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checkhand/venv/lib/python3.11/site-packages/scipy/stats/tests/test_multivariate.py
51hhh 7903ed308f 提交
2025-08-11 12:24:21 +08:00

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"""
Test functions for multivariate normal distributions.
"""
import pickle
from dataclasses import dataclass
from numpy.testing import (assert_allclose, assert_almost_equal,
assert_array_almost_equal, assert_equal,
assert_array_less, assert_)
import pytest
from pytest import raises as assert_raises
from .test_continuous_basic import check_distribution_rvs
import numpy as np
import scipy.linalg
from scipy.stats._multivariate import (_PSD,
_lnB,
multivariate_normal_frozen)
from scipy.stats import (multivariate_normal, multivariate_hypergeom,
matrix_normal, special_ortho_group, ortho_group,
random_correlation, unitary_group, dirichlet,
beta, wishart, multinomial, invwishart, chi2,
invgamma, norm, uniform, ks_2samp, kstest, binom,
hypergeom, multivariate_t, cauchy, normaltest,
random_table, uniform_direction, vonmises_fisher,
dirichlet_multinomial, vonmises)
from scipy.stats import _covariance, Covariance
from scipy.stats._continuous_distns import _norm_pdf as norm_pdf
from scipy import stats
from scipy.integrate import tanhsinh, cubature, quad
from scipy.integrate import romb, qmc_quad, dblquad, tplquad
from scipy.special import multigammaln
import scipy.special as special
from .common_tests import check_random_state_property
from .data._mvt import _qsimvtv
from unittest.mock import patch
def assert_close(res, ref, *args, **kwargs):
res, ref = np.asarray(res), np.asarray(ref)
assert_allclose(res, ref, *args, **kwargs)
assert_equal(res.shape, ref.shape)
class TestCovariance:
def test_input_validation(self):
message = "The input `precision` must be a square, two-dimensional..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaPrecision(np.ones(2))
message = "`precision.shape` must equal `covariance.shape`."
with pytest.raises(ValueError, match=message):
_covariance.CovViaPrecision(np.eye(3), covariance=np.eye(2))
message = "The input `diagonal` must be a one-dimensional array..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaDiagonal("alpaca")
message = "The input `cholesky` must be a square, two-dimensional..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaCholesky(np.ones(2))
message = "The input `eigenvalues` must be a one-dimensional..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaEigendecomposition(("alpaca", np.eye(2)))
message = "The input `eigenvectors` must be a square..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaEigendecomposition((np.ones(2), "alpaca"))
message = "The shapes of `eigenvalues` and `eigenvectors` must be..."
with pytest.raises(ValueError, match=message):
_covariance.CovViaEigendecomposition(([1, 2, 3], np.eye(2)))
_covariance_preprocessing = {"Diagonal": np.diag,
"Precision": np.linalg.inv,
"Cholesky": np.linalg.cholesky,
"Eigendecomposition": np.linalg.eigh,
"PSD": lambda x:
_PSD(x, allow_singular=True)}
_all_covariance_types = np.array(list(_covariance_preprocessing))
_matrices = {"diagonal full rank": np.diag([1, 2, 3]),
"general full rank": [[5, 1, 3], [1, 6, 4], [3, 4, 7]],
"diagonal singular": np.diag([1, 0, 3]),
"general singular": [[5, -1, 0], [-1, 5, 0], [0, 0, 0]]}
_cov_types = {"diagonal full rank": _all_covariance_types,
"general full rank": _all_covariance_types[1:],
"diagonal singular": _all_covariance_types[[0, -2, -1]],
"general singular": _all_covariance_types[-2:]}
@pytest.mark.parametrize("cov_type_name", _all_covariance_types[:-1])
def test_factories(self, cov_type_name):
A = np.diag([1, 2, 3])
x = [-4, 2, 5]
cov_type = getattr(_covariance, f"CovVia{cov_type_name}")
preprocessing = self._covariance_preprocessing[cov_type_name]
factory = getattr(Covariance, f"from_{cov_type_name.lower()}")
res = factory(preprocessing(A))
ref = cov_type(preprocessing(A))
assert type(res) is type(ref)
assert_allclose(res.whiten(x), ref.whiten(x))
@pytest.mark.parametrize("matrix_type", list(_matrices))
@pytest.mark.parametrize("cov_type_name", _all_covariance_types)
def test_covariance(self, matrix_type, cov_type_name):
message = (f"CovVia{cov_type_name} does not support {matrix_type} "
"matrices")
if cov_type_name not in self._cov_types[matrix_type]:
pytest.skip(message)
A = self._matrices[matrix_type]
cov_type = getattr(_covariance, f"CovVia{cov_type_name}")
preprocessing = self._covariance_preprocessing[cov_type_name]
psd = _PSD(A, allow_singular=True)
# test properties
cov_object = cov_type(preprocessing(A))
assert_close(cov_object.log_pdet, psd.log_pdet)
assert_equal(cov_object.rank, psd.rank)
assert_equal(cov_object.shape, np.asarray(A).shape)
assert_close(cov_object.covariance, np.asarray(A))
# test whitening/coloring 1D x
rng = np.random.default_rng(5292808890472453840)
x = rng.random(size=3)
res = cov_object.whiten(x)
ref = x @ psd.U
# res != ref in general; but res @ res == ref @ ref
assert_close(res @ res, ref @ ref)
if hasattr(cov_object, "_colorize") and "singular" not in matrix_type:
# CovViaPSD does not have _colorize
assert_close(cov_object.colorize(res), x)
# test whitening/coloring 3D x
x = rng.random(size=(2, 4, 3))
res = cov_object.whiten(x)
ref = x @ psd.U
assert_close((res**2).sum(axis=-1), (ref**2).sum(axis=-1))
if hasattr(cov_object, "_colorize") and "singular" not in matrix_type:
assert_close(cov_object.colorize(res), x)
# gh-19197 reported that multivariate normal `rvs` produced incorrect
# results when a singular Covariance object was produce using
# `from_eigenvalues`. This was due to an issue in `colorize` with
# singular covariance matrices. Check this edge case, which is skipped
# in the previous tests.
if hasattr(cov_object, "_colorize"):
res = cov_object.colorize(np.eye(len(A)))
assert_close(res.T @ res, A)
@pytest.mark.parametrize("size", [None, tuple(), 1, (2, 4, 3)])
@pytest.mark.parametrize("matrix_type", list(_matrices))
@pytest.mark.parametrize("cov_type_name", _all_covariance_types)
def test_mvn_with_covariance(self, size, matrix_type, cov_type_name):
message = (f"CovVia{cov_type_name} does not support {matrix_type} "
"matrices")
if cov_type_name not in self._cov_types[matrix_type]:
pytest.skip(message)
A = self._matrices[matrix_type]
cov_type = getattr(_covariance, f"CovVia{cov_type_name}")
preprocessing = self._covariance_preprocessing[cov_type_name]
mean = [0.1, 0.2, 0.3]
cov_object = cov_type(preprocessing(A))
mvn = multivariate_normal
dist0 = multivariate_normal(mean, A, allow_singular=True)
dist1 = multivariate_normal(mean, cov_object, allow_singular=True)
rng = np.random.default_rng(5292808890472453840)
x = rng.multivariate_normal(mean, A, size=size)
rng = np.random.default_rng(5292808890472453840)
x1 = mvn.rvs(mean, cov_object, size=size, random_state=rng)
rng = np.random.default_rng(5292808890472453840)
x2 = mvn(mean, cov_object, seed=rng).rvs(size=size)
if isinstance(cov_object, _covariance.CovViaPSD):
assert_close(x1, np.squeeze(x)) # for backward compatibility
assert_close(x2, np.squeeze(x))
else:
assert_equal(x1.shape, x.shape)
assert_equal(x2.shape, x.shape)
assert_close(x2, x1)
assert_close(mvn.pdf(x, mean, cov_object), dist0.pdf(x))
assert_close(dist1.pdf(x), dist0.pdf(x))
assert_close(mvn.logpdf(x, mean, cov_object), dist0.logpdf(x))
assert_close(dist1.logpdf(x), dist0.logpdf(x))
assert_close(mvn.entropy(mean, cov_object), dist0.entropy())
assert_close(dist1.entropy(), dist0.entropy())
@pytest.mark.parametrize("size", [tuple(), (2, 4, 3)])
@pytest.mark.parametrize("cov_type_name", _all_covariance_types)
def test_mvn_with_covariance_cdf(self, size, cov_type_name):
# This is split from the test above because it's slow to be running
# with all matrix types, and there's no need because _mvn.mvnun
# does the calculation. All Covariance needs to do is pass is
# provide the `covariance` attribute.
matrix_type = "diagonal full rank"
A = self._matrices[matrix_type]
cov_type = getattr(_covariance, f"CovVia{cov_type_name}")
preprocessing = self._covariance_preprocessing[cov_type_name]
mean = [0.1, 0.2, 0.3]
cov_object = cov_type(preprocessing(A))
mvn = multivariate_normal
dist0 = multivariate_normal(mean, A, allow_singular=True)
dist1 = multivariate_normal(mean, cov_object, allow_singular=True)
rng = np.random.default_rng(5292808890472453840)
x = rng.multivariate_normal(mean, A, size=size)
assert_close(mvn.cdf(x, mean, cov_object), dist0.cdf(x))
assert_close(dist1.cdf(x), dist0.cdf(x))
assert_close(mvn.logcdf(x, mean, cov_object), dist0.logcdf(x))
assert_close(dist1.logcdf(x), dist0.logcdf(x))
def test_covariance_instantiation(self):
message = "The `Covariance` class cannot be instantiated directly."
with pytest.raises(NotImplementedError, match=message):
Covariance()
@pytest.mark.filterwarnings("ignore::RuntimeWarning") # matrix not PSD
def test_gh9942(self):
# Originally there was a mistake in the `multivariate_normal_frozen`
# `rvs` method that caused all covariance objects to be processed as
# a `_CovViaPSD`. Ensure that this is resolved.
A = np.diag([1, 2, -1e-8])
n = A.shape[0]
mean = np.zeros(n)
# Error if the matrix is processed as a `_CovViaPSD`
with pytest.raises(ValueError, match="The input matrix must be..."):
multivariate_normal(mean, A).rvs()
# No error if it is provided as a `CovViaEigendecomposition`
seed = 3562050283508273023
rng1 = np.random.default_rng(seed)
rng2 = np.random.default_rng(seed)
cov = Covariance.from_eigendecomposition(np.linalg.eigh(A))
rv = multivariate_normal(mean, cov)
res = rv.rvs(random_state=rng1)
ref = multivariate_normal.rvs(mean, cov, random_state=rng2)
assert_equal(res, ref)
def test_gh19197(self):
# gh-19197 reported that multivariate normal `rvs` produced incorrect
# results when a singular Covariance object was produce using
# `from_eigenvalues`. Check that this specific issue is resolved;
# a more general test is included in `test_covariance`.
mean = np.ones(2)
cov = Covariance.from_eigendecomposition((np.zeros(2), np.eye(2)))
dist = scipy.stats.multivariate_normal(mean=mean, cov=cov)
rvs = dist.rvs(size=None)
assert_equal(rvs, mean)
cov = scipy.stats.Covariance.from_eigendecomposition(
(np.array([1., 0.]), np.array([[1., 0.], [0., 400.]])))
dist = scipy.stats.multivariate_normal(mean=mean, cov=cov)
rvs = dist.rvs(size=None)
assert rvs[0] != mean[0]
assert rvs[1] == mean[1]
def _random_covariance(dim, evals, rng, singular=False):
# Generates random covariance matrix with dimensionality `dim` and
# eigenvalues `evals` using provided Generator `rng`. Randomly sets
# some evals to zero if `singular` is True.
A = rng.random((dim, dim))
A = A @ A.T
_, v = np.linalg.eigh(A)
if singular:
zero_eigs = rng.normal(size=dim) > 0
evals[zero_eigs] = 0
cov = v @ np.diag(evals) @ v.T
return cov
def _sample_orthonormal_matrix(n):
M = np.random.randn(n, n)
u, s, v = scipy.linalg.svd(M)
return u
@dataclass
class MVNProblem:
"""Instantiate a multivariate normal integration problem with special structure.
When covariance matrix is a correlation matrix where the off-diagonal entries
``covar[i, j] == lambdas[i]*lambdas[j]`` for ``i != j``, then the multidimensional
integral reduces to a simpler univariate integral that can be numerically integrated
easily.
The ``generate_*()`` classmethods provide a few options for creating variations
of this problem.
References
----------
.. [1] Tong, Y.L. "The Multivariate Normal Distribution".
Springer-Verlag. p192. 1990.
"""
ndim : int
low : np.ndarray
high : np.ndarray
lambdas : np.ndarray
covar : np.ndarray
target_val : float
target_err : float
#: The `generator_halves()` case has an analytically-known true value that we'll
#: record here. It remain None for most cases, though.
true_val : float | None = None
def __init__(self, ndim, low, high, lambdas):
super().__init__()
self.ndim = ndim
self.low = low
self.high = high
self.lambdas = lambdas
self.covar = np.outer(self.lambdas, self.lambdas)
np.fill_diagonal(self.covar, 1.0)
self.find_target()
@classmethod
def generate_semigeneral(cls, ndim, rng=None):
"""Random lambdas, random upper bounds, infinite lower bounds.
"""
rng = np.random.default_rng(rng)
low = np.full(ndim, -np.inf)
high = rng.uniform(0.0, np.sqrt(ndim), size=ndim)
lambdas = rng.uniform(-1.0, 1.0, size=ndim)
self = cls(
ndim=ndim,
low=low,
high=high,
lambdas=lambdas,
)
return self
@classmethod
def generate_constant(cls, ndim, rng=None):
"""Constant off-diagonal covariance, random upper bounds, infinite lower bounds.
"""
rng = np.random.default_rng(rng)
low = np.full(ndim, -np.inf)
high = rng.uniform(0.0, np.sqrt(ndim), size=ndim)
sigma = np.sqrt(rng.uniform(0.0, 1.0))
lambdas = np.full(ndim, sigma)
self = cls(
ndim=ndim,
low=low,
high=high,
lambdas=lambdas,
)
return self
@classmethod
def generate_halves(cls, ndim, rng=None):
"""Off-diagonal covariance of 0.5, negative orthant bounds.
True analytically-derived answer is 1/(ndim+1).
"""
low = np.full(ndim, -np.inf)
high = np.zeros(ndim)
lambdas = np.sqrt(0.5)
self = cls(
ndim=ndim,
low=low,
high=high,
lambdas=lambdas,
)
self.true_val = 1 / (ndim+1)
return self
def find_target(self, **kwds):
"""Perform the simplified integral and store the results.
"""
d = dict(
a=-9.0,
b=+9.0,
)
d.update(kwds)
self.target_val, self.target_err = quad(self.univariate_func, **d)
def _univariate_term(self, t):
"""The parameter-specific term of the univariate integrand,
for separate plotting.
"""
denom = np.sqrt(1 - self.lambdas**2)
return np.prod(
special.ndtr((self.high + self.lambdas*t[:, np.newaxis]) / denom) -
special.ndtr((self.low + self.lambdas*t[:, np.newaxis]) / denom),
axis=1,
)
def univariate_func(self, t):
"""Univariate integrand.
"""
t = np.atleast_1d(t)
return np.squeeze(norm_pdf(t) * self._univariate_term(t))
def plot_integrand(self):
"""Plot the univariate integrand and its component terms for understanding.
"""
from matplotlib import pyplot as plt
t = np.linspace(-9.0, 9.0, 1001)
plt.plot(t, norm_pdf(t), label=r'$\phi(t)$')
plt.plot(t, self._univariate_term(t), label=r'$f(t)$')
plt.plot(t, self.univariate_func(t), label=r'$f(t)*phi(t)$')
plt.legend()
@dataclass
class SingularMVNProblem:
"""Instantiate a multivariate normal integration problem with a special singular
covariance structure.
When covariance matrix is a correlation matrix where the off-diagonal entries
``covar[i, j] == -lambdas[i]*lambdas[j]`` for ``i != j``, and
``sum(lambdas**2 / (1+lambdas**2)) == 1``, then the matrix is singular, and
the multidimensional integral reduces to a simpler univariate integral that
can be numerically integrated fairly easily.
The lower bound must be infinite, though the upper bounds can be general.
References
----------
.. [1] Kwong, K.-S. (1995). "Evaluation of the one-sided percentage points of the
singular multivariate normal distribution." Journal of Statistical
Computation and Simulation, 51(2-4), 121-135. doi:10.1080/00949659508811627
"""
ndim : int
low : np.ndarray
high : np.ndarray
lambdas : np.ndarray
covar : np.ndarray
target_val : float
target_err : float
def __init__(self, ndim, high, lambdas):
self.ndim = ndim
self.high = high
self.lambdas = lambdas
self.low = np.full(ndim, -np.inf)
self.covar = -np.outer(self.lambdas, self.lambdas)
np.fill_diagonal(self.covar, 1.0)
self.find_target()
@classmethod
def generate_semiinfinite(cls, ndim, rng=None):
"""Singular lambdas, random upper bounds.
"""
rng = np.random.default_rng(rng)
high = rng.uniform(0.0, np.sqrt(ndim), size=ndim)
p = rng.dirichlet(np.full(ndim, 1.0))
lambdas = np.sqrt(p / (1-p)) * rng.choice([-1.0, 1.0], size=ndim)
self = cls(
ndim=ndim,
high=high,
lambdas=lambdas,
)
return self
def find_target(self, **kwds):
d = dict(
a=-9.0,
b=+9.0,
)
d.update(kwds)
self.target_val, self.target_err = quad(self.univariate_func, **d)
def _univariate_term(self, t):
denom = np.sqrt(1 + self.lambdas**2)
i1 = np.prod(
special.ndtr((self.high - 1j*self.lambdas*t[:, np.newaxis]) / denom),
axis=1,
)
i2 = np.prod(
special.ndtr((-self.high + 1j*self.lambdas*t[:, np.newaxis]) / denom),
axis=1,
)
# The imaginary part is an odd function, so it can be ignored; it will integrate
# out to 0.
return (i1 - (-1)**self.ndim * i2).real
def univariate_func(self, t):
t = np.atleast_1d(t)
return (norm_pdf(t) * self._univariate_term(t)).squeeze()
def plot_integrand(self):
"""Plot the univariate integrand and its component terms for understanding.
"""
from matplotlib import pyplot as plt
t = np.linspace(-9.0, 9.0, 1001)
plt.plot(t, norm_pdf(t), label=r'$\phi(t)$')
plt.plot(t, self._univariate_term(t), label=r'$f(t)$')
plt.plot(t, self.univariate_func(t), label=r'$f(t)*phi(t)$')
plt.ylim(-0.1, 1.1)
plt.legend()
class TestMultivariateNormal:
def test_input_shape(self):
mu = np.arange(3)
cov = np.identity(2)
assert_raises(ValueError, multivariate_normal.pdf, (0, 1), mu, cov)
assert_raises(ValueError, multivariate_normal.pdf, (0, 1, 2), mu, cov)
assert_raises(ValueError, multivariate_normal.cdf, (0, 1), mu, cov)
assert_raises(ValueError, multivariate_normal.cdf, (0, 1, 2), mu, cov)
def test_scalar_values(self):
np.random.seed(1234)
# When evaluated on scalar data, the pdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on a single vector, the pdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on scalar data, the cdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
cdf = multivariate_normal.cdf(x, mean, cov)
assert_equal(cdf.ndim, 0)
# When evaluated on a single vector, the cdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
cdf = multivariate_normal.cdf(x, mean, cov)
assert_equal(cdf.ndim, 0)
def test_logpdf(self):
# Check that the log of the pdf is in fact the logpdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logpdf(x, mean, cov)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_logpdf_default_values(self):
# Check that the log of the pdf is in fact the logpdf
# with default parameters Mean=None and cov = 1
np.random.seed(1234)
x = np.random.randn(5)
d1 = multivariate_normal.logpdf(x)
d2 = multivariate_normal.pdf(x)
# check whether default values are being used
d3 = multivariate_normal.logpdf(x, None, 1)
d4 = multivariate_normal.pdf(x, None, 1)
assert_allclose(d1, np.log(d2))
assert_allclose(d3, np.log(d4))
def test_logcdf(self):
# Check that the log of the cdf is in fact the logcdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logcdf(x, mean, cov)
d2 = multivariate_normal.cdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_logcdf_default_values(self):
# Check that the log of the cdf is in fact the logcdf
# with default parameters Mean=None and cov = 1
np.random.seed(1234)
x = np.random.randn(5)
d1 = multivariate_normal.logcdf(x)
d2 = multivariate_normal.cdf(x)
# check whether default values are being used
d3 = multivariate_normal.logcdf(x, None, 1)
d4 = multivariate_normal.cdf(x, None, 1)
assert_allclose(d1, np.log(d2))
assert_allclose(d3, np.log(d4))
def test_rank(self):
# Check that the rank is detected correctly.
np.random.seed(1234)
n = 4
mean = np.random.randn(n)
for expected_rank in range(1, n + 1):
s = np.random.randn(n, expected_rank)
cov = np.dot(s, s.T)
distn = multivariate_normal(mean, cov, allow_singular=True)
assert_equal(distn.cov_object.rank, expected_rank)
def test_degenerate_distributions(self):
for n in range(1, 5):
z = np.random.randn(n)
for k in range(1, n):
# Sample a small covariance matrix.
s = np.random.randn(k, k)
cov_kk = np.dot(s, s.T)
# Embed the small covariance matrix into a larger singular one.
cov_nn = np.zeros((n, n))
cov_nn[:k, :k] = cov_kk
# Embed part of the vector in the same way
x = np.zeros(n)
x[:k] = z[:k]
# Define a rotation of the larger low rank matrix.
u = _sample_orthonormal_matrix(n)
cov_rr = np.dot(u, np.dot(cov_nn, u.T))
y = np.dot(u, x)
# Check some identities.
distn_kk = multivariate_normal(np.zeros(k), cov_kk,
allow_singular=True)
distn_nn = multivariate_normal(np.zeros(n), cov_nn,
allow_singular=True)
distn_rr = multivariate_normal(np.zeros(n), cov_rr,
allow_singular=True)
assert_equal(distn_kk.cov_object.rank, k)
assert_equal(distn_nn.cov_object.rank, k)
assert_equal(distn_rr.cov_object.rank, k)
pdf_kk = distn_kk.pdf(x[:k])
pdf_nn = distn_nn.pdf(x)
pdf_rr = distn_rr.pdf(y)
assert_allclose(pdf_kk, pdf_nn)
assert_allclose(pdf_kk, pdf_rr)
logpdf_kk = distn_kk.logpdf(x[:k])
logpdf_nn = distn_nn.logpdf(x)
logpdf_rr = distn_rr.logpdf(y)
assert_allclose(logpdf_kk, logpdf_nn)
assert_allclose(logpdf_kk, logpdf_rr)
# Add an orthogonal component and find the density
y_orth = y + u[:, -1]
pdf_rr_orth = distn_rr.pdf(y_orth)
logpdf_rr_orth = distn_rr.logpdf(y_orth)
# Ensure that this has zero probability
assert_equal(pdf_rr_orth, 0.0)
assert_equal(logpdf_rr_orth, -np.inf)
def test_degenerate_array(self):
# Test that we can generate arrays of random variate from a degenerate
# multivariate normal, and that the pdf for these samples is non-zero
# (i.e. samples from the distribution lie on the subspace)
k = 10
for n in range(2, 6):
for r in range(1, n):
mn = np.zeros(n)
u = _sample_orthonormal_matrix(n)[:, :r]
vr = np.dot(u, u.T)
X = multivariate_normal.rvs(mean=mn, cov=vr, size=k)
pdf = multivariate_normal.pdf(X, mean=mn, cov=vr,
allow_singular=True)
assert_equal(pdf.size, k)
assert np.all(pdf > 0.0)
logpdf = multivariate_normal.logpdf(X, mean=mn, cov=vr,
allow_singular=True)
assert_equal(logpdf.size, k)
assert np.all(logpdf > -np.inf)
def test_large_pseudo_determinant(self):
# Check that large pseudo-determinants are handled appropriately.
# Construct a singular diagonal covariance matrix
# whose pseudo determinant overflows double precision.
large_total_log = 1000.0
npos = 100
nzero = 2
large_entry = np.exp(large_total_log / npos)
n = npos + nzero
cov = np.zeros((n, n), dtype=float)
np.fill_diagonal(cov, large_entry)
cov[-nzero:, -nzero:] = 0
# Check some determinants.
assert_equal(scipy.linalg.det(cov), 0)
assert_equal(scipy.linalg.det(cov[:npos, :npos]), np.inf)
assert_allclose(np.linalg.slogdet(cov[:npos, :npos]),
(1, large_total_log))
# Check the pseudo-determinant.
psd = _PSD(cov)
assert_allclose(psd.log_pdet, large_total_log)
def test_broadcasting(self):
rng = np.random.RandomState(1234)
n = 4
# Construct a random covariance matrix.
data = rng.randn(n, n)
cov = np.dot(data, data.T)
mean = rng.randn(n)
# Construct an ndarray which can be interpreted as
# a 2x3 array whose elements are random data vectors.
X = rng.randn(2, 3, n)
# Check that multiple data points can be evaluated at once.
desired_pdf = multivariate_normal.pdf(X, mean, cov)
desired_cdf = multivariate_normal.cdf(X, mean, cov)
for i in range(2):
for j in range(3):
actual = multivariate_normal.pdf(X[i, j], mean, cov)
assert_allclose(actual, desired_pdf[i,j])
# Repeat for cdf
actual = multivariate_normal.cdf(X[i, j], mean, cov)
assert_allclose(actual, desired_cdf[i,j], rtol=1e-3)
def test_normal_1D(self):
# The probability density function for a 1D normal variable should
# agree with the standard normal distribution in scipy.stats.distributions
x = np.linspace(0, 2, 10)
mean, cov = 1.2, 0.9
scale = cov**0.5
d1 = norm.pdf(x, mean, scale)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, d2)
# The same should hold for the cumulative distribution function
d1 = norm.cdf(x, mean, scale)
d2 = multivariate_normal.cdf(x, mean, cov)
assert_allclose(d1, d2)
def test_marginalization(self):
# Integrating out one of the variables of a 2D Gaussian should
# yield a 1D Gaussian
mean = np.array([2.5, 3.5])
cov = np.array([[.5, 0.2], [0.2, .6]])
n = 2 ** 8 + 1 # Number of samples
delta = 6 / (n - 1) # Grid spacing
v = np.linspace(0, 6, n)
xv, yv = np.meshgrid(v, v)
pos = np.empty((n, n, 2))
pos[:, :, 0] = xv
pos[:, :, 1] = yv
pdf = multivariate_normal.pdf(pos, mean, cov)
# Marginalize over x and y axis
margin_x = romb(pdf, delta, axis=0)
margin_y = romb(pdf, delta, axis=1)
# Compare with standard normal distribution
gauss_x = norm.pdf(v, loc=mean[0], scale=cov[0, 0] ** 0.5)
gauss_y = norm.pdf(v, loc=mean[1], scale=cov[1, 1] ** 0.5)
assert_allclose(margin_x, gauss_x, rtol=1e-2, atol=1e-2)
assert_allclose(margin_y, gauss_y, rtol=1e-2, atol=1e-2)
def test_frozen(self):
# The frozen distribution should agree with the regular one
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
norm_frozen = multivariate_normal(mean, cov)
assert_allclose(norm_frozen.pdf(x), multivariate_normal.pdf(x, mean, cov))
assert_allclose(norm_frozen.logpdf(x),
multivariate_normal.logpdf(x, mean, cov))
assert_allclose(norm_frozen.cdf(x), multivariate_normal.cdf(x, mean, cov))
assert_allclose(norm_frozen.logcdf(x),
multivariate_normal.logcdf(x, mean, cov))
@pytest.mark.parametrize(
'covariance',
[
np.eye(2),
Covariance.from_diagonal([1, 1]),
]
)
def test_frozen_multivariate_normal_exposes_attributes(self, covariance):
mean = np.ones((2,))
cov_should_be = np.eye(2)
norm_frozen = multivariate_normal(mean, covariance)
assert np.allclose(norm_frozen.mean, mean)
assert np.allclose(norm_frozen.cov, cov_should_be)
def test_pseudodet_pinv(self):
# Make sure that pseudo-inverse and pseudo-det agree on cutoff
# Assemble random covariance matrix with large and small eigenvalues
np.random.seed(1234)
n = 7
x = np.random.randn(n, n)
cov = np.dot(x, x.T)
s, u = scipy.linalg.eigh(cov)
s = np.full(n, 0.5)
s[0] = 1.0
s[-1] = 1e-7
cov = np.dot(u, np.dot(np.diag(s), u.T))
# Set cond so that the lowest eigenvalue is below the cutoff
cond = 1e-5
psd = _PSD(cov, cond=cond)
psd_pinv = _PSD(psd.pinv, cond=cond)
# Check that the log pseudo-determinant agrees with the sum
# of the logs of all but the smallest eigenvalue
assert_allclose(psd.log_pdet, np.sum(np.log(s[:-1])))
# Check that the pseudo-determinant of the pseudo-inverse
# agrees with 1 / pseudo-determinant
assert_allclose(-psd.log_pdet, psd_pinv.log_pdet)
def test_exception_nonsquare_cov(self):
cov = [[1, 2, 3], [4, 5, 6]]
assert_raises(ValueError, _PSD, cov)
def test_exception_nonfinite_cov(self):
cov_nan = [[1, 0], [0, np.nan]]
assert_raises(ValueError, _PSD, cov_nan)
cov_inf = [[1, 0], [0, np.inf]]
assert_raises(ValueError, _PSD, cov_inf)
def test_exception_non_psd_cov(self):
cov = [[1, 0], [0, -1]]
assert_raises(ValueError, _PSD, cov)
def test_exception_singular_cov(self):
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.ones((5, 5))
e = np.linalg.LinAlgError
assert_raises(e, multivariate_normal, mean, cov)
assert_raises(e, multivariate_normal.pdf, x, mean, cov)
assert_raises(e, multivariate_normal.logpdf, x, mean, cov)
assert_raises(e, multivariate_normal.cdf, x, mean, cov)
assert_raises(e, multivariate_normal.logcdf, x, mean, cov)
# Message used to be "singular matrix", but this is more accurate.
# See gh-15508
cov = [[1., 0.], [1., 1.]]
msg = "When `allow_singular is False`, the input matrix"
with pytest.raises(np.linalg.LinAlgError, match=msg):
multivariate_normal(cov=cov)
def test_R_values(self):
# Compare the multivariate pdf with some values precomputed
# in R version 3.0.1 (2013-05-16) on Mac OS X 10.6.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_pdf <- dmnorm(cbind(x,y,z), mu, Sigma)
r_pdf = np.array([0.0002214706, 0.0013819953, 0.0049138692,
0.0103803050, 0.0140250800])
x = np.linspace(0, 2, 5)
y = 3 * x - 2
z = x + np.cos(y)
r = np.array([x, y, z]).T
mean = np.array([1, 3, 2], 'd')
cov = np.array([[1, 2, 0], [2, 5, .5], [0, .5, 3]], 'd')
pdf = multivariate_normal.pdf(r, mean, cov)
assert_allclose(pdf, r_pdf, atol=1e-10)
# Compare the multivariate cdf with some values precomputed
# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_cdf <- pmnorm(cbind(x,y,z), mu, Sigma)
r_cdf = np.array([0.0017866215, 0.0267142892, 0.0857098761,
0.1063242573, 0.2501068509])
cdf = multivariate_normal.cdf(r, mean, cov)
assert_allclose(cdf, r_cdf, atol=2e-5)
# Also test bivariate cdf with some values precomputed
# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > mu <- c(1, 3)
# > Sigma <- matrix(c(1,2,2,5), 2, 2)
# > r_cdf2 <- pmnorm(cbind(x,y), mu, Sigma)
r_cdf2 = np.array([0.01262147, 0.05838989, 0.18389571,
0.40696599, 0.66470577])
r2 = np.array([x, y]).T
mean2 = np.array([1, 3], 'd')
cov2 = np.array([[1, 2], [2, 5]], 'd')
cdf2 = multivariate_normal.cdf(r2, mean2, cov2)
assert_allclose(cdf2, r_cdf2, atol=1e-5)
def test_multivariate_normal_rvs_zero_covariance(self):
mean = np.zeros(2)
covariance = np.zeros((2, 2))
model = multivariate_normal(mean, covariance, allow_singular=True)
sample = model.rvs()
assert_equal(sample, [0, 0])
def test_rvs_shape(self):
# Check that rvs parses the mean and covariance correctly, and returns
# an array of the right shape
N = 300
d = 4
sample = multivariate_normal.rvs(mean=np.zeros(d), cov=1, size=N)
assert_equal(sample.shape, (N, d))
sample = multivariate_normal.rvs(mean=None,
cov=np.array([[2, .1], [.1, 1]]),
size=N)
assert_equal(sample.shape, (N, 2))
u = multivariate_normal(mean=0, cov=1)
sample = u.rvs(N)
assert_equal(sample.shape, (N, ))
def test_large_sample(self):
# Generate large sample and compare sample mean and sample covariance
# with mean and covariance matrix.
rng = np.random.RandomState(2846)
n = 3
mean = rng.randn(n)
M = rng.randn(n, n)
cov = np.dot(M, M.T)
size = 5000
sample = multivariate_normal.rvs(mean, cov, size, random_state=rng)
assert_allclose(np.cov(sample.T), cov, rtol=1e-1)
assert_allclose(sample.mean(0), mean, rtol=1e-1)
def test_entropy(self):
rng = np.random.RandomState(2846)
n = 3
mean = rng.randn(n)
M = rng.randn(n, n)
cov = np.dot(M, M.T)
rv = multivariate_normal(mean, cov)
# Check that frozen distribution agrees with entropy function
assert_almost_equal(rv.entropy(), multivariate_normal.entropy(mean, cov))
# Compare entropy with manually computed expression involving
# the sum of the logs of the eigenvalues of the covariance matrix
eigs = np.linalg.eig(cov)[0]
desired = 1 / 2 * (n * (np.log(2 * np.pi) + 1) + np.sum(np.log(eigs)))
assert_almost_equal(desired, rv.entropy())
def test_lnB(self):
alpha = np.array([1, 1, 1])
desired = .5 # e^lnB = 1/2 for [1, 1, 1]
assert_almost_equal(np.exp(_lnB(alpha)), desired)
def test_cdf_with_lower_limit_arrays(self):
# test CDF with lower limit in several dimensions
rng = np.random.default_rng(2408071309372769818)
mean = [0, 0]
cov = np.eye(2)
a = rng.random((4, 3, 2))*6 - 3
b = rng.random((4, 3, 2))*6 - 3
cdf1 = multivariate_normal.cdf(b, mean, cov, lower_limit=a)
cdf2a = multivariate_normal.cdf(b, mean, cov)
cdf2b = multivariate_normal.cdf(a, mean, cov)
ab1 = np.concatenate((a[..., 0:1], b[..., 1:2]), axis=-1)
ab2 = np.concatenate((a[..., 1:2], b[..., 0:1]), axis=-1)
cdf2ab1 = multivariate_normal.cdf(ab1, mean, cov)
cdf2ab2 = multivariate_normal.cdf(ab2, mean, cov)
cdf2 = cdf2a + cdf2b - cdf2ab1 - cdf2ab2
assert_allclose(cdf1, cdf2)
def test_cdf_with_lower_limit_consistency(self):
# check that multivariate normal CDF functions are consistent
rng = np.random.default_rng(2408071309372769818)
mean = rng.random(3)
cov = rng.random((3, 3))
cov = cov @ cov.T
a = rng.random((2, 3))*6 - 3
b = rng.random((2, 3))*6 - 3
cdf1 = multivariate_normal.cdf(b, mean, cov, lower_limit=a)
cdf2 = multivariate_normal(mean, cov).cdf(b, lower_limit=a)
cdf3 = np.exp(multivariate_normal.logcdf(b, mean, cov, lower_limit=a))
cdf4 = np.exp(multivariate_normal(mean, cov).logcdf(b, lower_limit=a))
assert_allclose(cdf2, cdf1, rtol=1e-4)
assert_allclose(cdf3, cdf1, rtol=1e-4)
assert_allclose(cdf4, cdf1, rtol=1e-4)
def test_cdf_signs(self):
# check that sign of output is correct when np.any(lower > x)
mean = np.zeros(3)
cov = np.eye(3)
b = [[1, 1, 1], [0, 0, 0], [1, 0, 1], [0, 1, 0]]
a = [[0, 0, 0], [1, 1, 1], [0, 1, 0], [1, 0, 1]]
# when odd number of elements of b < a, output is negative
expected_signs = np.array([1, -1, -1, 1])
cdf = multivariate_normal.cdf(b, mean, cov, lower_limit=a)
assert_allclose(cdf, cdf[0]*expected_signs)
@pytest.mark.slow
def test_cdf_vs_cubature(self):
ndim = 3
rng = np.random.default_rng(123)
a = rng.uniform(size=(ndim, ndim))
cov = a.T @ a
m = rng.uniform(size=ndim)
dist = multivariate_normal(mean=m, cov=cov)
x = rng.uniform(low=-3, high=3, size=(ndim,))
cdf = dist.cdf(x)
dist_i = multivariate_normal(mean=[0]*ndim, cov=cov)
cdf_i = cubature(dist_i.pdf, [-np.inf]*ndim, x - m).estimate
assert_allclose(cdf, cdf_i, atol=5e-6)
def test_cdf_known(self):
# https://github.com/scipy/scipy/pull/17410#issuecomment-1312628547
for ndim in range(2, 12):
cov = np.full((ndim, ndim), 0.5)
np.fill_diagonal(cov, 1.)
dist = multivariate_normal([0]*ndim, cov=cov)
assert_allclose(
dist.cdf([0]*ndim),
1. / (1. + ndim),
atol=5e-5
)
@pytest.mark.parametrize("ndim", range(2, 10))
@pytest.mark.parametrize("seed", [0xdeadbeef, 0xdd24528764c9773579731c6b022b48e2])
def test_cdf_vs_univariate(self, seed, ndim):
rng = np.random.default_rng(seed)
case = MVNProblem.generate_semigeneral(ndim=ndim, rng=rng)
assert (case.low == -np.inf).all()
dist = multivariate_normal(mean=[0]*ndim, cov=case.covar)
cdf_val = dist.cdf(case.high, rng=rng)
assert_allclose(cdf_val, case.target_val, atol=5e-5)
@pytest.mark.parametrize("ndim", range(2, 11))
@pytest.mark.parametrize("seed", [0xdeadbeef, 0xdd24528764c9773579731c6b022b48e2])
def test_cdf_vs_univariate_2(self, seed, ndim):
rng = np.random.default_rng(seed)
case = MVNProblem.generate_constant(ndim=ndim, rng=rng)
assert (case.low == -np.inf).all()
dist = multivariate_normal(mean=[0]*ndim, cov=case.covar)
cdf_val = dist.cdf(case.high, rng=rng)
assert_allclose(cdf_val, case.target_val, atol=5e-5)
@pytest.mark.parametrize("ndim", range(4, 11))
@pytest.mark.parametrize("seed", [0xdeadbeef, 0xdd24528764c9773579731c6b022b48e4])
def test_cdf_vs_univariate_singular(self, seed, ndim):
# NB: ndim = 2, 3 has much poorer accuracy than ndim > 3 for many seeds.
# No idea why.
rng = np.random.default_rng(seed)
case = SingularMVNProblem.generate_semiinfinite(ndim=ndim, rng=rng)
assert (case.low == -np.inf).all()
dist = multivariate_normal(mean=[0]*ndim, cov=case.covar, allow_singular=True,
# default maxpts is too slow, limit it here
maxpts=10_000*case.covar.shape[0]
)
cdf_val = dist.cdf(case.high, rng=rng)
assert_allclose(cdf_val, case.target_val, atol=1e-3)
def test_mean_cov(self):
# test the interaction between a Covariance object and mean
P = np.diag(1 / np.array([1, 2, 3]))
cov_object = _covariance.CovViaPrecision(P)
message = "`cov` represents a covariance matrix in 3 dimensions..."
with pytest.raises(ValueError, match=message):
multivariate_normal.entropy([0, 0], cov_object)
with pytest.raises(ValueError, match=message):
multivariate_normal([0, 0], cov_object)
x = [0.5, 0.5, 0.5]
ref = multivariate_normal.pdf(x, [0, 0, 0], cov_object)
assert_equal(multivariate_normal.pdf(x, cov=cov_object), ref)
ref = multivariate_normal.pdf(x, [1, 1, 1], cov_object)
assert_equal(multivariate_normal.pdf(x, 1, cov=cov_object), ref)
def test_fit_wrong_fit_data_shape(self):
data = [1, 3]
error_msg = "`x` must be two-dimensional."
with pytest.raises(ValueError, match=error_msg):
multivariate_normal.fit(data)
@pytest.mark.parametrize('dim', (3, 5))
def test_fit_correctness(self, dim):
rng = np.random.default_rng(4385269356937404)
x = rng.random((100, dim))
mean_est, cov_est = multivariate_normal.fit(x)
mean_ref, cov_ref = np.mean(x, axis=0), np.cov(x.T, ddof=0)
assert_allclose(mean_est, mean_ref, atol=1e-15)
assert_allclose(cov_est, cov_ref, rtol=1e-15)
def test_fit_both_parameters_fixed(self):
data = np.full((2, 1), 3)
mean_fixed = 1.
cov_fixed = np.atleast_2d(1.)
mean, cov = multivariate_normal.fit(data, fix_mean=mean_fixed,
fix_cov=cov_fixed)
assert_equal(mean, mean_fixed)
assert_equal(cov, cov_fixed)
@pytest.mark.parametrize('fix_mean', [np.zeros((2, 2)),
np.zeros((3, ))])
def test_fit_fix_mean_input_validation(self, fix_mean):
msg = ("`fix_mean` must be a one-dimensional array the same "
"length as the dimensionality of the vectors `x`.")
with pytest.raises(ValueError, match=msg):
multivariate_normal.fit(np.eye(2), fix_mean=fix_mean)
@pytest.mark.parametrize('fix_cov', [np.zeros((2, )),
np.zeros((3, 2)),
np.zeros((4, 4))])
def test_fit_fix_cov_input_validation_dimension(self, fix_cov):
msg = ("`fix_cov` must be a two-dimensional square array "
"of same side length as the dimensionality of the "
"vectors `x`.")
with pytest.raises(ValueError, match=msg):
multivariate_normal.fit(np.eye(3), fix_cov=fix_cov)
def test_fit_fix_cov_not_positive_semidefinite(self):
error_msg = "`fix_cov` must be symmetric positive semidefinite."
with pytest.raises(ValueError, match=error_msg):
fix_cov = np.array([[1., 0.], [0., -1.]])
multivariate_normal.fit(np.eye(2), fix_cov=fix_cov)
def test_fit_fix_mean(self):
rng = np.random.default_rng(4385269356937404)
loc = rng.random(3)
A = rng.random((3, 3))
cov = np.dot(A, A.T)
samples = multivariate_normal.rvs(mean=loc, cov=cov, size=100,
random_state=rng)
mean_free, cov_free = multivariate_normal.fit(samples)
logp_free = multivariate_normal.logpdf(samples, mean=mean_free,
cov=cov_free).sum()
mean_fix, cov_fix = multivariate_normal.fit(samples, fix_mean=loc)
assert_equal(mean_fix, loc)
logp_fix = multivariate_normal.logpdf(samples, mean=mean_fix,
cov=cov_fix).sum()
# test that fixed parameters result in lower likelihood than free
# parameters
assert logp_fix < logp_free
# test that a small perturbation of the resulting parameters
# has lower likelihood than the estimated parameters
A = rng.random((3, 3))
m = 1e-8 * np.dot(A, A.T)
cov_perturbed = cov_fix + m
logp_perturbed = (multivariate_normal.logpdf(samples,
mean=mean_fix,
cov=cov_perturbed)
).sum()
assert logp_perturbed < logp_fix
def test_fit_fix_cov(self):
rng = np.random.default_rng(4385269356937404)
loc = rng.random(3)
A = rng.random((3, 3))
cov = np.dot(A, A.T)
samples = multivariate_normal.rvs(mean=loc, cov=cov,
size=100, random_state=rng)
mean_free, cov_free = multivariate_normal.fit(samples)
logp_free = multivariate_normal.logpdf(samples, mean=mean_free,
cov=cov_free).sum()
mean_fix, cov_fix = multivariate_normal.fit(samples, fix_cov=cov)
assert_equal(mean_fix, np.mean(samples, axis=0))
assert_equal(cov_fix, cov)
logp_fix = multivariate_normal.logpdf(samples, mean=mean_fix,
cov=cov_fix).sum()
# test that fixed parameters result in lower likelihood than free
# parameters
assert logp_fix < logp_free
# test that a small perturbation of the resulting parameters
# has lower likelihood than the estimated parameters
mean_perturbed = mean_fix + 1e-8 * rng.random(3)
logp_perturbed = (multivariate_normal.logpdf(samples,
mean=mean_perturbed,
cov=cov_fix)
).sum()
assert logp_perturbed < logp_fix
class TestMatrixNormal:
def test_bad_input(self):
# Check that bad inputs raise errors
num_rows = 4
num_cols = 3
M = np.full((num_rows,num_cols), 0.3)
U = 0.5 * np.identity(num_rows) + np.full((num_rows, num_rows), 0.5)
V = 0.7 * np.identity(num_cols) + np.full((num_cols, num_cols), 0.3)
# Incorrect dimensions
assert_raises(ValueError, matrix_normal, np.zeros((5,4,3)))
assert_raises(ValueError, matrix_normal, M, np.zeros(10), V)
assert_raises(ValueError, matrix_normal, M, U, np.zeros(10))
assert_raises(ValueError, matrix_normal, M, U, U)
assert_raises(ValueError, matrix_normal, M, V, V)
assert_raises(ValueError, matrix_normal, M.T, U, V)
e = np.linalg.LinAlgError
# Singular covariance for the rvs method of a non-frozen instance
assert_raises(e, matrix_normal.rvs,
M, U, np.ones((num_cols, num_cols)))
assert_raises(e, matrix_normal.rvs,
M, np.ones((num_rows, num_rows)), V)
# Singular covariance for a frozen instance
assert_raises(e, matrix_normal, M, U, np.ones((num_cols, num_cols)))
assert_raises(e, matrix_normal, M, np.ones((num_rows, num_rows)), V)
def test_default_inputs(self):
# Check that default argument handling works
num_rows = 4
num_cols = 3
M = np.full((num_rows,num_cols), 0.3)
U = 0.5 * np.identity(num_rows) + np.full((num_rows, num_rows), 0.5)
V = 0.7 * np.identity(num_cols) + np.full((num_cols, num_cols), 0.3)
Z = np.zeros((num_rows, num_cols))
Zr = np.zeros((num_rows, 1))
Zc = np.zeros((1, num_cols))
Ir = np.identity(num_rows)
Ic = np.identity(num_cols)
I1 = np.identity(1)
assert_equal(matrix_normal.rvs(mean=M, rowcov=U, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(mean=M).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(rowcov=U).shape,
(num_rows, 1))
assert_equal(matrix_normal.rvs(colcov=V).shape,
(1, num_cols))
assert_equal(matrix_normal.rvs(mean=M, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(mean=M, rowcov=U).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(rowcov=U, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal(mean=M).rowcov, Ir)
assert_equal(matrix_normal(mean=M).colcov, Ic)
assert_equal(matrix_normal(rowcov=U).mean, Zr)
assert_equal(matrix_normal(rowcov=U).colcov, I1)
assert_equal(matrix_normal(colcov=V).mean, Zc)
assert_equal(matrix_normal(colcov=V).rowcov, I1)
assert_equal(matrix_normal(mean=M, rowcov=U).colcov, Ic)
assert_equal(matrix_normal(mean=M, colcov=V).rowcov, Ir)
assert_equal(matrix_normal(rowcov=U, colcov=V).mean, Z)
def test_covariance_expansion(self):
# Check that covariance can be specified with scalar or vector
num_rows = 4
num_cols = 3
M = np.full((num_rows, num_cols), 0.3)
Uv = np.full(num_rows, 0.2)
Us = 0.2
Vv = np.full(num_cols, 0.1)
Vs = 0.1
Ir = np.identity(num_rows)
Ic = np.identity(num_cols)
assert_equal(matrix_normal(mean=M, rowcov=Uv, colcov=Vv).rowcov,
0.2*Ir)
assert_equal(matrix_normal(mean=M, rowcov=Uv, colcov=Vv).colcov,
0.1*Ic)
assert_equal(matrix_normal(mean=M, rowcov=Us, colcov=Vs).rowcov,
0.2*Ir)
assert_equal(matrix_normal(mean=M, rowcov=Us, colcov=Vs).colcov,
0.1*Ic)
def test_frozen_matrix_normal(self):
for i in range(1,5):
for j in range(1,5):
M = np.full((i,j), 0.3)
U = 0.5 * np.identity(i) + np.full((i,i), 0.5)
V = 0.7 * np.identity(j) + np.full((j,j), 0.3)
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
rvs1 = frozen.rvs(random_state=1234)
rvs2 = matrix_normal.rvs(mean=M, rowcov=U, colcov=V,
random_state=1234)
assert_equal(rvs1, rvs2)
X = frozen.rvs(random_state=1234)
pdf1 = frozen.pdf(X)
pdf2 = matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)
assert_equal(pdf1, pdf2)
logpdf1 = frozen.logpdf(X)
logpdf2 = matrix_normal.logpdf(X, mean=M, rowcov=U, colcov=V)
assert_equal(logpdf1, logpdf2)
def test_matches_multivariate(self):
# Check that the pdfs match those obtained by vectorising and
# treating as a multivariate normal.
for i in range(1,5):
for j in range(1,5):
M = np.full((i,j), 0.3)
U = 0.5 * np.identity(i) + np.full((i,i), 0.5)
V = 0.7 * np.identity(j) + np.full((j,j), 0.3)
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X = frozen.rvs(random_state=1234)
pdf1 = frozen.pdf(X)
logpdf1 = frozen.logpdf(X)
entropy1 = frozen.entropy()
vecX = X.T.flatten()
vecM = M.T.flatten()
cov = np.kron(V,U)
pdf2 = multivariate_normal.pdf(vecX, mean=vecM, cov=cov)
logpdf2 = multivariate_normal.logpdf(vecX, mean=vecM, cov=cov)
entropy2 = multivariate_normal.entropy(mean=vecM, cov=cov)
assert_allclose(pdf1, pdf2, rtol=1E-10)
assert_allclose(logpdf1, logpdf2, rtol=1E-10)
assert_allclose(entropy1, entropy2)
def test_array_input(self):
# Check array of inputs has the same output as the separate entries.
num_rows = 4
num_cols = 3
M = np.full((num_rows,num_cols), 0.3)
U = 0.5 * np.identity(num_rows) + np.full((num_rows, num_rows), 0.5)
V = 0.7 * np.identity(num_cols) + np.full((num_cols, num_cols), 0.3)
N = 10
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X1 = frozen.rvs(size=N, random_state=1234)
X2 = frozen.rvs(size=N, random_state=4321)
X = np.concatenate((X1[np.newaxis,:,:,:],X2[np.newaxis,:,:,:]), axis=0)
assert_equal(X.shape, (2, N, num_rows, num_cols))
array_logpdf = frozen.logpdf(X)
assert_equal(array_logpdf.shape, (2, N))
for i in range(2):
for j in range(N):
separate_logpdf = matrix_normal.logpdf(X[i,j], mean=M,
rowcov=U, colcov=V)
assert_allclose(separate_logpdf, array_logpdf[i,j], 1E-10)
def test_moments(self):
# Check that the sample moments match the parameters
num_rows = 4
num_cols = 3
M = np.full((num_rows,num_cols), 0.3)
U = 0.5 * np.identity(num_rows) + np.full((num_rows, num_rows), 0.5)
V = 0.7 * np.identity(num_cols) + np.full((num_cols, num_cols), 0.3)
N = 1000
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X = frozen.rvs(size=N, random_state=1234)
sample_mean = np.mean(X,axis=0)
assert_allclose(sample_mean, M, atol=0.1)
sample_colcov = np.cov(X.reshape(N*num_rows,num_cols).T)
assert_allclose(sample_colcov, V, atol=0.1)
sample_rowcov = np.cov(np.swapaxes(X,1,2).reshape(
N*num_cols,num_rows).T)
assert_allclose(sample_rowcov, U, atol=0.1)
def test_samples(self):
# Regression test to ensure that we always generate the same stream of
# random variates.
actual = matrix_normal.rvs(
mean=np.array([[1, 2], [3, 4]]),
rowcov=np.array([[4, -1], [-1, 2]]),
colcov=np.array([[5, 1], [1, 10]]),
random_state=np.random.default_rng(0),
size=2
)
expected = np.array(
[[[1.56228264238181, -1.24136424071189],
[2.46865788392114, 6.22964440489445]],
[[3.86405716144353, 10.73714311429529],
[2.59428444080606, 5.79987854490876]]]
)
assert_allclose(actual, expected)
class TestDirichlet:
def test_frozen_dirichlet(self):
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
assert_equal(d.var(), dirichlet.var(alpha))
assert_equal(d.mean(), dirichlet.mean(alpha))
assert_equal(d.entropy(), dirichlet.entropy(alpha))
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
assert_equal(d.pdf(x[:-1]), dirichlet.pdf(x[:-1], alpha))
assert_equal(d.logpdf(x[:-1]), dirichlet.logpdf(x[:-1], alpha))
def test_numpy_rvs_shape_compatibility(self):
np.random.seed(2846)
alpha = np.array([1.0, 2.0, 3.0])
x = np.random.dirichlet(alpha, size=7)
assert_equal(x.shape, (7, 3))
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
dirichlet.pdf(x.T, alpha)
dirichlet.pdf(x.T[:-1], alpha)
dirichlet.logpdf(x.T, alpha)
dirichlet.logpdf(x.T[:-1], alpha)
def test_alpha_with_zeros(self):
np.random.seed(2846)
alpha = [1.0, 0.0, 3.0]
# don't pass invalid alpha to np.random.dirichlet
x = np.random.dirichlet(np.maximum(1e-9, alpha), size=7).T
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_with_negative_entries(self):
np.random.seed(2846)
alpha = [1.0, -2.0, 3.0]
# don't pass invalid alpha to np.random.dirichlet
x = np.random.dirichlet(np.maximum(1e-9, alpha), size=7).T
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_zeros(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, 0.0, 0.2, 0.7])
dirichlet.pdf(x, alpha)
dirichlet.logpdf(x, alpha)
alpha = np.array([1.0, 1.0, 1.0, 1.0])
assert_almost_equal(dirichlet.pdf(x, alpha), 6)
assert_almost_equal(dirichlet.logpdf(x, alpha), np.log(6))
def test_data_with_zeros_and_small_alpha(self):
alpha = np.array([1.0, 0.5, 3.0, 4.0])
x = np.array([0.1, 0.0, 0.2, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_negative_entries(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, -0.1, 0.3, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_too_large_entries(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, 1.1, 0.3, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_too_deep_c(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.full((2, 7, 7), 1 / 14)
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_too_deep(self):
alpha = np.array([[1.0, 2.0], [3.0, 4.0]])
x = np.full((2, 2, 7), 1 / 4)
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_correct_depth(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.full((3, 7), 1 / 3)
dirichlet.pdf(x, alpha)
dirichlet.logpdf(x, alpha)
def test_non_simplex_data(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.full((3, 7), 1 / 2)
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_vector_too_short(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.full((2, 7), 1 / 2)
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_vector_too_long(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.full((5, 7), 1 / 5)
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_mean_var_cov(self):
# Reference values calculated by hand and confirmed with Mathematica, e.g.
# `Covariance[DirichletDistribution[{ 1, 0.8, 0.2, 10^-300}]]`
alpha = np.array([1., 0.8, 0.2])
d = dirichlet(alpha)
expected_mean = [0.5, 0.4, 0.1]
expected_var = [1. / 12., 0.08, 0.03]
expected_cov = [
[ 1. / 12, -1. / 15, -1. / 60],
[-1. / 15, 2. / 25, -1. / 75],
[-1. / 60, -1. / 75, 3. / 100],
]
assert_array_almost_equal(d.mean(), expected_mean)
assert_array_almost_equal(d.var(), expected_var)
assert_array_almost_equal(d.cov(), expected_cov)
def test_scalar_values(self):
alpha = np.array([0.2])
d = dirichlet(alpha)
# For alpha of length 1, mean and var should be scalar instead of array
assert_equal(d.mean().ndim, 0)
assert_equal(d.var().ndim, 0)
assert_equal(d.pdf([1.]).ndim, 0)
assert_equal(d.logpdf([1.]).ndim, 0)
def test_K_and_K_minus_1_calls_equal(self):
# Test that calls with K and K-1 entries yield the same results.
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
assert_almost_equal(d.pdf(x[:-1]), d.pdf(x))
def test_multiple_entry_calls(self):
# Test that calls with multiple x vectors as matrix work
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
num_tests = 10
num_multiple = 5
xm = None
for i in range(num_tests):
for m in range(num_multiple):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
if xm is not None:
xm = np.vstack((xm, x))
else:
xm = x
rm = d.pdf(xm.T)
rs = None
for xs in xm:
r = d.pdf(xs)
if rs is not None:
rs = np.append(rs, r)
else:
rs = r
assert_array_almost_equal(rm, rs)
def test_2D_dirichlet_is_beta(self):
np.random.seed(2846)
alpha = np.random.uniform(10e-10, 100, 2)
d = dirichlet(alpha)
b = beta(alpha[0], alpha[1])
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, 2)
x /= np.sum(x)
assert_almost_equal(b.pdf(x), d.pdf([x]))
assert_almost_equal(b.mean(), d.mean()[0])
assert_almost_equal(b.var(), d.var()[0])
def test_multivariate_normal_dimensions_mismatch():
# Regression test for GH #3493. Check that setting up a PDF with a mean of
# length M and a covariance matrix of size (N, N), where M != N, raises a
# ValueError with an informative error message.
mu = np.array([0.0, 0.0])
sigma = np.array([[1.0]])
assert_raises(ValueError, multivariate_normal, mu, sigma)
# A simple check that the right error message was passed along. Checking
# that the entire message is there, word for word, would be somewhat
# fragile, so we just check for the leading part.
try:
multivariate_normal(mu, sigma)
except ValueError as e:
msg = "Dimension mismatch"
assert_equal(str(e)[:len(msg)], msg)
class TestWishart:
def test_scale_dimensions(self):
# Test that we can call the Wishart with various scale dimensions
# Test case: dim=1, scale=1
true_scale = np.array(1, ndmin=2)
scales = [
1, # scalar
[1], # iterable
np.array(1), # 0-dim
np.r_[1], # 1-dim
np.array(1, ndmin=2) # 2-dim
]
for scale in scales:
w = wishart(1, scale)
assert_equal(w.scale, true_scale)
assert_equal(w.scale.shape, true_scale.shape)
# Test case: dim=2, scale=[[1,0]
# [0,2]
true_scale = np.array([[1,0],
[0,2]])
scales = [
[1,2], # iterable
np.r_[1,2], # 1-dim
np.array([[1,0], # 2-dim
[0,2]])
]
for scale in scales:
w = wishart(2, scale)
assert_equal(w.scale, true_scale)
assert_equal(w.scale.shape, true_scale.shape)
# We cannot call with a df < dim - 1
assert_raises(ValueError, wishart, 1, np.eye(2))
# But we can call with dim - 1 < df < dim
wishart(1.1, np.eye(2)) # no error
# see gh-5562
# We cannot call with a 3-dimension array
scale = np.array(1, ndmin=3)
assert_raises(ValueError, wishart, 1, scale)
def test_quantile_dimensions(self):
# Test that we can call the Wishart rvs with various quantile dimensions
# If dim == 1, consider x.shape = [1,1,1]
X = [
1, # scalar
[1], # iterable
np.array(1), # 0-dim
np.r_[1], # 1-dim
np.array(1, ndmin=2), # 2-dim
np.array([1], ndmin=3) # 3-dim
]
w = wishart(1,1)
density = w.pdf(np.array(1, ndmin=3))
for x in X:
assert_equal(w.pdf(x), density)
# If dim == 1, consider x.shape = [1,1,*]
X = [
[1,2,3], # iterable
np.r_[1,2,3], # 1-dim
np.array([1,2,3], ndmin=3) # 3-dim
]
w = wishart(1,1)
density = w.pdf(np.array([1,2,3], ndmin=3))
for x in X:
assert_equal(w.pdf(x), density)
# If dim == 2, consider x.shape = [2,2,1]
# where x[:,:,*] = np.eye(1)*2
X = [
2, # scalar
[2,2], # iterable
np.array(2), # 0-dim
np.r_[2,2], # 1-dim
np.array([[2,0],
[0,2]]), # 2-dim
np.array([[2,0],
[0,2]])[:,:,np.newaxis] # 3-dim
]
w = wishart(2,np.eye(2))
density = w.pdf(np.array([[2,0],
[0,2]])[:,:,np.newaxis])
for x in X:
assert_equal(w.pdf(x), density)
def test_frozen(self):
# Test that the frozen and non-frozen Wishart gives the same answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim-1) // 2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim-1) // 2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
w = wishart(df, scale)
assert_equal(w.var(), wishart.var(df, scale))
assert_equal(w.mean(), wishart.mean(df, scale))
assert_equal(w.mode(), wishart.mode(df, scale))
assert_equal(w.entropy(), wishart.entropy(df, scale))
assert_equal(w.pdf(x), wishart.pdf(x, df, scale))
def test_wishart_2D_rvs(self):
dim = 3
df = 10
# Construct a simple non-diagonal positive definite matrix
scale = np.eye(dim)
scale[0,1] = 0.5
scale[1,0] = 0.5
# Construct frozen Wishart random variables
w = wishart(df, scale)
# Get the generated random variables from a known seed
rng = np.random.RandomState(248042)
w_rvs = wishart.rvs(df, scale, random_state=rng)
rng = np.random.RandomState(248042)
frozen_w_rvs = w.rvs(random_state=rng)
# Manually calculate what it should be, based on the Bartlett (1933)
# decomposition of a Wishart into D A A' D', where D is the Cholesky
# factorization of the scale matrix and A is the lower triangular matrix
# with the square root of chi^2 variates on the diagonal and N(0,1)
# variates in the lower triangle.
rng = np.random.RandomState(248042)
covariances = rng.normal(size=3)
variances = np.r_[
rng.chisquare(df),
rng.chisquare(df-1),
rng.chisquare(df-2),
]**0.5
# Construct the lower-triangular A matrix
A = np.diag(variances)
A[np.tril_indices(dim, k=-1)] = covariances
# Wishart random variate
D = np.linalg.cholesky(scale)
DA = D.dot(A)
manual_w_rvs = np.dot(DA, DA.T)
# Test for equality
assert_allclose(w_rvs, manual_w_rvs)
assert_allclose(frozen_w_rvs, manual_w_rvs)
def test_1D_is_chisquared(self):
# The 1-dimensional Wishart with an identity scale matrix is just a
# chi-squared distribution.
# Test variance, mean, entropy, pdf
# Kolgomorov-Smirnov test for rvs
rng = np.random.default_rng(482974)
sn = 500
dim = 1
scale = np.eye(dim)
df_range = np.arange(1, 10, 2, dtype=float)
X = np.linspace(0.1,10,num=10)
for df in df_range:
w = wishart(df, scale)
c = chi2(df)
# Statistics
assert_allclose(w.var(), c.var())
assert_allclose(w.mean(), c.mean())
assert_allclose(w.entropy(), c.entropy())
# PDF
assert_allclose(w.pdf(X), c.pdf(X))
# rvs
rvs = w.rvs(size=sn, random_state=rng)
args = (df,)
alpha = 0.01
check_distribution_rvs('chi2', args, alpha, rvs)
def test_is_scaled_chisquared(self):
# The 2-dimensional Wishart with an arbitrary scale matrix can be
# transformed to a scaled chi-squared distribution.
# For :math:`S \sim W_p(V,n)` and :math:`\lambda \in \mathbb{R}^p` we have
# :math:`\lambda' S \lambda \sim \lambda' V \lambda \times \chi^2(n)`
rng = np.random.default_rng(482974)
sn = 500
df = 10
dim = 4
# Construct an arbitrary positive definite matrix
scale = np.diag(np.arange(4)+1)
scale[np.tril_indices(4, k=-1)] = np.arange(6)
scale = np.dot(scale.T, scale)
# Use :math:`\lambda = [1, \dots, 1]'`
lamda = np.ones((dim,1))
sigma_lamda = lamda.T.dot(scale).dot(lamda).squeeze()
w = wishart(df, sigma_lamda)
c = chi2(df, scale=sigma_lamda)
# Statistics
assert_allclose(w.var(), c.var())
assert_allclose(w.mean(), c.mean())
assert_allclose(w.entropy(), c.entropy())
# PDF
X = np.linspace(0.1,10,num=10)
assert_allclose(w.pdf(X), c.pdf(X))
# rvs
rvs = w.rvs(size=sn, random_state=rng)
args = (df,0,sigma_lamda)
alpha = 0.01
check_distribution_rvs('chi2', args, alpha, rvs)
class TestMultinomial:
def test_logpmf(self):
vals1 = multinomial.logpmf((3,4), 7, (0.3, 0.7))
assert_allclose(vals1, -1.483270127243324, rtol=1e-8)
vals2 = multinomial.logpmf([3, 4], 0, [.3, .7])
assert vals2 == -np.inf
vals3 = multinomial.logpmf([0, 0], 0, [.3, .7])
assert vals3 == 0
vals4 = multinomial.logpmf([3, 4], 0, [-2, 3])
assert_allclose(vals4, np.nan, rtol=1e-8)
def test_reduces_binomial(self):
# test that the multinomial pmf reduces to the binomial pmf in the 2d
# case
val1 = multinomial.logpmf((3, 4), 7, (0.3, 0.7))
val2 = binom.logpmf(3, 7, 0.3)
assert_allclose(val1, val2, rtol=1e-8)
val1 = multinomial.pmf((6, 8), 14, (0.1, 0.9))
val2 = binom.pmf(6, 14, 0.1)
assert_allclose(val1, val2, rtol=1e-8)
def test_R(self):
# test against the values produced by this R code
# (https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Multinom.html)
# X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
# X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
# X
# apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5)))
n, p = 3, [1./8, 2./8, 5./8]
r_vals = {(0, 0, 3): 0.244140625, (1, 0, 2): 0.146484375,
(2, 0, 1): 0.029296875, (3, 0, 0): 0.001953125,
(0, 1, 2): 0.292968750, (1, 1, 1): 0.117187500,
(2, 1, 0): 0.011718750, (0, 2, 1): 0.117187500,
(1, 2, 0): 0.023437500, (0, 3, 0): 0.015625000}
for x in r_vals:
assert_allclose(multinomial.pmf(x, n, p), r_vals[x], atol=1e-14)
@pytest.mark.parametrize("n", [0, 3])
def test_rvs_np(self, n):
# test that .rvs agrees w/numpy
message = "Some rows of `p` do not sum to 1.0 within..."
with pytest.warns(FutureWarning, match=message):
rndm = np.random.RandomState(123)
sc_rvs = multinomial.rvs(n, [1/4.]*3, size=7, random_state=123)
np_rvs = rndm.multinomial(n, [1/4.]*3, size=7)
assert_equal(sc_rvs, np_rvs)
with pytest.warns(FutureWarning, match=message):
rndm = np.random.RandomState(123)
sc_rvs = multinomial.rvs(n, [1/4.]*5, size=7, random_state=123)
np_rvs = rndm.multinomial(n, [1/4.]*5, size=7)
assert_equal(sc_rvs, np_rvs)
def test_pmf(self):
vals0 = multinomial.pmf((5,), 5, (1,))
assert_allclose(vals0, 1, rtol=1e-8)
vals1 = multinomial.pmf((3,4), 7, (.3, .7))
assert_allclose(vals1, .22689449999999994, rtol=1e-8)
vals2 = multinomial.pmf([[[3,5],[0,8]], [[-1, 9], [1, 1]]], 8,
(.1, .9))
assert_allclose(vals2, [[.03306744, .43046721], [0, 0]], rtol=1e-8)
x = np.empty((0,2), dtype=np.float64)
vals3 = multinomial.pmf(x, 4, (.3, .7))
assert_equal(vals3, np.empty([], dtype=np.float64))
vals4 = multinomial.pmf([1,2], 4, (.3, .7))
assert_allclose(vals4, 0, rtol=1e-8)
vals5 = multinomial.pmf([3, 3, 0], 6, [2/3.0, 1/3.0, 0])
assert_allclose(vals5, 0.219478737997, rtol=1e-8)
vals5 = multinomial.pmf([0, 0, 0], 0, [2/3.0, 1/3.0, 0])
assert vals5 == 1
vals6 = multinomial.pmf([2, 1, 0], 0, [2/3.0, 1/3.0, 0])
assert vals6 == 0
def test_pmf_broadcasting(self):
vals0 = multinomial.pmf([1, 2], 3, [[.1, .9], [.2, .8]])
assert_allclose(vals0, [.243, .384], rtol=1e-8)
vals1 = multinomial.pmf([1, 2], [3, 4], [.1, .9])
assert_allclose(vals1, [.243, 0], rtol=1e-8)
vals2 = multinomial.pmf([[[1, 2], [1, 1]]], 3, [.1, .9])
assert_allclose(vals2, [[.243, 0]], rtol=1e-8)
vals3 = multinomial.pmf([1, 2], [[[3], [4]]], [.1, .9])
assert_allclose(vals3, [[[.243], [0]]], rtol=1e-8)
vals4 = multinomial.pmf([[1, 2], [1,1]], [[[[3]]]], [.1, .9])
assert_allclose(vals4, [[[[.243, 0]]]], rtol=1e-8)
@pytest.mark.parametrize("n", [0, 5])
def test_cov(self, n):
cov1 = multinomial.cov(n, (.2, .3, .5))
cov2 = [[n*.2*.8, -n*.2*.3, -n*.2*.5],
[-n*.3*.2, n*.3*.7, -n*.3*.5],
[-n*.5*.2, -n*.5*.3, n*.5*.5]]
assert_allclose(cov1, cov2, rtol=1e-8)
def test_cov_broadcasting(self):
cov1 = multinomial.cov(5, [[.1, .9], [.2, .8]])
cov2 = [[[.45, -.45],[-.45, .45]], [[.8, -.8], [-.8, .8]]]
assert_allclose(cov1, cov2, rtol=1e-8)
cov3 = multinomial.cov([4, 5], [.1, .9])
cov4 = [[[.36, -.36], [-.36, .36]], [[.45, -.45], [-.45, .45]]]
assert_allclose(cov3, cov4, rtol=1e-8)
cov5 = multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
cov6 = [[[4*.3*.7, -4*.3*.7], [-4*.3*.7, 4*.3*.7]],
[[5*.4*.6, -5*.4*.6], [-5*.4*.6, 5*.4*.6]]]
assert_allclose(cov5, cov6, rtol=1e-8)
@pytest.mark.parametrize("n", [0, 2])
def test_entropy(self, n):
# this is equivalent to a binomial distribution with n=2, so the
# entropy .77899774929 is easily computed "by hand"
ent0 = multinomial.entropy(n, [.2, .8])
assert_allclose(ent0, binom.entropy(n, .2), rtol=1e-8)
def test_entropy_broadcasting(self):
ent0 = multinomial.entropy([2, 3], [.2, .8])
assert_allclose(ent0, [binom.entropy(2, .2), binom.entropy(3, .2)],
rtol=1e-8)
ent1 = multinomial.entropy([7, 8], [[.3, .7], [.4, .6]])
assert_allclose(ent1, [binom.entropy(7, .3), binom.entropy(8, .4)],
rtol=1e-8)
ent2 = multinomial.entropy([[7], [8]], [[.3, .7], [.4, .6]])
assert_allclose(ent2,
[[binom.entropy(7, .3), binom.entropy(7, .4)],
[binom.entropy(8, .3), binom.entropy(8, .4)]],
rtol=1e-8)
@pytest.mark.parametrize("n", [0, 5])
def test_mean(self, n):
mean1 = multinomial.mean(n, [.2, .8])
assert_allclose(mean1, [n*.2, n*.8], rtol=1e-8)
def test_mean_broadcasting(self):
mean1 = multinomial.mean([5, 6], [.2, .8])
assert_allclose(mean1, [[5*.2, 5*.8], [6*.2, 6*.8]], rtol=1e-8)
def test_frozen(self):
# The frozen distribution should agree with the regular one
np.random.seed(1234)
n = 12
pvals = (.1, .2, .3, .4)
x = [[0,0,0,12],[0,0,1,11],[0,1,1,10],[1,1,1,9],[1,1,2,8]]
x = np.asarray(x, dtype=np.float64)
mn_frozen = multinomial(n, pvals)
assert_allclose(mn_frozen.pmf(x), multinomial.pmf(x, n, pvals))
assert_allclose(mn_frozen.logpmf(x), multinomial.logpmf(x, n, pvals))
assert_allclose(mn_frozen.entropy(), multinomial.entropy(n, pvals))
def test_gh_11860(self):
# gh-11860 reported cases in which the adjustments made by multinomial
# to the last element of `p` can cause `nan`s even when the input is
# essentially valid. Check that a pathological case returns a finite,
# nonzero result. (This would fail in main before the PR.)
n = 88
rng = np.random.default_rng(8879715917488330089)
p = rng.random(n)
p[-1] = 1e-30
p /= np.sum(p)
x = np.ones(n)
logpmf = multinomial.logpmf(x, n, p)
assert np.isfinite(logpmf)
@pytest.mark.parametrize('dtype', [np.float32, np.float64])
def test_gh_22565(self, dtype):
# Same issue as gh-11860 above, essentially, but the original
# fix didn't completely solve the problem.
n = 19
p = np.asarray([0.2, 0.2, 0.2, 0.2, 0.2], dtype=dtype)
res1 = multinomial.pmf(x=[1, 2, 5, 7, 4], n=n, p=p)
res2 = multinomial.pmf(x=[1, 2, 4, 5, 7], n=n, p=p)
np.testing.assert_allclose(res1, res2, rtol=1e-15)
class TestInvwishart:
def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
def test_1D_is_invgamma(self):
# The 1-dimensional inverse Wishart with an identity scale matrix is
# just an inverse gamma distribution.
# Test variance, mean, pdf, entropy
# Kolgomorov-Smirnov test for rvs
rng = np.random.RandomState(482974)
sn = 500
dim = 1
scale = np.eye(dim)
df_range = np.arange(5, 20, 2, dtype=float)
X = np.linspace(0.1,10,num=10)
for df in df_range:
iw = invwishart(df, scale)
ig = invgamma(df/2, scale=1./2)
# Statistics
assert_allclose(iw.var(), ig.var())
assert_allclose(iw.mean(), ig.mean())
# PDF
assert_allclose(iw.pdf(X), ig.pdf(X))
# rvs
rvs = iw.rvs(size=sn, random_state=rng)
args = (df/2, 0, 1./2)
alpha = 0.01
check_distribution_rvs('invgamma', args, alpha, rvs)
# entropy
assert_allclose(iw.entropy(), ig.entropy())
def test_invwishart_2D_rvs(self):
dim = 3
df = 10
# Construct a simple non-diagonal positive definite matrix
scale = np.eye(dim)
scale[0,1] = 0.5
scale[1,0] = 0.5
# Construct frozen inverse-Wishart random variables
iw = invwishart(df, scale)
# Get the generated random variables from a known seed
rng = np.random.RandomState(608072)
iw_rvs = invwishart.rvs(df, scale, random_state=rng)
rng = np.random.RandomState(608072)
frozen_iw_rvs = iw.rvs(random_state=rng)
# Manually calculate what it should be, based on the decomposition in
# https://arxiv.org/abs/2310.15884 of an invers-Wishart into L L',
# where L A = D, D is the Cholesky factorization of the scale matrix,
# and A is the lower triangular matrix with the square root of chi^2
# variates on the diagonal and N(0,1) variates in the lower triangle.
# the diagonal chi^2 variates in this A are reversed compared to those
# in the Bartlett decomposition A for Wishart rvs.
rng = np.random.RandomState(608072)
covariances = rng.normal(size=3)
variances = np.r_[
rng.chisquare(df-2),
rng.chisquare(df-1),
rng.chisquare(df),
]**0.5
# Construct the lower-triangular A matrix
A = np.diag(variances)
A[np.tril_indices(dim, k=-1)] = covariances
# inverse-Wishart random variate
D = np.linalg.cholesky(scale)
L = np.linalg.solve(A.T, D.T).T
manual_iw_rvs = np.dot(L, L.T)
# Test for equality
assert_allclose(iw_rvs, manual_iw_rvs)
assert_allclose(frozen_iw_rvs, manual_iw_rvs)
def test_sample_mean(self):
"""Test that sample mean consistent with known mean."""
# Construct an arbitrary positive definite scale matrix
df = 10
sample_size = 20_000
for dim in [1, 5]:
scale = np.diag(np.arange(dim) + 1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim - 1) / 2)
scale = np.dot(scale.T, scale)
dist = invwishart(df, scale)
Xmean_exp = dist.mean()
Xvar_exp = dist.var()
Xmean_std = (Xvar_exp / sample_size)**0.5 # asymptotic SE of mean estimate
X = dist.rvs(size=sample_size, random_state=1234)
Xmean_est = X.mean(axis=0)
ntests = dim*(dim + 1)//2
fail_rate = 0.01 / ntests # correct for multiple tests
max_diff = norm.ppf(1 - fail_rate / 2)
assert np.allclose(
(Xmean_est - Xmean_exp) / Xmean_std,
0,
atol=max_diff,
)
def test_logpdf_4x4(self):
"""Regression test for gh-8844."""
X = np.array([[2, 1, 0, 0.5],
[1, 2, 0.5, 0.5],
[0, 0.5, 3, 1],
[0.5, 0.5, 1, 2]])
Psi = np.array([[9, 7, 3, 1],
[7, 9, 5, 1],
[3, 5, 8, 2],
[1, 1, 2, 9]])
nu = 6
prob = invwishart.logpdf(X, nu, Psi)
# Explicit calculation from the formula on wikipedia.
p = X.shape[0]
sig, logdetX = np.linalg.slogdet(X)
sig, logdetPsi = np.linalg.slogdet(Psi)
M = np.linalg.solve(X, Psi)
expected = ((nu/2)*logdetPsi
- (nu*p/2)*np.log(2)
- multigammaln(nu/2, p)
- (nu + p + 1)/2*logdetX
- 0.5*M.trace())
assert_allclose(prob, expected)
class TestSpecialOrthoGroup:
def test_reproducibility(self):
x = special_ortho_group.rvs(3, random_state=np.random.default_rng(514))
expected = np.array([[-0.93200988, 0.01533561, -0.36210826],
[0.35742128, 0.20446501, -0.91128705],
[0.06006333, -0.97875374, -0.19604469]])
assert_array_almost_equal(x, expected)
def test_invalid_dim(self):
assert_raises(ValueError, special_ortho_group.rvs, None)
assert_raises(ValueError, special_ortho_group.rvs, (2, 2))
assert_raises(ValueError, special_ortho_group.rvs, -1)
assert_raises(ValueError, special_ortho_group.rvs, 2.5)
def test_frozen_matrix(self):
dim = 7
frozen = special_ortho_group(dim)
rvs1 = frozen.rvs(random_state=1234)
rvs2 = special_ortho_group.rvs(dim, random_state=1234)
assert_equal(rvs1, rvs2)
def test_det_and_ortho(self):
xs = [special_ortho_group.rvs(dim)
for dim in range(2,12)
for i in range(3)]
# Test that determinants are always +1
dets = [np.linalg.det(x) for x in xs]
assert_allclose(dets, [1.]*30, rtol=1e-13)
# Test that these are orthogonal matrices
for x in xs:
assert_array_almost_equal(np.dot(x, x.T),
np.eye(x.shape[0]))
def test_haar(self):
# Test that the distribution is constant under rotation
# Every column should have the same distribution
# Additionally, the distribution should be invariant under another rotation
# Generate samples
dim = 5
samples = 1000 # Not too many, or the test takes too long
ks_prob = .05
xs = special_ortho_group.rvs(
dim, size=samples, random_state=np.random.default_rng(513)
)
# Dot a few rows (0, 1, 2) with unit vectors (0, 2, 4, 3),
# effectively picking off entries in the matrices of xs.
# These projections should all have the same distribution,
# establishing rotational invariance. We use the two-sided
# KS test to confirm this.
# We could instead test that angles between random vectors
# are uniformly distributed, but the below is sufficient.
# It is not feasible to consider all pairs, so pick a few.
els = ((0,0), (0,2), (1,4), (2,3))
#proj = {(er, ec): [x[er][ec] for x in xs] for er, ec in els}
proj = {(er, ec): sorted([x[er][ec] for x in xs]) for er, ec in els}
pairs = [(e0, e1) for e0 in els for e1 in els if e0 > e1]
ks_tests = [ks_2samp(proj[p0], proj[p1])[1] for (p0, p1) in pairs]
assert_array_less([ks_prob]*len(pairs), ks_tests)
def test_one_by_one(self):
# Test that the distribution is a delta function at the identity matrix
# when dim=1
assert_allclose(special_ortho_group.rvs(1, size=1000), 1, rtol=1e-13)
def test_zero_by_zero(self):
assert_equal(special_ortho_group.rvs(0, size=4).shape, (4, 0, 0))
class TestOrthoGroup:
def test_reproducibility(self):
seed = 514
rng = np.random.RandomState(seed)
x = ortho_group.rvs(3, random_state=rng)
x2 = ortho_group.rvs(3, random_state=seed)
# Note this matrix has det -1, distinguishing O(N) from SO(N)
assert_almost_equal(np.linalg.det(x), -1)
expected = np.array([[0.381686, -0.090374, 0.919863],
[0.905794, -0.161537, -0.391718],
[-0.183993, -0.98272, -0.020204]])
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_dim(self):
assert_raises(ValueError, ortho_group.rvs, None)
assert_raises(ValueError, ortho_group.rvs, (2, 2))
assert_raises(ValueError, ortho_group.rvs, -1)
assert_raises(ValueError, ortho_group.rvs, 2.5)
def test_frozen_matrix(self):
dim = 7
frozen = ortho_group(dim)
frozen_seed = ortho_group(dim, seed=1234)
rvs1 = frozen.rvs(random_state=1234)
rvs2 = ortho_group.rvs(dim, random_state=1234)
rvs3 = frozen_seed.rvs(size=1)
assert_equal(rvs1, rvs2)
assert_equal(rvs1, rvs3)
def test_det_and_ortho(self):
xs = [[ortho_group.rvs(dim)
for i in range(10)]
for dim in range(2,12)]
# Test that abs determinants are always +1
dets = np.array([[np.linalg.det(x) for x in xx] for xx in xs])
assert_allclose(np.fabs(dets), np.ones(dets.shape), rtol=1e-13)
# Test that these are orthogonal matrices
for xx in xs:
for x in xx:
assert_array_almost_equal(np.dot(x, x.T),
np.eye(x.shape[0]))
@pytest.mark.parametrize("dim", [2, 5, 10, 20])
def test_det_distribution_gh18272(self, dim):
# Test that positive and negative determinants are equally likely.
rng = np.random.default_rng(6796248956179332344)
dist = ortho_group(dim=dim)
rvs = dist.rvs(size=5000, random_state=rng)
dets = scipy.linalg.det(rvs)
k = np.sum(dets > 0)
n = len(dets)
res = stats.binomtest(k, n)
low, high = res.proportion_ci(confidence_level=0.95)
assert low < 0.5 < high
def test_haar(self):
# Test that the distribution is constant under rotation
# Every column should have the same distribution
# Additionally, the distribution should be invariant under another rotation
# Generate samples
dim = 5
samples = 1000 # Not too many, or the test takes too long
ks_prob = .05
rng = np.random.RandomState(518) # Note that the test is sensitive to seed too
xs = ortho_group.rvs(dim, size=samples, random_state=rng)
# Dot a few rows (0, 1, 2) with unit vectors (0, 2, 4, 3),
# effectively picking off entries in the matrices of xs.
# These projections should all have the same distribution,
# establishing rotational invariance. We use the two-sided
# KS test to confirm this.
# We could instead test that angles between random vectors
# are uniformly distributed, but the below is sufficient.
# It is not feasible to consider all pairs, so pick a few.
els = ((0,0), (0,2), (1,4), (2,3))
#proj = {(er, ec): [x[er][ec] for x in xs] for er, ec in els}
proj = {(er, ec): sorted([x[er][ec] for x in xs]) for er, ec in els}
pairs = [(e0, e1) for e0 in els for e1 in els if e0 > e1]
ks_tests = [ks_2samp(proj[p0], proj[p1])[1] for (p0, p1) in pairs]
assert_array_less([ks_prob]*len(pairs), ks_tests)
def test_one_by_one(self):
# Test that the 1x1 distribution gives ±1 with equal probability.
dim = 1
xs = ortho_group.rvs(dim, size=5000, random_state=np.random.default_rng(514))
assert_allclose(np.abs(xs), 1, rtol=1e-13)
k = np.sum(xs > 0)
n = len(xs)
res = stats.binomtest(k, n)
low, high = res.proportion_ci(confidence_level=0.95)
assert low < 0.5 < high
def test_zero_by_zero(self):
assert_equal(special_ortho_group.rvs(0, size=4).shape, (4, 0, 0))
@pytest.mark.slow
def test_pairwise_distances(self):
# Test that the distribution of pairwise distances is close to correct.
rng = np.random.RandomState(514)
def random_ortho(dim, random_state=None):
u, _s, v = np.linalg.svd(rng.normal(size=(dim, dim)))
return np.dot(u, v)
for dim in range(2, 6):
def generate_test_statistics(rvs, N=1000, eps=1e-10):
stats = np.array([
np.sum((rvs(dim=dim, random_state=rng) -
rvs(dim=dim, random_state=rng))**2)
for _ in range(N)
])
# Add a bit of noise to account for numeric accuracy.
stats += np.random.uniform(-eps, eps, size=stats.shape)
return stats
expected = generate_test_statistics(random_ortho)
actual = generate_test_statistics(scipy.stats.ortho_group.rvs)
_D, p = scipy.stats.ks_2samp(expected, actual)
assert_array_less(.05, p)
class TestRandomCorrelation:
def test_reproducibility(self):
rng = np.random.RandomState(514)
eigs = (.5, .8, 1.2, 1.5)
x = random_correlation.rvs(eigs, random_state=rng)
x2 = random_correlation.rvs(eigs, random_state=514)
expected = np.array([[1., -0.184851, 0.109017, -0.227494],
[-0.184851, 1., 0.231236, 0.326669],
[0.109017, 0.231236, 1., -0.178912],
[-0.227494, 0.326669, -0.178912, 1.]])
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_eigs(self):
assert_raises(ValueError, random_correlation.rvs, None)
assert_raises(ValueError, random_correlation.rvs, 'test')
assert_raises(ValueError, random_correlation.rvs, 2.5)
assert_raises(ValueError, random_correlation.rvs, [2.5])
assert_raises(ValueError, random_correlation.rvs, [[1,2],[3,4]])
assert_raises(ValueError, random_correlation.rvs, [2.5, -.5])
assert_raises(ValueError, random_correlation.rvs, [1, 2, .1])
def test_frozen_matrix(self):
eigs = (.5, .8, 1.2, 1.5)
frozen = random_correlation(eigs)
frozen_seed = random_correlation(eigs, seed=514)
rvs1 = random_correlation.rvs(eigs, random_state=514)
rvs2 = frozen.rvs(random_state=514)
rvs3 = frozen_seed.rvs()
assert_equal(rvs1, rvs2)
assert_equal(rvs1, rvs3)
def test_definition(self):
# Test the definition of a correlation matrix in several dimensions:
#
# 1. Det is product of eigenvalues (and positive by construction
# in examples)
# 2. 1's on diagonal
# 3. Matrix is symmetric
def norm(i, e):
return i*e/sum(e)
rng = np.random.RandomState(123)
eigs = [norm(i, rng.uniform(size=i)) for i in range(2, 6)]
eigs.append([4,0,0,0])
ones = [[1.]*len(e) for e in eigs]
xs = [random_correlation.rvs(e, random_state=rng) for e in eigs]
# Test that determinants are products of eigenvalues
# These are positive by construction
# Could also test that the eigenvalues themselves are correct,
# but this seems sufficient.
dets = [np.fabs(np.linalg.det(x)) for x in xs]
dets_known = [np.prod(e) for e in eigs]
assert_allclose(dets, dets_known, rtol=1e-13, atol=1e-13)
# Test for 1's on the diagonal
diags = [np.diag(x) for x in xs]
for a, b in zip(diags, ones):
assert_allclose(a, b, rtol=1e-13)
# Correlation matrices are symmetric
for x in xs:
assert_allclose(x, x.T, rtol=1e-13)
def test_to_corr(self):
# Check some corner cases in to_corr
# ajj == 1
m = np.array([[0.1, 0], [0, 1]], dtype=float)
m = random_correlation._to_corr(m)
assert_allclose(m, np.array([[1, 0], [0, 0.1]]))
# Floating point overflow; fails to compute the correct
# rotation, but should still produce some valid rotation
# rather than infs/nans
with np.errstate(over='ignore'):
g = np.array([[0, 1], [-1, 0]])
m0 = np.array([[1e300, 0], [0, np.nextafter(1, 0)]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m, g.T.dot(m0).dot(g))
m0 = np.array([[0.9, 1e300], [1e300, 1.1]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m, g.T.dot(m0).dot(g))
# Zero discriminant; should set the first diag entry to 1
m0 = np.array([[2, 1], [1, 2]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m[0,0], 1)
# Slightly negative discriminant; should be approx correct still
m0 = np.array([[2 + 1e-7, 1], [1, 2]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m[0,0], 1)
class TestUniformDirection:
@pytest.mark.parametrize("dim", [1, 3])
@pytest.mark.parametrize("size", [None, 1, 5, (5, 4)])
def test_samples(self, dim, size):
# test that samples have correct shape and norm 1
rng = np.random.default_rng(2777937887058094419)
uniform_direction_dist = uniform_direction(dim, seed=rng)
samples = uniform_direction_dist.rvs(size)
mean, cov = np.zeros(dim), np.eye(dim)
expected_shape = rng.multivariate_normal(mean, cov, size=size).shape
assert samples.shape == expected_shape
norms = np.linalg.norm(samples, axis=-1)
assert_allclose(norms, 1.)
@pytest.mark.parametrize("dim", [None, 0, (2, 2), 2.5])
def test_invalid_dim(self, dim):
message = ("Dimension of vector must be specified, "
"and must be an integer greater than 0.")
with pytest.raises(ValueError, match=message):
uniform_direction.rvs(dim)
def test_frozen_distribution(self):
dim = 5
frozen = uniform_direction(dim)
frozen_seed = uniform_direction(dim, seed=514)
rvs1 = frozen.rvs(random_state=514)
rvs2 = uniform_direction.rvs(dim, random_state=514)
rvs3 = frozen_seed.rvs()
assert_equal(rvs1, rvs2)
assert_equal(rvs1, rvs3)
@pytest.mark.parametrize("dim", [2, 5, 8])
def test_uniform(self, dim):
rng = np.random.default_rng(1036978481269651776)
spherical_dist = uniform_direction(dim, seed=rng)
# generate random, orthogonal vectors
v1, v2 = spherical_dist.rvs(size=2)
v2 -= v1 @ v2 * v1
v2 /= np.linalg.norm(v2)
assert_allclose(v1 @ v2, 0, atol=1e-14) # orthogonal
# generate data and project onto orthogonal vectors
samples = spherical_dist.rvs(size=10000)
s1 = samples @ v1
s2 = samples @ v2
angles = np.arctan2(s1, s2)
# test that angles follow a uniform distribution
# normalize angles to range [0, 1]
angles += np.pi
angles /= 2*np.pi
# perform KS test
uniform_dist = uniform()
kstest_result = kstest(angles, uniform_dist.cdf)
assert kstest_result.pvalue > 0.05
class TestUnitaryGroup:
def test_reproducibility(self):
rng = np.random.RandomState(514)
x = unitary_group.rvs(3, random_state=rng)
x2 = unitary_group.rvs(3, random_state=514)
expected = np.array(
[[0.308771+0.360312j, 0.044021+0.622082j, 0.160327+0.600173j],
[0.732757+0.297107j, 0.076692-0.4614j, -0.394349+0.022613j],
[-0.148844+0.357037j, -0.284602-0.557949j, 0.607051+0.299257j]]
)
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_dim(self):
assert_raises(ValueError, unitary_group.rvs, None)
assert_raises(ValueError, unitary_group.rvs, (2, 2))
assert_raises(ValueError, unitary_group.rvs, -1)
assert_raises(ValueError, unitary_group.rvs, 2.5)
def test_frozen_matrix(self):
dim = 7
frozen = unitary_group(dim)
frozen_seed = unitary_group(dim, seed=514)
rvs1 = frozen.rvs(random_state=514)
rvs2 = unitary_group.rvs(dim, random_state=514)
rvs3 = frozen_seed.rvs(size=1)
assert_equal(rvs1, rvs2)
assert_equal(rvs1, rvs3)
def test_unitarity(self):
xs = [unitary_group.rvs(dim)
for dim in range(2,12)
for i in range(3)]
# Test that these are unitary matrices
for x in xs:
assert_allclose(np.dot(x, x.conj().T), np.eye(x.shape[0]), atol=1e-15)
def test_haar(self):
# Test that the eigenvalues, which lie on the unit circle in
# the complex plane, are uncorrelated.
# Generate samples
for dim in (1, 5):
samples = 1000 # Not too many, or the test takes too long
# Note that the test is sensitive to seed too
xs = unitary_group.rvs(
dim, size=samples, random_state=np.random.default_rng(514)
)
# The angles "x" of the eigenvalues should be uniformly distributed
# Overall this seems to be a necessary but weak test of the distribution.
eigs = np.vstack([scipy.linalg.eigvals(x) for x in xs])
x = np.arctan2(eigs.imag, eigs.real)
res = kstest(x.ravel(), uniform(-np.pi, 2*np.pi).cdf)
assert_(res.pvalue > 0.05)
def test_zero_by_zero(self):
assert_equal(unitary_group.rvs(0, size=4).shape, (4, 0, 0))
class TestMultivariateT:
# These tests were created by running vpa(mvtpdf(...)) in MATLAB. The
# function takes no `mu` parameter. The tests were run as
#
# >> ans = vpa(mvtpdf(x - mu, shape, df));
#
PDF_TESTS = [(
# x
[
[1, 2],
[4, 1],
[2, 1],
[2, 4],
[1, 4],
[4, 1],
[3, 2],
[3, 3],
[4, 4],
[5, 1],
],
# loc
[0, 0],
# shape
[
[1, 0],
[0, 1]
],
# df
4,
# ans
[
0.013972450422333741737457302178882,
0.0010998721906793330026219646100571,
0.013972450422333741737457302178882,
0.00073682844024025606101402363634634,
0.0010998721906793330026219646100571,
0.0010998721906793330026219646100571,
0.0020732579600816823488240725481546,
0.00095660371505271429414668515889275,
0.00021831953784896498569831346792114,
0.00037725616140301147447000396084604
]
), (
# x
[
[0.9718, 0.1298, 0.8134],
[0.4922, 0.5522, 0.7185],
[0.3010, 0.1491, 0.5008],
[0.5971, 0.2585, 0.8940],
[0.5434, 0.5287, 0.9507],
],
# loc
[-1, 1, 50],
# shape
[
[1.0000, 0.5000, 0.2500],
[0.5000, 1.0000, -0.1000],
[0.2500, -0.1000, 1.0000],
],
# df
8,
# ans
[
0.00000000000000069609279697467772867405511133763,
0.00000000000000073700739052207366474839369535934,
0.00000000000000069522909962669171512174435447027,
0.00000000000000074212293557998314091880208889767,
0.00000000000000077039675154022118593323030449058,
]
)]
@pytest.mark.parametrize("x, loc, shape, df, ans", PDF_TESTS)
def test_pdf_correctness(self, x, loc, shape, df, ans):
dist = multivariate_t(loc, shape, df, seed=0)
val = dist.pdf(x)
assert_array_almost_equal(val, ans)
@pytest.mark.parametrize("x, loc, shape, df, ans", PDF_TESTS)
def test_logpdf_correct(self, x, loc, shape, df, ans):
dist = multivariate_t(loc, shape, df, seed=0)
val1 = dist.pdf(x)
val2 = dist.logpdf(x)
assert_array_almost_equal(np.log(val1), val2)
# https://github.com/scipy/scipy/issues/10042#issuecomment-576795195
def test_mvt_with_df_one_is_cauchy(self):
x = [9, 7, 4, 1, -3, 9, 0, -3, -1, 3]
val = multivariate_t.pdf(x, df=1)
ans = cauchy.pdf(x)
assert_array_almost_equal(val, ans)
def test_mvt_with_high_df_is_approx_normal(self):
# `normaltest` returns the chi-squared statistic and the associated
# p-value. The null hypothesis is that `x` came from a normal
# distribution, so a low p-value represents rejecting the null, i.e.
# that it is unlikely that `x` came a normal distribution.
P_VAL_MIN = 0.1
dist = multivariate_t(0, 1, df=100000, seed=1)
samples = dist.rvs(size=100000)
_, p = normaltest(samples)
assert (p > P_VAL_MIN)
dist = multivariate_t([-2, 3], [[10, -1], [-1, 10]], df=100000,
seed=42)
samples = dist.rvs(size=100000)
_, p = normaltest(samples)
assert ((p > P_VAL_MIN).all())
@pytest.mark.thread_unsafe
@patch('scipy.stats.multivariate_normal._logpdf')
def test_mvt_with_inf_df_calls_normal(self, mock):
dist = multivariate_t(0, 1, df=np.inf, seed=7)
assert isinstance(dist, multivariate_normal_frozen)
multivariate_t.pdf(0, df=np.inf)
assert mock.call_count == 1
multivariate_t.logpdf(0, df=np.inf)
assert mock.call_count == 2
def test_shape_correctness(self):
# pdf and logpdf should return scalar when the
# number of samples in x is one.
dim = 4
loc = np.zeros(dim)
shape = np.eye(dim)
df = 4.5
x = np.zeros(dim)
res = multivariate_t(loc, shape, df).pdf(x)
assert np.isscalar(res)
res = multivariate_t(loc, shape, df).logpdf(x)
assert np.isscalar(res)
# pdf() and logpdf() should return probabilities of shape
# (n_samples,) when x has n_samples.
n_samples = 7
x = np.random.random((n_samples, dim))
res = multivariate_t(loc, shape, df).pdf(x)
assert (res.shape == (n_samples,))
res = multivariate_t(loc, shape, df).logpdf(x)
assert (res.shape == (n_samples,))
# rvs() should return scalar unless a size argument is applied.
res = multivariate_t(np.zeros(1), np.eye(1), 1).rvs()
assert np.isscalar(res)
# rvs() should return vector of shape (size,) if size argument
# is applied.
size = 7
res = multivariate_t(np.zeros(1), np.eye(1), 1).rvs(size=size)
assert (res.shape == (size,))
def test_default_arguments(self):
dist = multivariate_t()
assert_equal(dist.loc, [0])
assert_equal(dist.shape, [[1]])
assert (dist.df == 1)
DEFAULT_ARGS_TESTS = [
(None, None, None, 0, 1, 1),
(None, None, 7, 0, 1, 7),
(None, [[7, 0], [0, 7]], None, [0, 0], [[7, 0], [0, 7]], 1),
(None, [[7, 0], [0, 7]], 7, [0, 0], [[7, 0], [0, 7]], 7),
([7, 7], None, None, [7, 7], [[1, 0], [0, 1]], 1),
([7, 7], None, 7, [7, 7], [[1, 0], [0, 1]], 7),
([7, 7], [[7, 0], [0, 7]], None, [7, 7], [[7, 0], [0, 7]], 1),
([7, 7], [[7, 0], [0, 7]], 7, [7, 7], [[7, 0], [0, 7]], 7)
]
@pytest.mark.parametrize("loc, shape, df, loc_ans, shape_ans, df_ans",
DEFAULT_ARGS_TESTS)
def test_default_args(self, loc, shape, df, loc_ans, shape_ans, df_ans):
dist = multivariate_t(loc=loc, shape=shape, df=df)
assert_equal(dist.loc, loc_ans)
assert_equal(dist.shape, shape_ans)
assert (dist.df == df_ans)
ARGS_SHAPES_TESTS = [
(-1, 2, 3, [-1], [[2]], 3),
([-1], [2], 3, [-1], [[2]], 3),
(np.array([-1]), np.array([2]), 3, [-1], [[2]], 3)
]
@pytest.mark.parametrize("loc, shape, df, loc_ans, shape_ans, df_ans",
ARGS_SHAPES_TESTS)
def test_scalar_list_and_ndarray_arguments(self, loc, shape, df, loc_ans,
shape_ans, df_ans):
dist = multivariate_t(loc, shape, df)
assert_equal(dist.loc, loc_ans)
assert_equal(dist.shape, shape_ans)
assert_equal(dist.df, df_ans)
def test_argument_error_handling(self):
# `loc` should be a one-dimensional vector.
loc = [[1, 1]]
assert_raises(ValueError,
multivariate_t,
**dict(loc=loc))
# `shape` should be scalar or square matrix.
shape = [[1, 1], [2, 2], [3, 3]]
assert_raises(ValueError,
multivariate_t,
**dict(loc=loc, shape=shape))
# `df` should be greater than zero.
loc = np.zeros(2)
shape = np.eye(2)
df = -1
assert_raises(ValueError,
multivariate_t,
**dict(loc=loc, shape=shape, df=df))
df = 0
assert_raises(ValueError,
multivariate_t,
**dict(loc=loc, shape=shape, df=df))
def test_reproducibility(self):
rng = np.random.RandomState(4)
loc = rng.uniform(size=3)
shape = np.eye(3)
dist1 = multivariate_t(loc, shape, df=3, seed=2)
dist2 = multivariate_t(loc, shape, df=3, seed=2)
samples1 = dist1.rvs(size=10)
samples2 = dist2.rvs(size=10)
assert_equal(samples1, samples2)
def test_allow_singular(self):
# Make shape singular and verify error was raised.
args = dict(loc=[0,0], shape=[[0,0],[0,1]], df=1, allow_singular=False)
assert_raises(np.linalg.LinAlgError, multivariate_t, **args)
@pytest.mark.parametrize("size", [(10, 3), (5, 6, 4, 3)])
@pytest.mark.parametrize("dim", [2, 3, 4, 5])
@pytest.mark.parametrize("df", [1., 2., np.inf])
def test_rvs(self, size, dim, df):
dist = multivariate_t(np.zeros(dim), np.eye(dim), df)
rvs = dist.rvs(size=size)
assert rvs.shape == size + (dim, )
def test_cdf_signs(self):
# check that sign of output is correct when np.any(lower > x)
mean = np.zeros(3)
cov = np.eye(3)
df = 10
b = [[1, 1, 1], [0, 0, 0], [1, 0, 1], [0, 1, 0]]
a = [[0, 0, 0], [1, 1, 1], [0, 1, 0], [1, 0, 1]]
# when odd number of elements of b < a, output is negative
expected_signs = np.array([1, -1, -1, 1])
cdf = multivariate_normal.cdf(b, mean, cov, df, lower_limit=a)
assert_allclose(cdf, cdf[0]*expected_signs)
@pytest.mark.parametrize('dim', [1, 2, 5])
def test_cdf_against_multivariate_normal(self, dim):
# Check accuracy against MVN randomly-generated cases
self.cdf_against_mvn_test(dim)
@pytest.mark.parametrize('dim', [3, 6, 9])
def test_cdf_against_multivariate_normal_singular(self, dim):
# Check accuracy against MVN for randomly-generated singular cases
self.cdf_against_mvn_test(3, True)
def cdf_against_mvn_test(self, dim, singular=False):
# Check for accuracy in the limit that df -> oo and MVT -> MVN
rng = np.random.default_rng(413722918996573)
n = 3
w = 10**rng.uniform(-2, 1, size=dim)
cov = _random_covariance(dim, w, rng, singular)
mean = 10**rng.uniform(-1, 2, size=dim) * np.sign(rng.normal(size=dim))
a = -10**rng.uniform(-1, 2, size=(n, dim)) + mean
b = 10**rng.uniform(-1, 2, size=(n, dim)) + mean
res = stats.multivariate_t.cdf(b, mean, cov, df=10000, lower_limit=a,
allow_singular=True, random_state=rng)
ref = stats.multivariate_normal.cdf(b, mean, cov, allow_singular=True,
lower_limit=a)
assert_allclose(res, ref, atol=5e-4)
def test_cdf_against_univariate_t(self):
rng = np.random.default_rng(413722918996573)
cov = 2
mean = 0
x = rng.normal(size=10, scale=np.sqrt(cov))
df = 3
res = stats.multivariate_t.cdf(x, mean, cov, df, lower_limit=-np.inf,
random_state=rng)
ref = stats.t.cdf(x, df, mean, np.sqrt(cov))
incorrect = stats.norm.cdf(x, mean, np.sqrt(cov))
assert_allclose(res, ref, atol=5e-4) # close to t
assert np.all(np.abs(res - incorrect) > 1e-3) # not close to normal
@pytest.mark.parametrize("dim", [2, 3, 5, 10])
@pytest.mark.parametrize("seed", [3363958638, 7891119608, 3887698049,
5013150848, 1495033423, 6170824608])
@pytest.mark.parametrize("singular", [False, True])
def test_cdf_against_qsimvtv(self, dim, seed, singular):
if singular and seed != 3363958638:
pytest.skip('Agreement with qsimvtv is not great in singular case')
rng = np.random.default_rng(seed)
w = 10**rng.uniform(-2, 2, size=dim)
cov = _random_covariance(dim, w, rng, singular)
mean = rng.random(dim)
a = -rng.random(dim)
b = rng.random(dim)
df = rng.random() * 5
# no lower limit
res = stats.multivariate_t.cdf(b, mean, cov, df, random_state=rng,
allow_singular=True)
with np.errstate(invalid='ignore'):
ref = _qsimvtv(20000, df, cov, np.inf*a, b - mean, rng)[0]
assert_allclose(res, ref, atol=2e-4, rtol=1e-3)
# with lower limit
res = stats.multivariate_t.cdf(b, mean, cov, df, lower_limit=a,
random_state=rng, allow_singular=True)
with np.errstate(invalid='ignore'):
ref = _qsimvtv(20000, df, cov, a - mean, b - mean, rng)[0]
assert_allclose(res, ref, atol=1e-4, rtol=1e-3)
@pytest.mark.slow
def test_cdf_against_generic_integrators(self):
# Compare result against generic numerical integrators
dim = 3
rng = np.random.default_rng(41372291899657)
w = 10 ** rng.uniform(-1, 1, size=dim)
cov = _random_covariance(dim, w, rng, singular=True)
mean = rng.random(dim)
a = -rng.random(dim)
b = rng.random(dim)
df = rng.random() * 5
res = stats.multivariate_t.cdf(b, mean, cov, df, random_state=rng,
lower_limit=a)
def integrand(x):
return stats.multivariate_t.pdf(x.T, mean, cov, df)
ref = qmc_quad(integrand, a, b, qrng=stats.qmc.Halton(d=dim, seed=rng))
assert_allclose(res, ref.integral, rtol=1e-3)
def integrand(*zyx):
return stats.multivariate_t.pdf(zyx[::-1], mean, cov, df)
ref = tplquad(integrand, a[0], b[0], a[1], b[1], a[2], b[2])
assert_allclose(res, ref[0], rtol=1e-3)
def test_against_matlab(self):
# Test against matlab mvtcdf:
# C = [6.21786909 0.2333667 7.95506077;
# 0.2333667 29.67390923 16.53946426;
# 7.95506077 16.53946426 19.17725252]
# df = 1.9559939787727658
# mvtcdf([0, 0, 0], C, df) % 0.2523
rng = np.random.default_rng(2967390923)
cov = np.array([[ 6.21786909, 0.2333667 , 7.95506077],
[ 0.2333667 , 29.67390923, 16.53946426],
[ 7.95506077, 16.53946426, 19.17725252]])
df = 1.9559939787727658
dist = stats.multivariate_t(shape=cov, df=df)
res = dist.cdf([0, 0, 0], random_state=rng)
ref = 0.2523
assert_allclose(res, ref, rtol=1e-3)
def test_frozen(self):
seed = 4137229573
rng = np.random.default_rng(seed)
loc = rng.uniform(size=3)
x = rng.uniform(size=3) + loc
shape = np.eye(3)
df = rng.random()
args = (loc, shape, df)
rng_frozen = np.random.default_rng(seed)
rng_unfrozen = np.random.default_rng(seed)
dist = stats.multivariate_t(*args, seed=rng_frozen)
assert_equal(dist.cdf(x),
multivariate_t.cdf(x, *args, random_state=rng_unfrozen))
def test_vectorized(self):
dim = 4
n = (2, 3)
rng = np.random.default_rng(413722918996573)
A = rng.random(size=(dim, dim))
cov = A @ A.T
mean = rng.random(dim)
x = rng.random(n + (dim,))
df = rng.random() * 5
res = stats.multivariate_t.cdf(x, mean, cov, df, random_state=rng)
def _cdf_1d(x):
return _qsimvtv(10000, df, cov, -np.inf*x, x-mean, rng)[0]
ref = np.apply_along_axis(_cdf_1d, -1, x)
assert_allclose(res, ref, atol=1e-4, rtol=1e-3)
@pytest.mark.parametrize("dim", (3, 7))
def test_against_analytical(self, dim):
rng = np.random.default_rng(413722918996573)
A = scipy.linalg.toeplitz(c=[1] + [0.5] * (dim - 1))
res = stats.multivariate_t(shape=A).cdf([0] * dim, random_state=rng)
ref = 1 / (dim + 1)
assert_allclose(res, ref, rtol=5e-5)
def test_entropy_inf_df(self):
cov = np.eye(3, 3)
df = np.inf
mvt_entropy = stats.multivariate_t.entropy(shape=cov, df=df)
mvn_entropy = stats.multivariate_normal.entropy(None, cov)
assert mvt_entropy == mvn_entropy
@pytest.mark.parametrize("df", [1, 10, 100])
def test_entropy_1d(self, df):
mvt_entropy = stats.multivariate_t.entropy(shape=1., df=df)
t_entropy = stats.t.entropy(df=df)
assert_allclose(mvt_entropy, t_entropy, rtol=1e-13)
# entropy reference values were computed via numerical integration
#
# def integrand(x, y, mvt):
# vec = np.array([x, y])
# return mvt.logpdf(vec) * mvt.pdf(vec)
# def multivariate_t_entropy_quad_2d(df, cov):
# dim = cov.shape[0]
# loc = np.zeros((dim, ))
# mvt = stats.multivariate_t(loc, cov, df)
# limit = 100
# return -integrate.dblquad(integrand, -limit, limit, -limit, limit,
# args=(mvt, ))[0]
@pytest.mark.parametrize("df, cov, ref, tol",
[(10, np.eye(2, 2), 3.0378770664093313, 1e-14),
(100, np.array([[0.5, 1], [1, 10]]),
3.55102424550609, 1e-8)])
def test_entropy_vs_numerical_integration(self, df, cov, ref, tol):
loc = np.zeros((2, ))
mvt = stats.multivariate_t(loc, cov, df)
assert_allclose(mvt.entropy(), ref, rtol=tol)
@pytest.mark.parametrize(
"df, dim, ref, tol",
[
(10, 1, 1.5212624929756808, 1e-15),
(100, 1, 1.4289633653182439, 1e-13),
(500, 1, 1.420939531869349, 1e-14),
(1e20, 1, 1.4189385332046727, 1e-15),
(1e100, 1, 1.4189385332046727, 1e-15),
(10, 10, 15.069150450832911, 1e-15),
(1000, 10, 14.19936546446673, 1e-13),
(1e20, 10, 14.189385332046728, 1e-15),
(1e100, 10, 14.189385332046728, 1e-15),
(10, 100, 148.28902883192654, 1e-15),
(1000, 100, 141.99155538003762, 1e-14),
(1e20, 100, 141.8938533204673, 1e-15),
(1e100, 100, 141.8938533204673, 1e-15),
]
)
def test_extreme_entropy(self, df, dim, ref, tol):
# Reference values were calculated with mpmath:
# from mpmath import mp
# mp.dps = 500
#
# def mul_t_mpmath_entropy(dim, df=1):
# dim = mp.mpf(dim)
# df = mp.mpf(df)
# halfsum = (dim + df)/2
# half_df = df/2
#
# return float(
# -mp.loggamma(halfsum) + mp.loggamma(half_df)
# + dim / 2 * mp.log(df * mp.pi)
# + halfsum * (mp.digamma(halfsum) - mp.digamma(half_df))
# + 0.0
# )
mvt = stats.multivariate_t(shape=np.eye(dim), df=df)
assert_allclose(mvt.entropy(), ref, rtol=tol)
def test_entropy_with_covariance(self):
# Generated using np.randn(5, 5) and then rounding
# to two decimal places
_A = np.array([
[1.42, 0.09, -0.49, 0.17, 0.74],
[-1.13, -0.01, 0.71, 0.4, -0.56],
[1.07, 0.44, -0.28, -0.44, 0.29],
[-1.5, -0.94, -0.67, 0.73, -1.1],
[0.17, -0.08, 1.46, -0.32, 1.36]
])
# Set cov to be a symmetric positive semi-definite matrix
cov = _A @ _A.T
# Test the asymptotic case. For large degrees of freedom
# the entropy approaches the multivariate normal entropy.
df = 1e20
mul_t_entropy = stats.multivariate_t.entropy(shape=cov, df=df)
mul_norm_entropy = multivariate_normal(None, cov=cov).entropy()
assert_allclose(mul_t_entropy, mul_norm_entropy, rtol=1e-15)
# Test the regular case. For a dim of 5 the threshold comes out
# to be approximately 766.45. So using slightly
# different dfs on each site of the threshold, the entropies
# are being compared.
df1 = 765
df2 = 768
_entropy1 = stats.multivariate_t.entropy(shape=cov, df=df1)
_entropy2 = stats.multivariate_t.entropy(shape=cov, df=df2)
assert_allclose(_entropy1, _entropy2, rtol=1e-5)
class TestMultivariateHypergeom:
@pytest.mark.parametrize(
"x, m, n, expected",
[
# Ground truth value from R dmvhyper
([3, 4], [5, 10], 7, -1.119814),
# test for `n=0`
([3, 4], [5, 10], 0, -np.inf),
# test for `x < 0`
([-3, 4], [5, 10], 7, -np.inf),
# test for `m < 0` (RuntimeWarning issue)
([3, 4], [-5, 10], 7, np.nan),
# test for all `m < 0` and `x.sum() != n`
([[1, 2], [3, 4]], [[-4, -6], [-5, -10]],
[3, 7], [np.nan, np.nan]),
# test for `x < 0` and `m < 0` (RuntimeWarning issue)
([-3, 4], [-5, 10], 1, np.nan),
# test for `x > m`
([1, 11], [10, 1], 12, np.nan),
# test for `m < 0` (RuntimeWarning issue)
([1, 11], [10, -1], 12, np.nan),
# test for `n < 0`
([3, 4], [5, 10], -7, np.nan),
# test for `x.sum() != n`
([3, 3], [5, 10], 7, -np.inf)
]
)
def test_logpmf(self, x, m, n, expected):
vals = multivariate_hypergeom.logpmf(x, m, n)
assert_allclose(vals, expected, rtol=1e-6)
def test_reduces_hypergeom(self):
# test that the multivariate_hypergeom pmf reduces to the
# hypergeom pmf in the 2d case.
val1 = multivariate_hypergeom.pmf(x=[3, 1], m=[10, 5], n=4)
val2 = hypergeom.pmf(k=3, M=15, n=4, N=10)
assert_allclose(val1, val2, rtol=1e-8)
val1 = multivariate_hypergeom.pmf(x=[7, 3], m=[15, 10], n=10)
val2 = hypergeom.pmf(k=7, M=25, n=10, N=15)
assert_allclose(val1, val2, rtol=1e-8)
def test_rvs(self):
# test if `rvs` is unbiased and large sample size converges
# to the true mean.
rv = multivariate_hypergeom(m=[3, 5], n=4)
rvs = rv.rvs(size=1000, random_state=123)
assert_allclose(rvs.mean(0), rv.mean(), rtol=1e-2)
def test_rvs_broadcasting(self):
rv = multivariate_hypergeom(m=[[3, 5], [5, 10]], n=[4, 9])
rvs = rv.rvs(size=(1000, 2), random_state=123)
assert_allclose(rvs.mean(0), rv.mean(), rtol=1e-2)
@pytest.mark.parametrize('m, n', (
([0, 0, 20, 0, 0], 5), ([0, 0, 0, 0, 0], 0),
([0, 0], 0), ([0], 0)
))
def test_rvs_gh16171(self, m, n):
res = multivariate_hypergeom.rvs(m, n)
m = np.asarray(m)
res_ex = m.copy()
res_ex[m != 0] = n
assert_equal(res, res_ex)
@pytest.mark.parametrize(
"x, m, n, expected",
[
([5], [5], 5, 1),
([3, 4], [5, 10], 7, 0.3263403),
# Ground truth value from R dmvhyper
([[[3, 5], [0, 8]], [[-1, 9], [1, 1]]],
[5, 10], [[8, 8], [8, 2]],
[[0.3916084, 0.006993007], [0, 0.4761905]]),
# test with empty arrays.
(np.array([], dtype=int), np.array([], dtype=int), 0, []),
([1, 2], [4, 5], 5, 0),
# Ground truth value from R dmvhyper
([3, 3, 0], [5, 6, 7], 6, 0.01077354)
]
)
def test_pmf(self, x, m, n, expected):
vals = multivariate_hypergeom.pmf(x, m, n)
assert_allclose(vals, expected, rtol=1e-7)
@pytest.mark.parametrize(
"x, m, n, expected",
[
([3, 4], [[5, 10], [10, 15]], 7, [0.3263403, 0.3407531]),
([[1], [2]], [[3], [4]], [1, 3], [1., 0.]),
([[[1], [2]]], [[3], [4]], [1, 3], [[1., 0.]]),
([[1], [2]], [[[[3]]]], [1, 3], [[[1., 0.]]])
]
)
def test_pmf_broadcasting(self, x, m, n, expected):
vals = multivariate_hypergeom.pmf(x, m, n)
assert_allclose(vals, expected, rtol=1e-7)
def test_cov(self):
cov1 = multivariate_hypergeom.cov(m=[3, 7, 10], n=12)
cov2 = [[0.64421053, -0.26526316, -0.37894737],
[-0.26526316, 1.14947368, -0.88421053],
[-0.37894737, -0.88421053, 1.26315789]]
assert_allclose(cov1, cov2, rtol=1e-8)
def test_cov_broadcasting(self):
cov1 = multivariate_hypergeom.cov(m=[[7, 9], [10, 15]], n=[8, 12])
cov2 = [[[1.05, -1.05], [-1.05, 1.05]],
[[1.56, -1.56], [-1.56, 1.56]]]
assert_allclose(cov1, cov2, rtol=1e-8)
cov3 = multivariate_hypergeom.cov(m=[[4], [5]], n=[4, 5])
cov4 = [[[0.]], [[0.]]]
assert_allclose(cov3, cov4, rtol=1e-8)
cov5 = multivariate_hypergeom.cov(m=[7, 9], n=[8, 12])
cov6 = [[[1.05, -1.05], [-1.05, 1.05]],
[[0.7875, -0.7875], [-0.7875, 0.7875]]]
assert_allclose(cov5, cov6, rtol=1e-8)
def test_var(self):
# test with hypergeom
var0 = multivariate_hypergeom.var(m=[10, 5], n=4)
var1 = hypergeom.var(M=15, n=4, N=10)
assert_allclose(var0, var1, rtol=1e-8)
def test_var_broadcasting(self):
var0 = multivariate_hypergeom.var(m=[10, 5], n=[4, 8])
var1 = multivariate_hypergeom.var(m=[10, 5], n=4)
var2 = multivariate_hypergeom.var(m=[10, 5], n=8)
assert_allclose(var0[0], var1, rtol=1e-8)
assert_allclose(var0[1], var2, rtol=1e-8)
var3 = multivariate_hypergeom.var(m=[[10, 5], [10, 14]], n=[4, 8])
var4 = [[0.6984127, 0.6984127], [1.352657, 1.352657]]
assert_allclose(var3, var4, rtol=1e-8)
var5 = multivariate_hypergeom.var(m=[[5], [10]], n=[5, 10])
var6 = [[0.], [0.]]
assert_allclose(var5, var6, rtol=1e-8)
def test_mean(self):
# test with hypergeom
mean0 = multivariate_hypergeom.mean(m=[10, 5], n=4)
mean1 = hypergeom.mean(M=15, n=4, N=10)
assert_allclose(mean0[0], mean1, rtol=1e-8)
mean2 = multivariate_hypergeom.mean(m=[12, 8], n=10)
mean3 = [12.*10./20., 8.*10./20.]
assert_allclose(mean2, mean3, rtol=1e-8)
def test_mean_broadcasting(self):
mean0 = multivariate_hypergeom.mean(m=[[3, 5], [10, 5]], n=[4, 8])
mean1 = [[3.*4./8., 5.*4./8.], [10.*8./15., 5.*8./15.]]
assert_allclose(mean0, mean1, rtol=1e-8)
def test_mean_edge_cases(self):
mean0 = multivariate_hypergeom.mean(m=[0, 0, 0], n=0)
assert_equal(mean0, [0., 0., 0.])
mean1 = multivariate_hypergeom.mean(m=[1, 0, 0], n=2)
assert_equal(mean1, [np.nan, np.nan, np.nan])
mean2 = multivariate_hypergeom.mean(m=[[1, 0, 0], [1, 0, 1]], n=2)
assert_allclose(mean2, [[np.nan, np.nan, np.nan], [1., 0., 1.]],
rtol=1e-17)
mean3 = multivariate_hypergeom.mean(m=np.array([], dtype=int), n=0)
assert_equal(mean3, [])
assert_(mean3.shape == (0, ))
def test_var_edge_cases(self):
var0 = multivariate_hypergeom.var(m=[0, 0, 0], n=0)
assert_allclose(var0, [0., 0., 0.], rtol=1e-16)
var1 = multivariate_hypergeom.var(m=[1, 0, 0], n=2)
assert_equal(var1, [np.nan, np.nan, np.nan])
var2 = multivariate_hypergeom.var(m=[[1, 0, 0], [1, 0, 1]], n=2)
assert_allclose(var2, [[np.nan, np.nan, np.nan], [0., 0., 0.]],
rtol=1e-17)
var3 = multivariate_hypergeom.var(m=np.array([], dtype=int), n=0)
assert_equal(var3, [])
assert_(var3.shape == (0, ))
def test_cov_edge_cases(self):
cov0 = multivariate_hypergeom.cov(m=[1, 0, 0], n=1)
cov1 = [[0., 0., 0.], [0., 0., 0.], [0., 0., 0.]]
assert_allclose(cov0, cov1, rtol=1e-17)
cov3 = multivariate_hypergeom.cov(m=[0, 0, 0], n=0)
cov4 = [[0., 0., 0.], [0., 0., 0.], [0., 0., 0.]]
assert_equal(cov3, cov4)
cov5 = multivariate_hypergeom.cov(m=np.array([], dtype=int), n=0)
cov6 = np.array([], dtype=np.float64).reshape(0, 0)
assert_allclose(cov5, cov6, rtol=1e-17)
assert_(cov5.shape == (0, 0))
def test_frozen(self):
# The frozen distribution should agree with the regular one
np.random.seed(1234)
n = 12
m = [7, 9, 11, 13]
x = [[0, 0, 0, 12], [0, 0, 1, 11], [0, 1, 1, 10],
[1, 1, 1, 9], [1, 1, 2, 8]]
x = np.asarray(x, dtype=int)
mhg_frozen = multivariate_hypergeom(m, n)
assert_allclose(mhg_frozen.pmf(x),
multivariate_hypergeom.pmf(x, m, n))
assert_allclose(mhg_frozen.logpmf(x),
multivariate_hypergeom.logpmf(x, m, n))
assert_allclose(mhg_frozen.var(), multivariate_hypergeom.var(m, n))
assert_allclose(mhg_frozen.cov(), multivariate_hypergeom.cov(m, n))
def test_invalid_params(self):
assert_raises(ValueError, multivariate_hypergeom.pmf, 5, 10, 5)
assert_raises(ValueError, multivariate_hypergeom.pmf, 5, [10], 5)
assert_raises(ValueError, multivariate_hypergeom.pmf, [5, 4], [10], 5)
assert_raises(TypeError, multivariate_hypergeom.pmf, [5.5, 4.5],
[10, 15], 5)
assert_raises(TypeError, multivariate_hypergeom.pmf, [5, 4],
[10.5, 15.5], 5)
assert_raises(TypeError, multivariate_hypergeom.pmf, [5, 4],
[10, 15], 5.5)
class TestRandomTable:
def get_rng(self):
return np.random.default_rng(628174795866951638)
def test_process_parameters(self):
message = "`row` must be one-dimensional"
with pytest.raises(ValueError, match=message):
random_table([[1, 2]], [1, 2])
message = "`col` must be one-dimensional"
with pytest.raises(ValueError, match=message):
random_table([1, 2], [[1, 2]])
message = "each element of `row` must be non-negative"
with pytest.raises(ValueError, match=message):
random_table([1, -1], [1, 2])
message = "each element of `col` must be non-negative"
with pytest.raises(ValueError, match=message):
random_table([1, 2], [1, -2])
message = "sums over `row` and `col` must be equal"
with pytest.raises(ValueError, match=message):
random_table([1, 2], [1, 0])
message = "each element of `row` must be an integer"
with pytest.raises(ValueError, match=message):
random_table([2.1, 2.1], [1, 1, 2])
message = "each element of `col` must be an integer"
with pytest.raises(ValueError, match=message):
random_table([1, 2], [1.1, 1.1, 1])
row = [1, 3]
col = [2, 1, 1]
r, c, n = random_table._process_parameters([1, 3], [2, 1, 1])
assert_equal(row, r)
assert_equal(col, c)
assert n == np.sum(row)
@pytest.mark.parametrize("scale,method",
((1, "boyett"), (100, "patefield")))
def test_process_rvs_method_on_None(self, scale, method):
row = np.array([1, 3]) * scale
col = np.array([2, 1, 1]) * scale
ct = random_table
expected = ct.rvs(row, col, method=method, random_state=1)
got = ct.rvs(row, col, method=None, random_state=1)
assert_equal(expected, got)
def test_process_rvs_method_bad_argument(self):
row = [1, 3]
col = [2, 1, 1]
# order of items in set is random, so cannot check that
message = "'foo' not recognized, must be one of"
with pytest.raises(ValueError, match=message):
random_table.rvs(row, col, method="foo")
@pytest.mark.parametrize('frozen', (True, False))
@pytest.mark.parametrize('log', (True, False))
def test_pmf_logpmf(self, frozen, log):
# The pmf is tested through random sample generation
# with Boyett's algorithm, whose implementation is simple
# enough to verify manually for correctness.
rng = self.get_rng()
row = [2, 6]
col = [1, 3, 4]
rvs = random_table.rvs(row, col, size=1000,
method="boyett", random_state=rng)
obj = random_table(row, col) if frozen else random_table
method = getattr(obj, "logpmf" if log else "pmf")
if not frozen:
original_method = method
def method(x):
return original_method(x, row, col)
pmf = (lambda x: np.exp(method(x))) if log else method
unique_rvs, counts = np.unique(rvs, axis=0, return_counts=True)
# rough accuracy check
p = pmf(unique_rvs)
assert_allclose(p * len(rvs), counts, rtol=0.1)
# accept any iterable
p2 = pmf(list(unique_rvs[0]))
assert_equal(p2, p[0])
# accept high-dimensional input and 2d input
rvs_nd = rvs.reshape((10, 100) + rvs.shape[1:])
p = pmf(rvs_nd)
assert p.shape == (10, 100)
for i in range(p.shape[0]):
for j in range(p.shape[1]):
pij = p[i, j]
rvij = rvs_nd[i, j]
qij = pmf(rvij)
assert_equal(pij, qij)
# probability is zero if column marginal does not match
x = [[0, 1, 1], [2, 1, 3]]
assert_equal(np.sum(x, axis=-1), row)
p = pmf(x)
assert p == 0
# probability is zero if row marginal does not match
x = [[0, 1, 2], [1, 2, 2]]
assert_equal(np.sum(x, axis=-2), col)
p = pmf(x)
assert p == 0
# response to invalid inputs
message = "`x` must be at least two-dimensional"
with pytest.raises(ValueError, match=message):
pmf([1])
message = "`x` must contain only integral values"
with pytest.raises(ValueError, match=message):
pmf([[1.1]])
message = "`x` must contain only integral values"
with pytest.raises(ValueError, match=message):
pmf([[np.nan]])
message = "`x` must contain only non-negative values"
with pytest.raises(ValueError, match=message):
pmf([[-1]])
message = "shape of `x` must agree with `row`"
with pytest.raises(ValueError, match=message):
pmf([[1, 2, 3]])
message = "shape of `x` must agree with `col`"
with pytest.raises(ValueError, match=message):
pmf([[1, 2],
[3, 4]])
@pytest.mark.parametrize("method", ("boyett", "patefield"))
def test_rvs_mean(self, method):
# test if `rvs` is unbiased and large sample size converges
# to the true mean.
rng = self.get_rng()
row = [2, 6]
col = [1, 3, 4]
rvs = random_table.rvs(row, col, size=1000, method=method,
random_state=rng)
mean = random_table.mean(row, col)
assert_equal(np.sum(mean), np.sum(row))
assert_allclose(rvs.mean(0), mean, atol=0.05)
assert_equal(rvs.sum(axis=-1), np.broadcast_to(row, (1000, 2)))
assert_equal(rvs.sum(axis=-2), np.broadcast_to(col, (1000, 3)))
def test_rvs_cov(self):
# test if `rvs` generated with patefield and boyett algorithms
# produce approximately the same covariance matrix
rng = self.get_rng()
row = [2, 6]
col = [1, 3, 4]
rvs1 = random_table.rvs(row, col, size=10000, method="boyett",
random_state=rng)
rvs2 = random_table.rvs(row, col, size=10000, method="patefield",
random_state=rng)
cov1 = np.var(rvs1, axis=0)
cov2 = np.var(rvs2, axis=0)
assert_allclose(cov1, cov2, atol=0.02)
@pytest.mark.parametrize("method", ("boyett", "patefield"))
def test_rvs_size(self, method):
row = [2, 6]
col = [1, 3, 4]
# test size `None`
rv = random_table.rvs(row, col, method=method,
random_state=self.get_rng())
assert rv.shape == (2, 3)
# test size 1
rv2 = random_table.rvs(row, col, size=1, method=method,
random_state=self.get_rng())
assert rv2.shape == (1, 2, 3)
assert_equal(rv, rv2[0])
# test size 0
rv3 = random_table.rvs(row, col, size=0, method=method,
random_state=self.get_rng())
assert rv3.shape == (0, 2, 3)
# test other valid size
rv4 = random_table.rvs(row, col, size=20, method=method,
random_state=self.get_rng())
assert rv4.shape == (20, 2, 3)
rv5 = random_table.rvs(row, col, size=(4, 5), method=method,
random_state=self.get_rng())
assert rv5.shape == (4, 5, 2, 3)
assert_allclose(rv5.reshape(20, 2, 3), rv4, rtol=1e-15)
# test invalid size
message = "`size` must be a non-negative integer or `None`"
with pytest.raises(ValueError, match=message):
random_table.rvs(row, col, size=-1, method=method,
random_state=self.get_rng())
with pytest.raises(ValueError, match=message):
random_table.rvs(row, col, size=np.nan, method=method,
random_state=self.get_rng())
@pytest.mark.parametrize("method", ("boyett", "patefield"))
def test_rvs_method(self, method):
# This test assumes that pmf is correct and checks that random samples
# follow this probability distribution. This seems like a circular
# argument, since pmf is checked in test_pmf_logpmf with random samples
# generated with the rvs method. This test is not redundant, because
# test_pmf_logpmf intentionally uses rvs generation with Boyett only,
# but here we test both Boyett and Patefield.
row = [2, 6]
col = [1, 3, 4]
ct = random_table
rvs = ct.rvs(row, col, size=100000, method=method,
random_state=self.get_rng())
unique_rvs, counts = np.unique(rvs, axis=0, return_counts=True)
# generated frequencies should match expected frequencies
p = ct.pmf(unique_rvs, row, col)
assert_allclose(p * len(rvs), counts, rtol=0.02)
@pytest.mark.parametrize("method", ("boyett", "patefield"))
def test_rvs_with_zeros_in_col_row(self, method):
row = [0, 1, 0]
col = [1, 0, 0, 0]
d = random_table(row, col)
rv = d.rvs(1000, method=method, random_state=self.get_rng())
expected = np.zeros((1000, len(row), len(col)))
expected[...] = [[0, 0, 0, 0],
[1, 0, 0, 0],
[0, 0, 0, 0]]
assert_equal(rv, expected)
@pytest.mark.parametrize("method", (None, "boyett", "patefield"))
@pytest.mark.parametrize("col", ([], [0]))
@pytest.mark.parametrize("row", ([], [0]))
def test_rvs_with_edge_cases(self, method, row, col):
d = random_table(row, col)
rv = d.rvs(10, method=method, random_state=self.get_rng())
expected = np.zeros((10, len(row), len(col)))
assert_equal(rv, expected)
@pytest.mark.parametrize('v', (1, 2))
def test_rvs_rcont(self, v):
# This test checks the internal low-level interface.
# It is implicitly also checked by the other test_rvs* calls.
import scipy.stats._rcont as _rcont
row = np.array([1, 3], dtype=np.int64)
col = np.array([2, 1, 1], dtype=np.int64)
rvs = getattr(_rcont, f"rvs_rcont{v}")
ntot = np.sum(row)
result = rvs(row, col, ntot, 1, self.get_rng())
assert result.shape == (1, len(row), len(col))
assert np.sum(result) == ntot
def test_frozen(self):
row = [2, 6]
col = [1, 3, 4]
d = random_table(row, col, seed=self.get_rng())
sample = d.rvs()
expected = random_table.mean(row, col)
assert_equal(expected, d.mean())
expected = random_table.pmf(sample, row, col)
assert_equal(expected, d.pmf(sample))
expected = random_table.logpmf(sample, row, col)
assert_equal(expected, d.logpmf(sample))
@pytest.mark.parametrize("method", ("boyett", "patefield"))
def test_rvs_frozen(self, method):
row = [2, 6]
col = [1, 3, 4]
d = random_table(row, col, seed=self.get_rng())
expected = random_table.rvs(row, col, size=10, method=method,
random_state=self.get_rng())
got = d.rvs(size=10, method=method)
assert_equal(expected, got)
def check_pickling(distfn, args):
# check that a distribution instance pickles and unpickles
# pay special attention to the random_state property
# save the random_state (restore later)
rndm = distfn.random_state
distfn.random_state = 1234
distfn.rvs(*args, size=8)
s = pickle.dumps(distfn)
r0 = distfn.rvs(*args, size=8)
unpickled = pickle.loads(s)
r1 = unpickled.rvs(*args, size=8)
assert_equal(r0, r1)
# restore the random_state
distfn.random_state = rndm
@pytest.mark.thread_unsafe
def test_random_state_property(num_parallel_threads):
scale = np.eye(3)
scale[0, 1] = 0.5
scale[1, 0] = 0.5
dists = [
[multivariate_normal, ()],
[dirichlet, (np.array([1.]), )],
[wishart, (10, scale)],
[invwishart, (10, scale)],
[multinomial, (5, [0.5, 0.4, 0.1])],
[ortho_group, (2,)],
[special_ortho_group, (2,)]
]
for distfn, args in dists:
check_random_state_property(distfn, args)
check_pickling(distfn, args)
class TestVonMises_Fisher:
@pytest.mark.parametrize("dim", [2, 3, 4, 6])
@pytest.mark.parametrize("size", [None, 1, 5, (5, 4)])
def test_samples(self, dim, size):
# test that samples have correct shape and norm 1
rng = np.random.default_rng(2777937887058094419)
mu = np.full((dim, ), 1/np.sqrt(dim))
vmf_dist = vonmises_fisher(mu, 1, seed=rng)
samples = vmf_dist.rvs(size)
mean, cov = np.zeros(dim), np.eye(dim)
expected_shape = rng.multivariate_normal(mean, cov, size=size).shape
assert samples.shape == expected_shape
norms = np.linalg.norm(samples, axis=-1)
assert_allclose(norms, 1.)
@pytest.mark.parametrize("dim", [5, 8])
@pytest.mark.parametrize("kappa", [1e15, 1e20, 1e30])
def test_sampling_high_concentration(self, dim, kappa):
# test that no warnings are encountered for high values
rng = np.random.default_rng(2777937887058094419)
mu = np.full((dim, ), 1/np.sqrt(dim))
vmf_dist = vonmises_fisher(mu, kappa, seed=rng)
vmf_dist.rvs(10)
def test_two_dimensional_mu(self):
mu = np.ones((2, 2))
msg = "'mu' must have one-dimensional shape."
with pytest.raises(ValueError, match=msg):
vonmises_fisher(mu, 1)
def test_wrong_norm_mu(self):
mu = np.ones((2, ))
msg = "'mu' must be a unit vector of norm 1."
with pytest.raises(ValueError, match=msg):
vonmises_fisher(mu, 1)
def test_one_entry_mu(self):
mu = np.ones((1, ))
msg = "'mu' must have at least two entries."
with pytest.raises(ValueError, match=msg):
vonmises_fisher(mu, 1)
@pytest.mark.parametrize("kappa", [-1, (5, 3)])
def test_kappa_validation(self, kappa):
msg = "'kappa' must be a positive scalar."
with pytest.raises(ValueError, match=msg):
vonmises_fisher([1, 0], kappa)
@pytest.mark.parametrize("kappa", [0, 0.])
def test_kappa_zero(self, kappa):
msg = ("For 'kappa=0' the von Mises-Fisher distribution "
"becomes the uniform distribution on the sphere "
"surface. Consider using 'scipy.stats.uniform_direction' "
"instead.")
with pytest.raises(ValueError, match=msg):
vonmises_fisher([1, 0], kappa)
@pytest.mark.parametrize("method", [vonmises_fisher.pdf,
vonmises_fisher.logpdf])
def test_invalid_shapes_pdf_logpdf(self, method):
x = np.array([1., 0., 0])
msg = ("The dimensionality of the last axis of 'x' must "
"match the dimensionality of the von Mises Fisher "
"distribution.")
with pytest.raises(ValueError, match=msg):
method(x, [1, 0], 1)
@pytest.mark.parametrize("method", [vonmises_fisher.pdf,
vonmises_fisher.logpdf])
def test_unnormalized_input(self, method):
x = np.array([0.5, 0.])
msg = "'x' must be unit vectors of norm 1 along last dimension."
with pytest.raises(ValueError, match=msg):
method(x, [1, 0], 1)
# Expected values of the vonmises-fisher logPDF were computed via mpmath
# from mpmath import mp
# import numpy as np
# mp.dps = 50
# def logpdf_mpmath(x, mu, kappa):
# dim = mu.size
# halfdim = mp.mpf(0.5 * dim)
# kappa = mp.mpf(kappa)
# const = (kappa**(halfdim - mp.one)/((2*mp.pi)**halfdim * \
# mp.besseli(halfdim -mp.one, kappa)))
# return float(const * mp.exp(kappa*mp.fdot(x, mu)))
@pytest.mark.parametrize('x, mu, kappa, reference',
[(np.array([1., 0., 0.]), np.array([1., 0., 0.]),
1e-4, 0.0795854295583605),
(np.array([1., 0., 0]), np.array([0., 0., 1.]),
1e-4, 0.07957747141331854),
(np.array([1., 0., 0.]), np.array([1., 0., 0.]),
100, 15.915494309189533),
(np.array([1., 0., 0]), np.array([0., 0., 1.]),
100, 5.920684802611232e-43),
(np.array([1., 0., 0.]),
np.array([np.sqrt(0.98), np.sqrt(0.02), 0.]),
2000, 5.930499050746588e-07),
(np.array([1., 0., 0]), np.array([1., 0., 0.]),
2000, 318.3098861837907),
(np.array([1., 0., 0., 0., 0.]),
np.array([1., 0., 0., 0., 0.]),
2000, 101371.86957712633),
(np.array([1., 0., 0., 0., 0.]),
np.array([np.sqrt(0.98), np.sqrt(0.02), 0.,
0, 0.]),
2000, 0.00018886808182653578),
(np.array([1., 0., 0., 0., 0.]),
np.array([np.sqrt(0.8), np.sqrt(0.2), 0.,
0, 0.]),
2000, 2.0255393314603194e-87)])
def test_pdf_accuracy(self, x, mu, kappa, reference):
pdf = vonmises_fisher(mu, kappa).pdf(x)
assert_allclose(pdf, reference, rtol=1e-13)
# Expected values of the vonmises-fisher logPDF were computed via mpmath
# from mpmath import mp
# import numpy as np
# mp.dps = 50
# def logpdf_mpmath(x, mu, kappa):
# dim = mu.size
# halfdim = mp.mpf(0.5 * dim)
# kappa = mp.mpf(kappa)
# two = mp.mpf(2.)
# const = (kappa**(halfdim - mp.one)/((two*mp.pi)**halfdim * \
# mp.besseli(halfdim - mp.one, kappa)))
# return float(mp.log(const * mp.exp(kappa*mp.fdot(x, mu))))
@pytest.mark.parametrize('x, mu, kappa, reference',
[(np.array([1., 0., 0.]), np.array([1., 0., 0.]),
1e-4, -2.5309242486359573),
(np.array([1., 0., 0]), np.array([0., 0., 1.]),
1e-4, -2.5310242486359575),
(np.array([1., 0., 0.]), np.array([1., 0., 0.]),
100, 2.767293119578746),
(np.array([1., 0., 0]), np.array([0., 0., 1.]),
100, -97.23270688042125),
(np.array([1., 0., 0.]),
np.array([np.sqrt(0.98), np.sqrt(0.02), 0.]),
2000, -14.337987284534103),
(np.array([1., 0., 0]), np.array([1., 0., 0.]),
2000, 5.763025393132737),
(np.array([1., 0., 0., 0., 0.]),
np.array([1., 0., 0., 0., 0.]),
2000, 11.526550911307156),
(np.array([1., 0., 0., 0., 0.]),
np.array([np.sqrt(0.98), np.sqrt(0.02), 0.,
0, 0.]),
2000, -8.574461766359684),
(np.array([1., 0., 0., 0., 0.]),
np.array([np.sqrt(0.8), np.sqrt(0.2), 0.,
0, 0.]),
2000, -199.61906708886113)])
def test_logpdf_accuracy(self, x, mu, kappa, reference):
logpdf = vonmises_fisher(mu, kappa).logpdf(x)
assert_allclose(logpdf, reference, rtol=1e-14)
# Expected values of the vonmises-fisher entropy were computed via mpmath
# from mpmath import mp
# import numpy as np
# mp.dps = 50
# def entropy_mpmath(dim, kappa):
# mu = np.full((dim, ), 1/np.sqrt(dim))
# kappa = mp.mpf(kappa)
# halfdim = mp.mpf(0.5 * dim)
# logconstant = (mp.log(kappa**(halfdim - mp.one)
# /((2*mp.pi)**halfdim
# * mp.besseli(halfdim -mp.one, kappa)))
# return float(-logconstant - kappa * mp.besseli(halfdim, kappa)/
# mp.besseli(halfdim -1, kappa))
@pytest.mark.parametrize('dim, kappa, reference',
[(3, 1e-4, 2.531024245302624),
(3, 100, -1.7672931195787458),
(5, 5000, -11.359032310024453),
(8, 1, 3.4189526482545527)])
def test_entropy_accuracy(self, dim, kappa, reference):
mu = np.full((dim, ), 1/np.sqrt(dim))
entropy = vonmises_fisher(mu, kappa).entropy()
assert_allclose(entropy, reference, rtol=2e-14)
@pytest.mark.parametrize("method", [vonmises_fisher.pdf,
vonmises_fisher.logpdf])
def test_broadcasting(self, method):
# test that pdf and logpdf values are correctly broadcasted
testshape = (2, 2)
rng = np.random.default_rng(2777937887058094419)
x = uniform_direction(3).rvs(testshape, random_state=rng)
mu = np.full((3, ), 1/np.sqrt(3))
kappa = 5
result_all = method(x, mu, kappa)
assert result_all.shape == testshape
for i in range(testshape[0]):
for j in range(testshape[1]):
current_val = method(x[i, j, :], mu, kappa)
assert_allclose(current_val, result_all[i, j], rtol=1e-15)
def test_vs_vonmises_2d(self):
# test that in 2D, von Mises-Fisher yields the same results
# as the von Mises distribution
rng = np.random.default_rng(2777937887058094419)
mu = np.array([0, 1])
mu_angle = np.arctan2(mu[1], mu[0])
kappa = 20
vmf = vonmises_fisher(mu, kappa)
vonmises_dist = vonmises(loc=mu_angle, kappa=kappa)
vectors = uniform_direction(2).rvs(10, random_state=rng)
angles = np.arctan2(vectors[:, 1], vectors[:, 0])
assert_allclose(vonmises_dist.entropy(), vmf.entropy())
assert_allclose(vonmises_dist.pdf(angles), vmf.pdf(vectors))
assert_allclose(vonmises_dist.logpdf(angles), vmf.logpdf(vectors))
@pytest.mark.parametrize("dim", [2, 3, 6])
@pytest.mark.parametrize("kappa, mu_tol, kappa_tol",
[(1, 5e-2, 5e-2),
(10, 1e-2, 1e-2),
(100, 5e-3, 2e-2),
(1000, 1e-3, 2e-2)])
def test_fit_accuracy(self, dim, kappa, mu_tol, kappa_tol):
mu = np.full((dim, ), 1/np.sqrt(dim))
vmf_dist = vonmises_fisher(mu, kappa)
rng = np.random.default_rng(2777937887058094419)
n_samples = 10000
samples = vmf_dist.rvs(n_samples, random_state=rng)
mu_fit, kappa_fit = vonmises_fisher.fit(samples)
angular_error = np.arccos(mu.dot(mu_fit))
assert_allclose(angular_error, 0., atol=mu_tol, rtol=0)
assert_allclose(kappa, kappa_fit, rtol=kappa_tol)
def test_fit_error_one_dimensional_data(self):
x = np.zeros((3, ))
msg = "'x' must be two dimensional."
with pytest.raises(ValueError, match=msg):
vonmises_fisher.fit(x)
def test_fit_error_unnormalized_data(self):
x = np.ones((3, 3))
msg = "'x' must be unit vectors of norm 1 along last dimension."
with pytest.raises(ValueError, match=msg):
vonmises_fisher.fit(x)
def test_frozen_distribution(self):
mu = np.array([0, 0, 1])
kappa = 5
frozen = vonmises_fisher(mu, kappa)
frozen_seed = vonmises_fisher(mu, kappa, seed=514)
rvs1 = frozen.rvs(random_state=514)
rvs2 = vonmises_fisher.rvs(mu, kappa, random_state=514)
rvs3 = frozen_seed.rvs()
assert_equal(rvs1, rvs2)
assert_equal(rvs1, rvs3)
class TestDirichletMultinomial:
@classmethod
def get_params(self, m):
rng = np.random.default_rng(28469824356873456)
alpha = rng.uniform(0, 100, size=2)
x = rng.integers(1, 20, size=(m, 2))
n = x.sum(axis=-1)
return rng, m, alpha, n, x
def test_frozen(self):
rng = np.random.default_rng(28469824356873456)
alpha = rng.uniform(0, 100, 10)
x = rng.integers(0, 10, 10)
n = np.sum(x, axis=-1)
d = dirichlet_multinomial(alpha, n)
assert_equal(d.logpmf(x), dirichlet_multinomial.logpmf(x, alpha, n))
assert_equal(d.pmf(x), dirichlet_multinomial.pmf(x, alpha, n))
assert_equal(d.mean(), dirichlet_multinomial.mean(alpha, n))
assert_equal(d.var(), dirichlet_multinomial.var(alpha, n))
assert_equal(d.cov(), dirichlet_multinomial.cov(alpha, n))
def test_pmf_logpmf_against_R(self):
# # Compare PMF against R's extraDistr ddirmnon
# # library(extraDistr)
# # options(digits=16)
# ddirmnom(c(1, 2, 3), 6, c(3, 4, 5))
x = np.array([1, 2, 3])
n = np.sum(x)
alpha = np.array([3, 4, 5])
res = dirichlet_multinomial.pmf(x, alpha, n)
logres = dirichlet_multinomial.logpmf(x, alpha, n)
ref = 0.08484162895927638
assert_allclose(res, ref)
assert_allclose(logres, np.log(ref))
assert res.shape == logres.shape == ()
# library(extraDistr)
# options(digits=16)
# ddirmnom(c(4, 3, 2, 0, 2, 3, 5, 7, 4, 7), 37,
# c(45.01025314, 21.98739582, 15.14851365, 80.21588671,
# 52.84935481, 25.20905262, 53.85373737, 4.88568118,
# 89.06440654, 20.11359466))
rng = np.random.default_rng(28469824356873456)
alpha = rng.uniform(0, 100, 10)
x = rng.integers(0, 10, 10)
n = np.sum(x, axis=-1)
res = dirichlet_multinomial(alpha, n).pmf(x)
logres = dirichlet_multinomial.logpmf(x, alpha, n)
ref = 3.65409306285992e-16
assert_allclose(res, ref)
assert_allclose(logres, np.log(ref))
def test_pmf_logpmf_support(self):
# when the sum of the category counts does not equal the number of
# trials, the PMF is zero
rng, m, alpha, n, x = self.get_params(1)
n += 1
assert_equal(dirichlet_multinomial(alpha, n).pmf(x), 0)
assert_equal(dirichlet_multinomial(alpha, n).logpmf(x), -np.inf)
rng, m, alpha, n, x = self.get_params(10)
i = rng.random(size=10) > 0.5
x[i] = np.round(x[i] * 2) # sum of these x does not equal n
assert_equal(dirichlet_multinomial(alpha, n).pmf(x)[i], 0)
assert_equal(dirichlet_multinomial(alpha, n).logpmf(x)[i], -np.inf)
assert np.all(dirichlet_multinomial(alpha, n).pmf(x)[~i] > 0)
assert np.all(dirichlet_multinomial(alpha, n).logpmf(x)[~i] > -np.inf)
def test_dimensionality_one(self):
# if the dimensionality is one, there is only one possible outcome
n = 6 # number of trials
alpha = [10] # concentration parameters
x = np.asarray([n]) # counts
dist = dirichlet_multinomial(alpha, n)
assert_equal(dist.pmf(x), 1)
assert_equal(dist.pmf(x+1), 0)
assert_equal(dist.logpmf(x), 0)
assert_equal(dist.logpmf(x+1), -np.inf)
assert_equal(dist.mean(), n)
assert_equal(dist.var(), 0)
assert_equal(dist.cov(), 0)
def test_n_is_zero(self):
# similarly, only one possible outcome if n is zero
n = 0
alpha = np.asarray([1., 1.])
x = np.asarray([0, 0])
dist = dirichlet_multinomial(alpha, n)
assert_equal(dist.pmf(x), 1)
assert_equal(dist.pmf(x+1), 0)
assert_equal(dist.logpmf(x), 0)
assert_equal(dist.logpmf(x+1), -np.inf)
assert_equal(dist.mean(), [0, 0])
assert_equal(dist.var(), [0, 0])
assert_equal(dist.cov(), [[0, 0], [0, 0]])
@pytest.mark.parametrize('method_name', ['pmf', 'logpmf'])
def test_against_betabinom_pmf(self, method_name):
rng, m, alpha, n, x = self.get_params(100)
method = getattr(dirichlet_multinomial(alpha, n), method_name)
ref_method = getattr(stats.betabinom(n, *alpha.T), method_name)
res = method(x)
ref = ref_method(x.T[0])
assert_allclose(res, ref)
@pytest.mark.parametrize('method_name', ['mean', 'var'])
def test_against_betabinom_moments(self, method_name):
rng, m, alpha, n, x = self.get_params(100)
method = getattr(dirichlet_multinomial(alpha, n), method_name)
ref_method = getattr(stats.betabinom(n, *alpha.T), method_name)
res = method()[:, 0]
ref = ref_method()
assert_allclose(res, ref)
def test_moments(self):
rng = np.random.default_rng(28469824356873456)
dim = 5
n = rng.integers(1, 100)
alpha = rng.random(size=dim) * 10
dist = dirichlet_multinomial(alpha, n)
# Generate a random sample from the distribution using NumPy
m = 100000
p = rng.dirichlet(alpha, size=m)
x = rng.multinomial(n, p, size=m)
assert_allclose(dist.mean(), np.mean(x, axis=0), rtol=5e-3)
assert_allclose(dist.var(), np.var(x, axis=0), rtol=1e-2)
assert dist.mean().shape == dist.var().shape == (dim,)
cov = dist.cov()
assert cov.shape == (dim, dim)
assert_allclose(cov, np.cov(x.T), rtol=2e-2)
assert_equal(np.diag(cov), dist.var())
assert np.all(scipy.linalg.eigh(cov)[0] > 0) # positive definite
def test_input_validation(self):
# valid inputs
x0 = np.array([1, 2, 3])
n0 = np.sum(x0)
alpha0 = np.array([3, 4, 5])
text = "`x` must contain only non-negative integers."
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf([1, -1, 3], alpha0, n0)
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf([1, 2.1, 3], alpha0, n0)
text = "`alpha` must contain only positive values."
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf(x0, [3, 0, 4], n0)
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf(x0, [3, -1, 4], n0)
text = "`n` must be a non-negative integer."
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf(x0, alpha0, 49.1)
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf(x0, alpha0, -1)
x = np.array([1, 2, 3, 4])
alpha = np.array([3, 4, 5])
text = "`x` and `alpha` must be broadcastable."
with assert_raises(ValueError, match=text):
dirichlet_multinomial.logpmf(x, alpha, x.sum())
@pytest.mark.parametrize('method', ['pmf', 'logpmf'])
def test_broadcasting_pmf(self, method):
alpha = np.array([[3, 4, 5], [4, 5, 6], [5, 5, 7], [8, 9, 10]])
n = np.array([[6], [7], [8]])
x = np.array([[1, 2, 3], [2, 2, 3]]).reshape((2, 1, 1, 3))
method = getattr(dirichlet_multinomial, method)
res = method(x, alpha, n)
assert res.shape == (2, 3, 4)
for i in range(len(x)):
for j in range(len(n)):
for k in range(len(alpha)):
res_ijk = res[i, j, k]
ref = method(x[i].squeeze(), alpha[k].squeeze(), n[j].squeeze())
assert_allclose(res_ijk, ref)
@pytest.mark.parametrize('method_name', ['mean', 'var', 'cov'])
def test_broadcasting_moments(self, method_name):
alpha = np.array([[3, 4, 5], [4, 5, 6], [5, 5, 7], [8, 9, 10]])
n = np.array([[6], [7], [8]])
method = getattr(dirichlet_multinomial, method_name)
res = method(alpha, n)
assert res.shape == (3, 4, 3) if method_name != 'cov' else (3, 4, 3, 3)
for j in range(len(n)):
for k in range(len(alpha)):
res_ijk = res[j, k]
ref = method(alpha[k].squeeze(), n[j].squeeze())
assert_allclose(res_ijk, ref)
class TestNormalInverseGamma:
def test_marginal_x(self):
# According to [1], sqrt(a * lmbda / b) * (x - u) should follow a t-distribution
# with 2*a degrees of freedom. Test that this is true of the PDF and random
# variates.
rng = np.random.default_rng(8925849245)
mu, lmbda, a, b = rng.random(4)
norm_inv_gamma = stats.normal_inverse_gamma(mu, lmbda, a, b)
t = stats.t(2*a, loc=mu, scale=1/np.sqrt(a * lmbda / b))
# Test PDF
x = np.linspace(-5, 5, 11)
res = tanhsinh(lambda s2, x: norm_inv_gamma.pdf(x, s2), 0, np.inf, args=(x,))
ref = t.pdf(x)
assert_allclose(res.integral, ref)
# Test RVS
res = norm_inv_gamma.rvs(size=10000, random_state=rng)
_, pvalue = stats.ks_1samp(res[0], t.cdf)
assert pvalue > 0.1
def test_marginal_s2(self):
# According to [1], s2 should follow an inverse gamma distribution with
# shapes a, b (where b is the scale in our parameterization). Test that
# this is true of the PDF and random variates.
rng = np.random.default_rng(8925849245)
mu, lmbda, a, b = rng.random(4)
norm_inv_gamma = stats.normal_inverse_gamma(mu, lmbda, a, b)
inv_gamma = stats.invgamma(a, scale=b)
# Test PDF
s2 = np.linspace(0.1, 10, 10)
res = tanhsinh(lambda x, s2: norm_inv_gamma.pdf(x, s2),
-np.inf, np.inf, args=(s2,))
ref = inv_gamma.pdf(s2)
assert_allclose(res.integral, ref)
# Test RVS
res = norm_inv_gamma.rvs(size=10000, random_state=rng)
_, pvalue = stats.ks_1samp(res[1], inv_gamma.cdf)
assert pvalue > 0.1
def test_pdf_logpdf(self):
# Check that PDF and log-PDF are consistent
rng = np.random.default_rng(8925849245)
mu, lmbda, a, b = rng.random((4, 20)) - 0.25 # make some invalid
x, s2 = rng.random(size=(2, 20)) - 0.25
res = stats.normal_inverse_gamma(mu, lmbda, a, b).pdf(x, s2)
ref = stats.normal_inverse_gamma.logpdf(x, s2, mu, lmbda, a, b)
assert_allclose(res, np.exp(ref))
def test_invalid_and_special_cases(self):
# Test cases that are handled by input validation rather than the formulas
rng = np.random.default_rng(8925849245)
mu, lmbda, a, b = rng.random(4)
x, s2 = rng.random(2)
res = stats.normal_inverse_gamma(np.nan, lmbda, a, b).pdf(x, s2)
assert_equal(res, np.nan)
res = stats.normal_inverse_gamma(mu, -1, a, b).pdf(x, s2)
assert_equal(res, np.nan)
res = stats.normal_inverse_gamma(mu, lmbda, 0, b).pdf(x, s2)
assert_equal(res, np.nan)
res = stats.normal_inverse_gamma(mu, lmbda, a, -1).pdf(x, s2)
assert_equal(res, np.nan)
res = stats.normal_inverse_gamma(mu, lmbda, a, b).pdf(x, -1)
assert_equal(res, 0)
# PDF with out-of-support s2 is not zero if shape parameter is invalid
res = stats.normal_inverse_gamma(mu, [-1, np.nan], a, b).pdf(x, -1)
assert_equal(res, np.nan)
res = stats.normal_inverse_gamma(mu, -1, a, b).mean()
assert_equal(res, (np.nan, np.nan))
res = stats.normal_inverse_gamma(mu, lmbda, -1, b).var()
assert_equal(res, (np.nan, np.nan))
with pytest.raises(ValueError, match="Domain error in arguments..."):
stats.normal_inverse_gamma(mu, lmbda, a, -1).rvs()
def test_broadcasting(self):
# Test methods with broadcastable array parameters. Roughly speaking, the
# shapes should be the broadcasted shapes of all arguments, and the raveled
# outputs should be the same as the outputs with raveled inputs.
rng = np.random.default_rng(8925849245)
b = rng.random(2)
a = rng.random((3, 1)) + 2 # for defined moments
lmbda = rng.random((4, 1, 1))
mu = rng.random((5, 1, 1, 1))
s2 = rng.random((6, 1, 1, 1, 1))
x = rng.random((7, 1, 1, 1, 1, 1))
dist = stats.normal_inverse_gamma(mu, lmbda, a, b)
# Test PDF and log-PDF
broadcasted = np.broadcast_arrays(x, s2, mu, lmbda, a, b)
broadcasted_raveled = [np.ravel(arr) for arr in broadcasted]
res = dist.pdf(x, s2)
assert res.shape == broadcasted[0].shape
assert_allclose(res.ravel(),
stats.normal_inverse_gamma.pdf(*broadcasted_raveled))
res = dist.logpdf(x, s2)
assert res.shape == broadcasted[0].shape
assert_allclose(res.ravel(),
stats.normal_inverse_gamma.logpdf(*broadcasted_raveled))
# Test moments
broadcasted = np.broadcast_arrays(mu, lmbda, a, b)
broadcasted_raveled = [np.ravel(arr) for arr in broadcasted]
res = dist.mean()
assert res[0].shape == broadcasted[0].shape
assert_allclose((res[0].ravel(), res[1].ravel()),
stats.normal_inverse_gamma.mean(*broadcasted_raveled))
res = dist.var()
assert res[0].shape == broadcasted[0].shape
assert_allclose((res[0].ravel(), res[1].ravel()),
stats.normal_inverse_gamma.var(*broadcasted_raveled))
# Test RVS
size = (6, 5, 4, 3, 2)
rng = np.random.default_rng(2348923985324)
res = dist.rvs(size=size, random_state=rng)
rng = np.random.default_rng(2348923985324)
shape = 6, 5*4*3*2
ref = stats.normal_inverse_gamma.rvs(*broadcasted_raveled, size=shape,
random_state=rng)
assert_allclose((res[0].reshape(shape), res[1].reshape(shape)), ref)
@pytest.mark.slow
@pytest.mark.fail_slow(10)
def test_moments(self):
# Test moments against quadrature
rng = np.random.default_rng(8925849245)
mu, lmbda, a, b = rng.random(4)
a += 2 # ensure defined
dist = stats.normal_inverse_gamma(mu, lmbda, a, b)
res = dist.mean()
ref = dblquad(lambda s2, x: dist.pdf(x, s2) * x, -np.inf, np.inf, 0, np.inf)
assert_allclose(res[0], ref[0], rtol=1e-6)
ref = dblquad(lambda s2, x: dist.pdf(x, s2) * s2, -np.inf, np.inf, 0, np.inf)
assert_allclose(res[1], ref[0], rtol=1e-6)
@pytest.mark.parametrize('dtype', [np.int32, np.float16, np.float32, np.float64])
def test_dtype(self, dtype):
if np.__version__ < "2":
pytest.skip("Scalar dtypes only respected after NEP 50.")
rng = np.random.default_rng(8925849245)
x, s2, mu, lmbda, a, b = rng.uniform(3, 10, size=6).astype(dtype)
dtype_out = np.result_type(1.0, dtype)
dist = stats.normal_inverse_gamma(mu, lmbda, a, b)
assert dist.rvs()[0].dtype == dtype_out
assert dist.rvs()[1].dtype == dtype_out
assert dist.mean()[0].dtype == dtype_out
assert dist.mean()[1].dtype == dtype_out
assert dist.var()[0].dtype == dtype_out
assert dist.var()[1].dtype == dtype_out
assert dist.logpdf(x, s2).dtype == dtype_out
assert dist.pdf(x, s2).dtype == dtype_out
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