array_equal(a, 10)

    def test_one_bin(self):
        # Ticket 632
        hist, edges = histogram([1, 2, 3, 4], [1, 2])
        assert_array_equal(hist, [2, ])
        assert_array_equal(edges, [1, 2])
        assert_raises(ValueError, histogram, [1, 2], bins=0)
        h, e = histogram([1, 2], bins=1)
        assert_equal(h, np.array([2]))
        assert_allclose(e, np.array([1., 2.]))

    def test_normed(self):
        sup = suppress_warnings()
        with sup:
            rec = sup.record(np.VisibleDeprecationWarning, '.*normed.*')
            # Check that the integral of the density equals 1.
            n = 100
            v = np.random.rand(n)
            a, b = histogram(v, normed=True)
            area = np.sum(a * np.diff(b))
            assert_almost_equal(area, 1)
            assert_equal(len(rec), 1)

        sup = suppress_warnings()
        with sup:
            rec = sup.record(np.VisibleDeprecationWarning, '.*normed.*')
            # Check with non-constant bin widths (buggy but backwards
            # compatible)
            v = np.arange(10)
            bins = [0, 1, 5, 9, 10]
            a, b = histogram(v, bins, normed=True)
            area = np.sum(a * np.diff(b))
            assert_almost_equal(area, 1)
            assert_equal(len(rec), 1)

    def test_density(self):
        # Check that the integral of the density equals 1.
        n = 100
        v = np.random.rand(n)
        a, b = histogram(v, density=True)
        area = np.sum(a * np.diff(b))
        assert_almost_equal(area, 1)

        # Check with non-constant bin widths
        v = np.arange(10)
        bins = [0, 1, 3, 6, 10]
        a, b = histogram(v, bins, density=True)
        assert_array_equal(a, .1)
        assert_equal(np.sum(a * np.diff(b)), 1)

        # Test that passing False works too
        a, b = histogram(v, bins, density=False)
        assert_array_equal(a, [1, 2, 3, 4])

        # Variable bin widths are especially useful to deal with
        # infinities.
        v = np.arange(10)
        bins = [0, 1, 3, 6, np.inf]
        a, b = histogram(v, bins, density=True)
        assert_array_equal(a, [.1, .1, .1, 0.])

        # Taken from a bug report from N. Becker on the numpy-discussion
        # mailing list Aug. 6, 2010.
        counts, dmy = np.histogram(
            [1, 2, 3, 4], [0.5, 1.5, np.inf], density=True)
        assert_equal(counts, [.25, 0])

    def test_outliers(self):
        # Check that outliers are not tallied
        a = np.arange(10) + .5

        # Lower outliers
        h, b = histogram(a, range=[0, 9])
        assert_equal(h.sum(), 9)

        # Upper outliers
        h, b = histogram(a, range=[1, 10])
        assert_equal(h.sum(), 9)

        # Normalization
        h, b = histogram(a, range=[1, 9], density=True)
        assert_almost_equal((h * np.diff(b)).sum(), 1, decimal=15)

        # Weights
        w = np.arange(10) + .5
        h, b = histogram(a, range=[1, 9], weights=w, density=True)
        assert_equal((h * np.diff(b)).sum(), 1)

        h, b = histogram(a, bins=8, range=[1, 9], weights=w)
        assert_equal(h, w[1:-1])

    def test_arr_weights_mismatch(self):
        a = np.arange(10) + .5
        w = np.arange(11) + .5
        with assert_raises_regex(ValueError, "same shape as"):
            h, b = histogram(a, range=[1, 9], weights=w, density=True)


    def test_type(self):
        # Check the type of the returned histogram
        a = np.arange(10) + .5
        h, b = histogram(a)
        assert_(np.issubdtype(h.dtype, np.integer))

        h, b = histogram(a, density=True)
        assert_(np.issubdtype(h.dtype, np.floating))

        h, b = histogram(a, weights=np.ones(10, int))
        assert_(np.issubdtype(h.dtype, np.integer))

        h, b = histogram(a, weights=np.ones(10, float))
        assert_(np.issubdtype(h.dtype, np.floating))

    def test_f32_rounding(self):
        # gh-4799, check that the rounding of the edges works with float32
        x = np.array([276.318359, -69.593948, 21.329449], dtype=np.float32)
        y = np.array([5005.689453, 4481.327637, 6010.369629], dtype=np.float32)
        counts_hist, xedges, yedges = np.histogram2d(x, y, bins=100)
        assert_equal(counts_hist.sum(), 3.)

    def test_bool_conversion(self):
        # gh-12107
        # Reference integer histogram
        a = np.array([1, 1, 0], dtype=np.uint8)
        int_hist, int_edges = np.histogram(a)

        # Should raise an warning on booleans
        # Ensure that the histograms are equivalent, need to suppress
        # the warnings to get the actual outputs
        with suppress_warnings() as sup:
            rec = sup.record(RuntimeWarning, 'Converting input from .*')
            hist, edges = np.histogram([True, True, False])
            # A warning should be issued
            assert_equal(len(rec), 1)
            assert_array_equal(hist, int_hist)
            assert_array_equal(edges, int_edges)

    def test_weights(self):
        v = np.random.rand(100)
        w = np.ones(100) * 5
        a, b = histogram(v)
        na, nb = histogram(v, density=True)
        wa, wb = histogram(v, weights=w)
        nwa, nwb = histogram(v, weights=w, density=True)
        assert_array_almost_equal(a * 5, wa)
        assert_array_almost_equal(na, nwa)

        # Check weights are properly applied.
        v = np.linspace(0, 10, 10)
        w = np.concatenate((np.zeros(5), np.ones(5)))
        wa, wb = histogram(v, bins=np.arange(11), weights=w)
        assert_array_almost_equal(wa, w)

        # Check with integer weights
        wa, wb = histogram([1, 2, 2, 4], bins=4, weights=[4, 3, 2, 1])
        assert_array_equal(wa, [4, 5, 0, 1])
        wa, wb = histogram(
            [1, 2, 2, 4], bins=4, weights=[4, 3, 2, 1], density=True)
        assert_array_almost_equal(wa, np.array([4, 5, 0, 1]) / 10. / 3. * 4)

        # Check weights with non-uniform bin widths
        a, b = histogram(
            np.arange(9), [0, 1, 3, 6, 10],
            weights=[2, 1, 1, 1, 1, 1, 1, 1, 1], density=True)
        assert_almost_equal(a, [.2, .1, .1, .075])

    def test_exotic_weights(self):

        # Test the use of weights that are not integer or floats, but e.g.
        # complex numbers or object types.

        # Complex weights
        values = np.array([1.3, 2.5, 2.3])
        weights = np.array([1, -1, 2]) + 1j * np.array([2, 1, 2])

        # Check with custom bins
        wa, wb = histogram(values, bins=[0, 2, 3], weights=weights)
        assert_array_almost_equal(wa, np.array([1, 1]) + 1j * np.array([2, 3]))

        # Check with even bins
        wa, wb = histogram(values, bins=2, range=[1, 3], weights=weights)
        assert_array_almost_equal(wa, np.array([1, 1]) + 1j * np.array([2, 3]))

        # Decimal weights
        from decimal import Decimal
        values = np.array([1.3, 2.5, 2.3])
        weights = np.array([Decimal(1), Decimal(2), Decimal(3)])

        # Check with custom bins
        wa, wb = histogram(values, bins=[0, 2, 3], weights=weights)
        assert_array_almost_equal(wa, [Decimal(1), Decimal(5)])

        # Check with even bins
        wa, wb = histogram(values, bins=2, range=[1, 3], weights=weights)
        assert_array_almost_equal(wa, [Decimal(1), Decimal(5)])

    def test_no_side_effects(self):
        # This is a regression test that ensures that values passed to
        # ``histogram`` are unchanged.
        values = np.array([1.3, 2.5, 2.3])
        np.histogram(values, range=[-10, 10], bins=100)
        assert_array_almost_equal(values, [1.3, 2.5, 2.3])

    def test_empty(self):
        a, b = histogram([], bins=([0, 1]))
        assert_array_equal(a, np.array([0]))
        assert_array_equal(b, np.array([0, 1]))

    def test_error_binnum_type (self):
        # Tests if right Error is raised if bins argument is float
        vals = np.linspace(0.0, 1.0, num=100)
        histogram(vals, 5)
        assert_raises(TypeError, histogram, vals, 2.4)

    def test_finite_range(self):
        # Normal ranges should be fine
        vals = np.linspace(0.0, 1.0, num=100)
        histogram(vals, range=[0.25,0.75])
        assert_raises(ValueError, histogram, vals, range=[np.nan,0.75])
        assert_raises(ValueError, histogram, vals, range=[0.25,np.inf])

    def test_invalid_range(self):
        # start of range must be < end of range
        vals = np.linspace(0.0, 1.0, num=100)
        with assert_raises_regex(ValueError, "max must be larger than"):
            np.histogram(vals, range=[0.1, 0.01])

    def test_bin_edge_cases(self):
        # Ensure that floating-point computations correctly place edge cases.
        arr = np.array([337, 404, 739, 806, 1007, 1811, 2012])
        hist, edges = np.histogram(arr, bins=8296, range=(2, 2280))
        mask = hist > 0
        left_edges = edges[:-1][mask]
        right_edges = edges[1:][mask]
        for x, left, right in zip(arr, left_edges, right_edges):
            assert_(x >= left)
            assert_(x < right)

    def test_last_bin_inclusive_range(self):
        arr = np.array([0.,  0.,  0.,  1.,  2.,  3.,  3.,  4.,  5.])
        hist, edges = np.histogram(arr, bins=30, range=(-0.5, 5))
        assert_equal(hist[-1], 1)

    def test_bin_array_dims(self):
        # gracefully handle bins object > 1 dimension
        vals = np.linspace(0.0, 1.0, num=100)
        bins = np.array([[0, 0.5], [0.6, 1.0]])
        with assert_raises_regex(ValueError, "must be 1d"):
            np.histogram(vals, bins=bins)

    def test_unsigned_monotonicity_check(self):
        # Ensures ValueError is raised if bins not increasing monotonically
        # when bins contain unsigned values (see #9222)
        arr = np.array([2])
        bins = np.array([1, 3, 1], dtype='uint64')
        with assert_raises(ValueError):
            hist, edges = np.histogram(arr, bins=bins)

    def test_object_array_of_0d(self):
        # gh-7864
        assert_raises(ValueError,
            histogram, [np.array(0.4) for i in range(10)] + [-np.inf])
        assert_raises(ValueError,
            histogram, [np.array(0.4) for i in range(10)] + [np.inf])

        # these should not crash
        np.histogram([np.array(0.5) for i in range(10)] + [.500000000000001])
        np.histogram([np.array(0.5) for i in range(10)] + [.5])

    def test_some_nan_values(self):
        # gh-7503
        one_nan = np.array([0, 1, np.nan])
        all_nan = np.array([np.nan, np.nan])

        # the internal comparisons with NaN give warnings
        sup = suppress_warnings()
        sup.filter(RuntimeWarning)
        with sup:
            # can't infer range with nan
            assert_raises(ValueError, histogram, one_nan, bins='auto')
            assert_raises(ValueError, histogram, all_nan, bins='auto')

            # explicit range solves the problem
            h, b = histogram(one_nan, bins='auto', range=(0, 1))
            assert_equal(h.sum(), 2)  # nan is not counted
            h, b = histogram(all_nan, bins='auto', range=(0, 1))
            assert_equal(h.sum(), 0)  # nan is not counted

            # as does an explicit set of bins
            h, b = histogram(one_nan, bins=[0, 1])
            assert_equal(h.sum(), 2)  # nan is not counted
            h, b = histogram(all_nan, bins=[0, 1])
            assert_equal(h.sum(), 0)  # nan is not counted

    def test_datetime(self):
        begin = np.datetime64('2000-01-01', 'D')
        offsets = np.array([0, 0, 1, 1, 2, 3, 5, 10, 20])
        bins = np.array([0, 2, 7, 20])
        dates = begin + offsets
        date_bins = begin + bins

        td = np.dtype('timedelta64[D]')

        # Results should be the same for integer offsets or datetime values.
        # For now, only explicit bins are supported, since linspace does not
        # work on datetimes or timedeltas
        d_count, d_edge = histogram(dates, bins=date_bins)
        t_count, t_edge = histogram(offsets.astype(td), bins=bins.astype(td))
        i_count, i_edge = histogram(offsets, bins=bins)

        assert_equal(d_count, i_count)
        assert_equal(t_count, i_count)

        assert_equal((d_edge - begin).astype(int), i_edge)
        assert_equal(t_edge.astype(int), i_edge)

        assert_equal(d_edge.dtype, dates.dtype)
        assert_equal(t_edge.dtype, td)

    def do_signed_overflow_bounds(self, dtype):
        exponent = 8 * np.dtype(dtype).itemsize - 1
        arr = np.array([-2**exponent + 4, 2**exponent - 4], dtype=dtype)
        hist, e = histogram(arr, bins=2)
        assert_equal(e, [-2**exponent + 4, 0, 2**exponent - 4])
        assert_equal(hist, [1, 1])

    def test_signed_overflow_bounds(self):
        self.do_signed_overflow_bounds(np.byte)
        self.do_signed_overflow_bounds(np.short)
        self.do_signed_overflow_bounds(np.intc)
        self.do_signed_overflow_bounds(np.int_)
        self.do_signed_overflow_bounds(np.longlong)

    def do_precision_lower_bound(self, float_small, float_large):
        eps = np.finfo(float_large).eps

        arr = np.array([1.0], float_small)
        range = np.array([1.0 + eps, 2.0], float_large)

        # test is looking for behavior when the bounds change between dtypes
        if range.astype(float_small)[0] != 1:
            return

        # previously crashed
        count, x_loc = np.histogram(arr, bins=1, range=range)
        assert_equal(count, [1])

        # gh-10322 means that the type comes from arr - this may change
        assert_equal(x_loc.dtype, float_small)

    def do_precision_upper_bound(self, float_small, float_large):
        eps = np.finfo(float_large).eps

        arr = np.array([1.0], float_small)
        range = np.array([0.0, 1.0 - eps], float_large)

        # test is looking for behavior when the bounds change between dtypes
        if range.astype(float_small)[-1] != 1:
            return

        # previously crashed
        count, x_loc = np.histogram(arr, bins=1, range=range)
        assert_equal(count, [1])

        # gh-10322 means that the type comes from arr - this may change
        assert_equal(x_loc.dtype, float_small)

    def do_precision(self, float_small, float_large):
        self.do_precision_lower_bound(float_small, float_large)
        self.do_precision_upper_bound(float_small, float_large)

    def test_precision(self):
        # not looping results in a useful stack trace upon failure
        self.do_precision(np.half, np.single)
        self.do_precision(np.half, np.double)
        self.do_precision(np.half, np.longdouble)
        self.do_precision(np.single, np.double)
        self.do_precision(np.single, np.longdouble)
        self.do_precision(np.double, np.longdouble)

    def test_histogram_bin_edges(self):
        hist, e = histogram([1, 2, 3, 4], [1, 2])
        edges = histogram_bin_edges([1, 2, 3, 4], [1, 2])
        assert_array_equal(edges, e)

        arr = np.array([0.,  0.,  0.,  1.,  2.,  3.,  3.,  4.,  5.])
        hist, e = histogram(arr, bins=30, range=(-0.5, 5))
        edges = histogram_bin_edges(arr, bins=30, range=(-0.5, 5))
        assert_array_equal(edges, e)

        hist, e = histogram(arr, bins='auto', range=(0, 1))
        edges = histogram_bin_edges(arr, bins='auto', range=(0, 1))
        assert_array_equal(edges, e)

    @requires_memory(free_bytes=1e10)
    @pytest.mark.slow
    def test_big_arrays(self):
        sample = np.zeros([100000000, 3])
        xbins = 400
        ybins = 400
        zbins = np.arange(16000)
        hist = np.histogramdd(sample=sample, bins=(xbins, ybins, zbins))
        assert_equal(type(hist), type((1, 2)))


class TestHistogramOptimBinNums:
    """
    Provide test coverage when using provided estimators for optimal number of
    bins
    """

    def test_empty(self):
        estimator_list = ['fd', 'scott', 'rice', 'sturges',
                          'doane', 'sqrt', 'auto', 'stone']
        # check it can deal with empty data
        for estimator in estimator_list:
            a, b = histogram([], bins=estimator)
            assert_array_equal(a, np.array([0]))
            assert_array_equal(b, np.array([0, 1]))

    def test_simple(self):
        """
        Straightforward testing with a mixture of linspace data (for
        consistency). All test values have been precomputed and the values
        shouldn't change
        """
        # Some basic sanity checking, with some fixed data.
        # Checking for the correct number of bins
        basic_test = {50:   {'fd': 4,  'scott': 4,  'rice': 8,  'sturges': 7,
                             'doane': 8, 'sqrt': 8, 'auto': 7, 'stone': 2},
                      500:  {'fd': 8,  'scott': 8,  'rice': 16, 'sturges': 10,
                             'doane': 12, 'sqrt': 23, 'auto': 10, 'stone': 9},
                      5000: {'fd': 17, 'scott': 17, 'rice': 35, 'sturges': 14,
                             'doane': 17, 'sqrt': 71, 'auto': 17, 'stone': 20}}

        for testlen, expectedResults in basic_test.items():
            # Create some sort of non uniform data to test with
            # (2 peak uniform mixture)
            x1 = np.linspace(-10, -1, testlen // 5 * 2)
            x2 = np.linspace(1, 10, testlen // 5 * 3)
            x = np.concatenate((x1, x2))
            for estimator, numbins in expectedResults.items():
                a, b = np.histogram(x, estimator)
                assert_equal(len(a), numbins, err_msg="For the {0} estimator "
                             "with datasize of {1}".format(estimator, testlen))

    def test_small(self):
        """
        Smaller datasets have the potential to cause issues with the data
        adaptive methods, especially the FD method. All bin numbers have been
        precalculated.
        """
        small_dat = {1: {'fd': 1, 'scott': 1, 'rice': 1, 'sturges': 1,
                         'doane': 1, 'sqrt': 1, 'stone': 1},
                     2: {'fd': 2, 'scott': 1, 'rice': 3, 'sturges': 2,
                         'doane': 1, 'sqrt': 2, 'stone': 1},
                     3: {'fd': 2, 'scott': 2, 'rice': 3, 'sturges': 3,
                         'doane': 3, 'sqrt': 2, 'stone': 1}}

        for testlen, expectedResults in small_dat.items():
            testdat = np.arange(testlen)
            for estimator, expbins in expectedResults.items():
                a, b = np.histogram(testdat, estimator)
                assert_equal(len(a), expbins, err_msg="For the {0} estimator "
                             "with datasize of {1}".format(estimator, testlen))

    def test_incorrect_methods(self):
        """
        Check a Value Error is thrown when an unknown string is passed in
        """
        check_list = ['mad', 'freeman', 'histograms', 'IQR']
        for estimator in check_list:
            assert_raises(ValueError, histogram, [1, 2, 3], estimator)

    def test_novariance(self):
        """
        Check that methods handle no variance in data
        Primarily for Scott and FD as the SD and IQR are both 0 in this case
        """
        novar_dataset = np.ones(100)
        novar_resultdict = {'fd': 1, 'scott': 1, 'rice': 1, 'sturges': 1,
                            'doane': 1, 'sqrt': 1, 'auto': 1, 'stone': 1}

        for estimator, numbins in novar_resultdict.items():
            a, b = np.histogram(novar_dataset, estimator)
            assert_equal(len(a), numbins, err_msg="{0} estimator, "
                         "No Variance test".format(estimator))

    def test_limited_variance(self):
        """
        Check when IQR is 0, but variance exists, we return the sturges value
        and not the fd value.
        """
        lim_var_data = np.ones(1000)
        lim_var_data[:3] = 0
        lim_var_data[-4:] = 100

        edges_auto = histogram_bin_edges(lim_var_data, 'auto')
        assert_equal(edges_auto, np.linspace(0, 100, 12))

        edges_fd = histogram_bin_edges(lim_var_data, 'fd')
        assert_equal(edges_fd, np.array([0, 100]))

        edges_sturges = histogram_bin_edges(lim_var_data, 'sturges')
        assert_equal(edges_sturges, np.linspace(0, 100, 12))

    def test_outlier(self):
        """
        Check the FD, Scott and Doane with outliers.

        The FD estimates a smaller binwidth since it's less affected by
        outliers. Since the range is so (artificially) large, this means more
        bins, most of which will be empty, but the data of interest usually is
        unaffected. The Scott estimator is more affected and returns fewer bins,
        despite most of the variance being in one area of the data. The Doane
        estimator lies somewhere between the other two.
        """
        xcenter = np.linspace(-10, 10, 50)
        outlier_dataset = np.hstack((np.linspace(-110, -100, 5), xcenter))

        outlier_resultdict = {'fd': 21, 'scott': 5, 'doane': 11, 'stone': 6}

        for estimator, numbins in outlier_resultdict.items():
            a, b = np.histogram(outlier_dataset, estimator)
            assert_equal(len(a), numbins)

    def test_scott_vs_stone(self):
        """Verify that Scott's rule and Stone's rule converges for normally distributed data"""

        def nbins_ratio(seed, size):
            rng = np.random.RandomState(seed)
            x = rng.normal(loc=0, scale=2, size=size)
            a, b = len(np.histogram(x, 'stone')[0]), len(np.histogram(x, 'scott')[0])
            return a / (a + b)

        ll = [[nbins_ratio(seed, size) for size in np.geomspace(start=10, stop=100, num=4).round().astype(int)]
              for seed in range(10)]

        # the average difference between the two methods decreases as the dataset size increases.
        avg = abs(np.mean(ll, axis=0) - 0.5)
        assert_almost_equal(avg, [0.15, 0.09, 0.08, 0.03], decimal=2)

    def test_simple_range(self):
        """
        Straightforward testing with a mixture of linspace data (for
        consistency). Adding in a 3rd mixture that will then be
        completely ignored. All test values have been precomputed and
        the shouldn't change.
        """
        # some basic sanity checking, with some fixed data.
        # Checking for the correct number of bins
        basic_test = {
                      50:   {'fd': 8,  'scott': 8,  'rice': 15,
                             'sturges': 14, 'auto': 14, 'stone': 8},
                      500:  {'fd': 15, 'scott': 16, 'rice': 32,
                             'sturges': 20, 'auto': 20, 'stone': 80},
                      5000: {'fd': 33, 'scott': 33, 'rice': 69,
                             'sturges': 27, 'auto': 33, 'stone': 80}
                     }

        for testlen, expectedResults in basic_test.items():
            # create some sort of non uniform data to test with
            # (3 peak uniform mixture)
            x1 = np.linspace(-10, -1, testlen // 5 * 2)
            x2 = np.linspace(1, 10, testlen // 5 * 3)
            x3 = np.linspace(-100, -50, testlen)
            x = np.hstack((x1, x2, x3))
            for estimator, numbins in expectedResults.items():
                a, b = np.histogram(x, estimator, range = (-20, 20))
                msg = "For the {0} estimator".format(estimator)
                msg += " with datasize of {0}".format(testlen)
                assert_equal(len(a), numbins, err_msg=msg)

    @pytest.mark.parametrize("bins", ['auto', 'fd', 'doane', 'scott',
                                      'stone', 'rice', 'sturges'])
    def test_signed_integer_data(self, bins):
        # Regression test for gh-14379.
        a = np.array([-2, 0, 127], dtype=np.int8)
        hist, edges = np.histogram(a, bins=bins)
        hist32, edges32 = np.histogram(a.astype(np.int32), bins=bins)
        assert_array_equal(hist, hist32)
        assert_array_equal(edges, edges32)

    def test_simple_weighted(self):
        """
        Check that weighted data raises a TypeError
        """
        estimator_list = ['fd', 'scott', 'rice', 'sturges', 'auto']
        for estimator in estimator_list:
            assert_raises(TypeError, histogram, [1, 2, 3],
                          estimator, weights=[1, 2, 3])


class TestHistogramdd:

    def test_simple(self):
        x = np.array([[-.5, .5, 1.5], [-.5, 1.5, 2.5], [-.5, 2.5, .5],
                      [.5,  .5, 1.5], [.5,  1.5, 2.5], [.5,  2.5, 2.5]])
        H, edges = histogramdd(x, (2, 3, 3),
                               range=[[-1, 1], [0, 3], [0, 3]])
        answer = np.array([[[0, 1, 0], [0, 0, 1], [1, 0, 0]],
                           [[0, 1, 0], [0, 0, 1], [0, 0, 1]]])
        assert_array_equal(H, answer)

        # Check normalization
        ed = [[-2, 0, 2], [0, 1, 2, 3], [0, 1, 2, 3]]
        H, edges = histogramdd(x, bins=ed, density=True)
        assert_(np.all(H == answer / 12.))

        # Check that H has the correct shape.
        H, edges = histogramdd(x, (2, 3, 4),
                               range=[[-1, 1], [0, 3], [0, 4]],
                               density=True)
        answer = np.array([[[0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0]],
                           [[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 0]]])
        assert_array_almost_equal(H, answer / 6., 4)
        # Check that a sequence of arrays is accepted and H has the correct
        # shape.
        z = [np.squeeze(y) for y in np.split(x, 3, axis=1)]
        H, edges = histogramdd(
            z, bins=(4, 3, 2), range=[[-2, 2], [0, 3], [0, 2]])
        answer = np.array([[[0, 0], [0, 0], [0, 0]],
                           [[0, 1], [0, 0], [1, 0]],
                           [[0, 1], [0, 0], [0, 0]],
                           [[0, 0], [0, 0], [0, 0]]])
        assert_array_equal(H, answer)

        Z = np.zeros((5, 5, 5))
        Z[list(range(5)), list(range(5)), list(range(5))] = 1.
        H, edges = histogramdd([np.arange(5), np.arange(5), np.arange(5)], 5)
        assert_array_equal(H, Z)

    def test_shape_3d(self):
        # All possible permutations for bins of different lengths in 3D.
        bins = ((5, 4, 6), (6, 4, 5), (5, 6, 4), (4, 6, 5), (6, 5, 4),
                (4, 5, 6))
        r = np.random.rand(10, 3)
        for b in bins:
            H, edges = histogramdd(r, b)
            assert_(H.shape == b)

    def test_shape_4d(self):
        # All possible permutations for bins of different lengths in 4D.
        bins = ((7, 4, 5, 6), (4, 5, 7, 6), (5, 6, 4, 7), (7, 6, 5, 4),
                (5, 7, 6, 4), (4, 6, 7, 5), (6, 5, 7, 4), (7, 5, 4, 6),
                (7, 4, 6, 5), (6, 4, 7, 5), (6, 7, 5, 4), (4, 6, 5, 7),
                (4, 7, 5, 6), (5, 4, 6, 7), (5, 7, 4, 6), (6, 7, 4, 5),
                (6, 5, 4, 7), (4, 7, 6, 5), (4, 5, 6, 7), (7, 6, 4, 5),
                (5, 4, 7, 6), (5, 6, 7, 4), (6, 4, 5, 7), (7, 5, 6, 4))

        r = np.random.rand(10, 4)
        for b in bins:
            H, edges = histogramdd(r, b)
            assert_(H.shape == b)

    def test_weights(self):
        v = np.random.rand(100, 2)
        hist, edges = histogramdd(v)
        n_hist, edges = histogramdd(v, density=True)
        w_hist, edges = histogramdd(v, weights=np.ones(100))
        assert_array_equal(w_hist, hist)
        w_hist, edges = histogramdd(v, weights=np.ones(100) * 2, density=True)
        assert_array_equal(w_hist, n_hist)
        w_hist, edges = histogramdd(v, weights=np.ones(100, int) * 2)
        assert_array_equal(w_hist, 2 * hist)

    def test_identical_samples(self):
        x = np.zeros((10, 2), int)
        hist, edges = histogramdd(x, bins=2)
        assert_array_equal(edges[0], np.array([-0.5, 0., 0.5]))

    def test_empty(self):
        a, b = histogramdd([[], []], bins=([0, 1], [0, 1]))
        assert_array_max_ulp(a, np.array([[0.]]))
        a, b = np.histogramdd([[], [], []], bins=2)
        assert_array_max_ulp(a, np.zeros((2, 2, 2)))

    def test_bins_errors(self):
        # There are two ways to specify bins. Check for the right errors
        # when mixing those.
        x = np.arange(8).reshape(2, 4)
        assert_raises(ValueError, np.histogramdd, x, bins=[-1, 2, 4, 5])
        assert_raises(ValueError, np.histogramdd, x, bins=[1, 0.99, 1, 1])
        assert_raises(
            ValueError, np.histogramdd, x, bins=[1, 1, 1, [1, 2, 3, -3]])
        assert_(np.histogramdd(x, bins=[1, 1, 1, [1, 2, 3, 4]]))

    def test_inf_edges(self):
        # Test using +/-inf bin edges works. See #1788.
        with np.errstate(invalid='ignore'):
            x = np.arange(6).reshape(3, 2)
            expected = np.array([[1, 0], [0, 1], [0, 1]])
            h, e = np.histogramdd(x, bins=[3, [-np.inf, 2, 10]])
            assert_allclose(h, expected)
            h, e = np.histogramdd(x, bins=[3, np.array([-1, 2, np.inf])])
            assert_allclose(h, expected)
            h, e = np.histogramdd(x, bins=[3, [-np.inf, 3, np.inf]])
            assert_allclose(h, expected)

    def test_rightmost_binedge(self):
        # Test event very close to rightmost binedge. See Github issue #4266
        x = [0.9999999995]
        bins = [[0., 0.5, 1.0]]
        hist, _ = histogramdd(x, bins=bins)
        assert_(hist[0] == 0.0)
        assert_(hist[1] == 1.)
        x = [1.0]
        bins = [[0., 0.5, 1.0]]
        hist, _ = histogramdd(x, bins=bins)
        assert_(hist[0] == 0.0)
        assert_(hist[1] == 1.)
        x = [1.0000000001]
        bins = [[0., 0.5, 1.0]]
        hist, _ = histogramdd(x, bins=bins)
        assert_(hist[0] == 0.0)
        assert_(hist[1] == 0.0)
        x = [1.0001]
        bins = [[0., 0.5, 1.0]]
        hist, _ = histogramdd(x, bins=bins)
        assert_(hist[0] == 0.0)
        assert_(hist[1] == 0.0)

    def test_finite_range(self):
        vals = np.random.random((100, 3))
        histogramdd(vals, range=[[0.0, 1.0], [0.25, 0.75], [0.25, 0.5]])
        assert_raises(ValueError, histogramdd, vals,
                      range=[[0.0, 1.0], [0.25, 0.75], [0.25, np.inf]])
        assert_raises(ValueError, histogramdd, vals,
                      range=[[0.0, 1.0], [np.nan, 0.75], [0.25, 0.5]])

    def test_equal_edges(self):
        """ Test that adjacent entries in an edge array can be equal """
        x = np.array([0, 1, 2])
        y = np.array([0, 1, 2])
        x_edges = np.array([0, 2, 2])
        y_edges = 1
        hist, edges = histogramdd((x, y), bins=(x_edges, y_edges))

        hist_expected = np.array([
            [2.],
            [1.],  # x == 2 falls in the final bin
        ])
        assert_equal(hist, hist_expected)

    def test_edge_dtype(self):
        """ Test that if an edge array is input, its type is preserved """
        x = np.array([0, 10, 20])
        y = x / 10
        x_edges = np.array([0, 5, 15, 20])
        y_edges = x_edges / 10
        hist, edges = histogramdd((x, y), bins=(x_edges, y_edges))

        assert_equal(edges[0].dtype, x_edges.dtype)
        assert_equal(edges[1].dtype, y_edges.dtype)

    def test_large_integers(self):
        big = 2**60  # Too large to represent with a full precision float

        x = np.array([0], np.int64)
        x_edges = np.array([-1, +1], np.int64)
        y = big + x
        y_edges = big + x_edges

        hist, edges = histogramdd((x, y), bins=(x_edges, y_edges))

        assert_equal(hist[0, 0], 1)

    def test_density_non_uniform_2d(self):
        # Defines the following grid:
        #
        #    0 2     8
        #   0+-+-----+
        #    + |     +
        #    + |     +
        #   6+-+-----+
        #   8+-+-----+
        x_edges = np.array([0, 2, 8])
        y_edges = np.array([0, 6, 8])
        relative_areas = np.array([
            [3, 9],
            [1, 3]])

        # ensure the number of points in each region is proportional to its area
        x = np.array([1] + [1]*3 + [7]*3 + [7]*9)
        y = np.array([7] + [1]*3 + [7]*3 + [1]*9)

        # sanity check that the above worked as intended
        hist, edges = histogramdd((y, x), bins=(y_edges, x_edges))
        assert_equal(hist, relative_areas)

        # resulting histogram should be uniform, since counts and areas are proportional
        hist, edges = histogramdd((y, x), bins=(y_edges, x_edges), density=True)
        assert_equal(hist, 1 / (8*8))

    def test_density_non_uniform_1d(self):
        # compare to histogram to show the results are the same
        v = np.arange(10)
        bins = np.array([0, 1, 3, 6, 10])
        hist, edges = histogram(v, bins, density=True)
        hist_dd, edges_dd = histogramdd((v,), (bins,), density=True)
        assert_equal(hist, hist_dd)
        assert_equal(edges, edges_dd[0])

    def test_density_via_normed(self):
        # normed should simply alias to density argument
        v = np.arange(10)
        bins = np.array([0, 1, 3, 6, 10])
        hist, edges = histogram(v, bins, density=True)
        hist_dd, edges_dd = histogramdd((v,), (bins,), normed=True)
        assert_equal(hist, hist_dd)
        assert_equal(edges, edges_dd[0])

    def test_density_normed_redundancy(self):
        v = np.arange(10)
        bins = np.array([0, 1, 3, 6, 10])
        with assert_raises_regex(TypeError, "Cannot specify both"):
            hist_dd, edges_dd = histogramdd((v,), (bins,),
                                            density=True,
                                            normed=True)