def test_tukeylambda_stats_known_exact():
"""Compare results with some known exact formulas."""
# Some exact values of the Tukey Lambda variance and kurtosis:
# lambda var kurtosis
# 0 pi**2/3 6/5 (logistic distribution)
# 0.5 4 - pi (5/3 - pi/2)/(pi/4 - 1)**2 - 3
# 1 1/3 -6/5 (uniform distribution on (-1,1))
# 2 1/12 -6/5 (uniform distribution on (-1/2, 1/2))
# lambda = 0
var = tukeylambda_variance(0)
assert_allclose(var, np.pi**2 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(0)
assert_allclose(kurt, 1.2, atol=1e-10)
# lambda = 0.5
var = tukeylambda_variance(0.5)
assert_allclose(var, 4 - np.pi, atol=1e-12)
kurt = tukeylambda_kurtosis(0.5)
desired = (5./3 - np.pi/2) / (np.pi/4 - 1)**2 - 3
assert_allclose(kurt, desired, atol=1e-10)
# lambda = 1
var = tukeylambda_variance(1)
assert_allclose(var, 1.0 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(1)
assert_allclose(kurt, -1.2, atol=1e-10)
# lambda = 2
var = tukeylambda_variance(2)
assert_allclose(var, 1.0 / 12, atol=1e-12)
kurt = tukeylambda_kurtosis(2)
assert_allclose(kurt, -1.2, atol=1e-10)
def test_tukeylambda_stats_mpmath():
"""Compare results with some values that were computed using mpmath."""
a10 = dict(atol=1e-10, rtol=0)
a12 = dict(atol=1e-12, rtol=0)
data = [
# lambda variance kurtosis
[-0.1, 4.78050217874253547, 3.78559520346454510],
[-0.0649, 4.16428023599895777, 2.52019675947435718],
[-0.05, 3.93672267890775277, 2.13129793057777277],
[-0.001, 3.30128380390964882, 1.21452460083542988],
[ 0.001, 3.27850775649572176, 1.18560634779287585],
[ 0.03125, 2.95927803254615800, 0.804487555161819980],
[ 0.05, 2.78281053405464501, 0.611604043886644327],
[ 0.0649, 2.65282386754100551, 0.476834119532774540],
[ 1.2, 0.242153920578588346, -1.23428047169049726],
[ 10.0, 0.00095237579757703597, 2.37810697355144933],
[ 20.0, 0.00012195121951131043, 7.37654321002709531],
]
for lam, var_expected, kurt_expected in data:
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
# Test with vector arguments (most of the other tests are for single
# values).
lam, var_expected, kurt_expected = list(zip(*data))
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
def test_tukeylambda_stats_known_exact():
"""Compare results with some known exact formulas."""
# Some exact values of the Tukey Lambda variance and kurtosis:
# lambda var kurtosis
# 0 pi**2/3 6/5 (logistic distribution)
# 0.5 4 - pi (5/3 - pi/2)/(pi/4 - 1)**2 - 3
# 1 1/3 -6/5 (uniform distribution on (-1,1))
# 2 1/12 -6/5 (uniform distribution on (-1/2, 1/2))
# lambda = 0
var = tukeylambda_variance(0)
assert_allclose(var, np.pi**2 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(0)
assert_allclose(kurt, 1.2, atol=1e-10)
# lambda = 0.5
var = tukeylambda_variance(0.5)
assert_allclose(var, 4 - np.pi, atol=1e-12)
kurt = tukeylambda_kurtosis(0.5)
desired = (5. / 3 - np.pi / 2) / (np.pi / 4 - 1)**2 - 3
assert_allclose(kurt, desired, atol=1e-10)
# lambda = 1
var = tukeylambda_variance(1)
assert_allclose(var, 1.0 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(1)
assert_allclose(kurt, -1.2, atol=1e-10)
# lambda = 2
var = tukeylambda_variance(2)
assert_allclose(var, 1.0 / 12, atol=1e-12)
kurt = tukeylambda_kurtosis(2)
assert_allclose(kurt, -1.2, atol=1e-10)
def test_tukeylambda_stats_mpmath():
"""Compare results with some values that were computed using mpmath."""
a10 = dict(atol=1e-10, rtol=0)
a12 = dict(atol=1e-12, rtol=0)
data = [
# lambda variance kurtosis
[-0.1, 4.78050217874253547, 3.78559520346454510],
[-0.0649, 4.16428023599895777, 2.52019675947435718],
[-0.05, 3.93672267890775277, 2.13129793057777277],
[-0.001, 3.30128380390964882, 1.21452460083542988],
[0.001, 3.27850775649572176, 1.18560634779287585],
[0.03125, 2.95927803254615800, 0.804487555161819980],
[0.05, 2.78281053405464501, 0.611604043886644327],
[0.0649, 2.65282386754100551, 0.476834119532774540],
[1.2, 0.242153920578588346, -1.23428047169049726],
[10.0, 0.00095237579757703597, 2.37810697355144933],
[20.0, 0.00012195121951131043, 7.37654321002709531],
]
for lam, var_expected, kurt_expected in data:
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
# Test with vector arguments (most of the other tests are for single
# values).
lam, var_expected, kurt_expected = zip(*data)
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
def test_tukeylambda_stats_invalid():
"""Test values of lambda outside the domains of the functions."""
lam = [-1.0, -0.5]
var = tukeylambda_variance(lam)
assert_equal(var, np.array([np.nan, np.inf]))
lam = [-1.0, -0.25]
kurt = tukeylambda_kurtosis(lam)
assert_equal(kurt, np.array([np.nan, np.inf]))
def test_tukeylambda_stats_invalid():
"""Test values of lambda outside the domains of the functions."""
lam = [-1.0, -0.5]
var = tukeylambda_variance(lam)
assert_equal(var, np.array([np.nan, np.inf]))
lam = [-1.0, -0.25]
kurt = tukeylambda_kurtosis(lam)
assert_equal(kurt, np.array([np.nan, np.inf]))