1673 lines
63 KiB
Python
1673 lines
63 KiB
Python
"""
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Test functions for multivariate normal distributions.
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"""
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import pickle
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from numpy.testing import (assert_allclose, assert_almost_equal,
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assert_array_almost_equal, assert_equal,
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assert_array_less, assert_)
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import pytest
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from pytest import raises as assert_raises
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from .test_continuous_basic import check_distribution_rvs
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import numpy
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import numpy as np
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import scipy.linalg
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from scipy.stats._multivariate import _PSD, _lnB, _cho_inv_batch
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from scipy.stats import multivariate_normal
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from scipy.stats import matrix_normal
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from scipy.stats import special_ortho_group, ortho_group
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from scipy.stats import random_correlation
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from scipy.stats import unitary_group
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from scipy.stats import dirichlet, beta
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from scipy.stats import wishart, multinomial, invwishart, chi2, invgamma
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from scipy.stats import norm, uniform
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from scipy.stats import ks_2samp, kstest
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from scipy.stats import binom
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from scipy.integrate import romb
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from scipy.special import multigammaln
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from .common_tests import check_random_state_property
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class TestMultivariateNormal(object):
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def test_input_shape(self):
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mu = np.arange(3)
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cov = np.identity(2)
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assert_raises(ValueError, multivariate_normal.pdf, (0, 1), mu, cov)
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assert_raises(ValueError, multivariate_normal.pdf, (0, 1, 2), mu, cov)
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assert_raises(ValueError, multivariate_normal.cdf, (0, 1), mu, cov)
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assert_raises(ValueError, multivariate_normal.cdf, (0, 1, 2), mu, cov)
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def test_scalar_values(self):
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np.random.seed(1234)
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# When evaluated on scalar data, the pdf should return a scalar
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x, mean, cov = 1.5, 1.7, 2.5
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pdf = multivariate_normal.pdf(x, mean, cov)
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assert_equal(pdf.ndim, 0)
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# When evaluated on a single vector, the pdf should return a scalar
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
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pdf = multivariate_normal.pdf(x, mean, cov)
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assert_equal(pdf.ndim, 0)
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# When evaluated on scalar data, the cdf should return a scalar
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x, mean, cov = 1.5, 1.7, 2.5
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cdf = multivariate_normal.cdf(x, mean, cov)
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assert_equal(cdf.ndim, 0)
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# When evaluated on a single vector, the cdf should return a scalar
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
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cdf = multivariate_normal.cdf(x, mean, cov)
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assert_equal(cdf.ndim, 0)
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def test_logpdf(self):
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# Check that the log of the pdf is in fact the logpdf
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np.random.seed(1234)
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.abs(np.random.randn(5))
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d1 = multivariate_normal.logpdf(x, mean, cov)
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d2 = multivariate_normal.pdf(x, mean, cov)
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assert_allclose(d1, np.log(d2))
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def test_logpdf_default_values(self):
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# Check that the log of the pdf is in fact the logpdf
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# with default parameters Mean=None and cov = 1
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np.random.seed(1234)
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x = np.random.randn(5)
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d1 = multivariate_normal.logpdf(x)
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d2 = multivariate_normal.pdf(x)
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# check whether default values are being used
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d3 = multivariate_normal.logpdf(x, None, 1)
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d4 = multivariate_normal.pdf(x, None, 1)
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assert_allclose(d1, np.log(d2))
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assert_allclose(d3, np.log(d4))
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def test_logcdf(self):
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# Check that the log of the cdf is in fact the logcdf
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np.random.seed(1234)
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.abs(np.random.randn(5))
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d1 = multivariate_normal.logcdf(x, mean, cov)
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d2 = multivariate_normal.cdf(x, mean, cov)
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assert_allclose(d1, np.log(d2))
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def test_logcdf_default_values(self):
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# Check that the log of the cdf is in fact the logcdf
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# with default parameters Mean=None and cov = 1
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np.random.seed(1234)
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x = np.random.randn(5)
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d1 = multivariate_normal.logcdf(x)
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d2 = multivariate_normal.cdf(x)
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# check whether default values are being used
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d3 = multivariate_normal.logcdf(x, None, 1)
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d4 = multivariate_normal.cdf(x, None, 1)
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assert_allclose(d1, np.log(d2))
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assert_allclose(d3, np.log(d4))
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def test_rank(self):
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# Check that the rank is detected correctly.
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np.random.seed(1234)
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n = 4
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mean = np.random.randn(n)
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for expected_rank in range(1, n + 1):
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s = np.random.randn(n, expected_rank)
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cov = np.dot(s, s.T)
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distn = multivariate_normal(mean, cov, allow_singular=True)
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assert_equal(distn.cov_info.rank, expected_rank)
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def test_degenerate_distributions(self):
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def _sample_orthonormal_matrix(n):
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M = np.random.randn(n, n)
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u, s, v = scipy.linalg.svd(M)
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return u
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for n in range(1, 5):
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x = np.random.randn(n)
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for k in range(1, n + 1):
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# Sample a small covariance matrix.
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s = np.random.randn(k, k)
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cov_kk = np.dot(s, s.T)
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# Embed the small covariance matrix into a larger low rank matrix.
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cov_nn = np.zeros((n, n))
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cov_nn[:k, :k] = cov_kk
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# Define a rotation of the larger low rank matrix.
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u = _sample_orthonormal_matrix(n)
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cov_rr = np.dot(u, np.dot(cov_nn, u.T))
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y = np.dot(u, x)
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# Check some identities.
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distn_kk = multivariate_normal(np.zeros(k), cov_kk,
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allow_singular=True)
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distn_nn = multivariate_normal(np.zeros(n), cov_nn,
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allow_singular=True)
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distn_rr = multivariate_normal(np.zeros(n), cov_rr,
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allow_singular=True)
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assert_equal(distn_kk.cov_info.rank, k)
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assert_equal(distn_nn.cov_info.rank, k)
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assert_equal(distn_rr.cov_info.rank, k)
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pdf_kk = distn_kk.pdf(x[:k])
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pdf_nn = distn_nn.pdf(x)
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pdf_rr = distn_rr.pdf(y)
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assert_allclose(pdf_kk, pdf_nn)
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assert_allclose(pdf_kk, pdf_rr)
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logpdf_kk = distn_kk.logpdf(x[:k])
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logpdf_nn = distn_nn.logpdf(x)
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logpdf_rr = distn_rr.logpdf(y)
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assert_allclose(logpdf_kk, logpdf_nn)
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assert_allclose(logpdf_kk, logpdf_rr)
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def test_large_pseudo_determinant(self):
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# Check that large pseudo-determinants are handled appropriately.
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# Construct a singular diagonal covariance matrix
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# whose pseudo determinant overflows double precision.
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large_total_log = 1000.0
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npos = 100
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nzero = 2
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large_entry = np.exp(large_total_log / npos)
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n = npos + nzero
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cov = np.zeros((n, n), dtype=float)
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np.fill_diagonal(cov, large_entry)
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cov[-nzero:, -nzero:] = 0
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# Check some determinants.
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assert_equal(scipy.linalg.det(cov), 0)
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assert_equal(scipy.linalg.det(cov[:npos, :npos]), np.inf)
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assert_allclose(np.linalg.slogdet(cov[:npos, :npos]),
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(1, large_total_log))
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# Check the pseudo-determinant.
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psd = _PSD(cov)
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assert_allclose(psd.log_pdet, large_total_log)
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def test_broadcasting(self):
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np.random.seed(1234)
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n = 4
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# Construct a random covariance matrix.
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data = np.random.randn(n, n)
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cov = np.dot(data, data.T)
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mean = np.random.randn(n)
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# Construct an ndarray which can be interpreted as
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# a 2x3 array whose elements are random data vectors.
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X = np.random.randn(2, 3, n)
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# Check that multiple data points can be evaluated at once.
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desired_pdf = multivariate_normal.pdf(X, mean, cov)
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desired_cdf = multivariate_normal.cdf(X, mean, cov)
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for i in range(2):
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for j in range(3):
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actual = multivariate_normal.pdf(X[i, j], mean, cov)
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assert_allclose(actual, desired_pdf[i,j])
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# Repeat for cdf
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actual = multivariate_normal.cdf(X[i, j], mean, cov)
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assert_allclose(actual, desired_cdf[i,j], rtol=1e-3)
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def test_normal_1D(self):
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# The probability density function for a 1D normal variable should
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# agree with the standard normal distribution in scipy.stats.distributions
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x = np.linspace(0, 2, 10)
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mean, cov = 1.2, 0.9
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scale = cov**0.5
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d1 = norm.pdf(x, mean, scale)
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d2 = multivariate_normal.pdf(x, mean, cov)
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assert_allclose(d1, d2)
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# The same should hold for the cumulative distribution function
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d1 = norm.cdf(x, mean, scale)
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d2 = multivariate_normal.cdf(x, mean, cov)
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assert_allclose(d1, d2)
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def test_marginalization(self):
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# Integrating out one of the variables of a 2D Gaussian should
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# yield a 1D Gaussian
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mean = np.array([2.5, 3.5])
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cov = np.array([[.5, 0.2], [0.2, .6]])
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n = 2 ** 8 + 1 # Number of samples
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delta = 6 / (n - 1) # Grid spacing
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v = np.linspace(0, 6, n)
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xv, yv = np.meshgrid(v, v)
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pos = np.empty((n, n, 2))
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pos[:, :, 0] = xv
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pos[:, :, 1] = yv
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pdf = multivariate_normal.pdf(pos, mean, cov)
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# Marginalize over x and y axis
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margin_x = romb(pdf, delta, axis=0)
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margin_y = romb(pdf, delta, axis=1)
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# Compare with standard normal distribution
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gauss_x = norm.pdf(v, loc=mean[0], scale=cov[0, 0] ** 0.5)
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gauss_y = norm.pdf(v, loc=mean[1], scale=cov[1, 1] ** 0.5)
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assert_allclose(margin_x, gauss_x, rtol=1e-2, atol=1e-2)
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assert_allclose(margin_y, gauss_y, rtol=1e-2, atol=1e-2)
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def test_frozen(self):
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# The frozen distribution should agree with the regular one
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np.random.seed(1234)
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.abs(np.random.randn(5))
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norm_frozen = multivariate_normal(mean, cov)
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assert_allclose(norm_frozen.pdf(x), multivariate_normal.pdf(x, mean, cov))
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assert_allclose(norm_frozen.logpdf(x),
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multivariate_normal.logpdf(x, mean, cov))
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assert_allclose(norm_frozen.cdf(x), multivariate_normal.cdf(x, mean, cov))
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assert_allclose(norm_frozen.logcdf(x),
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multivariate_normal.logcdf(x, mean, cov))
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def test_pseudodet_pinv(self):
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# Make sure that pseudo-inverse and pseudo-det agree on cutoff
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# Assemble random covariance matrix with large and small eigenvalues
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np.random.seed(1234)
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n = 7
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x = np.random.randn(n, n)
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cov = np.dot(x, x.T)
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s, u = scipy.linalg.eigh(cov)
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s = np.full(n, 0.5)
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s[0] = 1.0
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s[-1] = 1e-7
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cov = np.dot(u, np.dot(np.diag(s), u.T))
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# Set cond so that the lowest eigenvalue is below the cutoff
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cond = 1e-5
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psd = _PSD(cov, cond=cond)
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psd_pinv = _PSD(psd.pinv, cond=cond)
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# Check that the log pseudo-determinant agrees with the sum
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# of the logs of all but the smallest eigenvalue
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assert_allclose(psd.log_pdet, np.sum(np.log(s[:-1])))
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# Check that the pseudo-determinant of the pseudo-inverse
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# agrees with 1 / pseudo-determinant
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assert_allclose(-psd.log_pdet, psd_pinv.log_pdet)
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def test_exception_nonsquare_cov(self):
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cov = [[1, 2, 3], [4, 5, 6]]
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assert_raises(ValueError, _PSD, cov)
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def test_exception_nonfinite_cov(self):
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cov_nan = [[1, 0], [0, np.nan]]
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assert_raises(ValueError, _PSD, cov_nan)
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cov_inf = [[1, 0], [0, np.inf]]
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assert_raises(ValueError, _PSD, cov_inf)
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def test_exception_non_psd_cov(self):
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cov = [[1, 0], [0, -1]]
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assert_raises(ValueError, _PSD, cov)
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def test_exception_singular_cov(self):
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np.random.seed(1234)
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x = np.random.randn(5)
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mean = np.random.randn(5)
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cov = np.ones((5, 5))
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e = np.linalg.LinAlgError
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assert_raises(e, multivariate_normal, mean, cov)
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assert_raises(e, multivariate_normal.pdf, x, mean, cov)
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assert_raises(e, multivariate_normal.logpdf, x, mean, cov)
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assert_raises(e, multivariate_normal.cdf, x, mean, cov)
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assert_raises(e, multivariate_normal.logcdf, x, mean, cov)
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def test_R_values(self):
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# Compare the multivariate pdf with some values precomputed
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# in R version 3.0.1 (2013-05-16) on Mac OS X 10.6.
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# The values below were generated by the following R-script:
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# > library(mnormt)
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# > x <- seq(0, 2, length=5)
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# > y <- 3*x - 2
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# > z <- x + cos(y)
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# > mu <- c(1, 3, 2)
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# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
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# > r_pdf <- dmnorm(cbind(x,y,z), mu, Sigma)
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r_pdf = np.array([0.0002214706, 0.0013819953, 0.0049138692,
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0.0103803050, 0.0140250800])
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x = np.linspace(0, 2, 5)
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y = 3 * x - 2
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z = x + np.cos(y)
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r = np.array([x, y, z]).T
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mean = np.array([1, 3, 2], 'd')
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cov = np.array([[1, 2, 0], [2, 5, .5], [0, .5, 3]], 'd')
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pdf = multivariate_normal.pdf(r, mean, cov)
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assert_allclose(pdf, r_pdf, atol=1e-10)
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# Compare the multivariate cdf with some values precomputed
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# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
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# The values below were generated by the following R-script:
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# > library(mnormt)
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# > x <- seq(0, 2, length=5)
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# > y <- 3*x - 2
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# > z <- x + cos(y)
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# > mu <- c(1, 3, 2)
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# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
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# > r_cdf <- pmnorm(cbind(x,y,z), mu, Sigma)
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r_cdf = np.array([0.0017866215, 0.0267142892, 0.0857098761,
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0.1063242573, 0.2501068509])
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cdf = multivariate_normal.cdf(r, mean, cov)
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assert_allclose(cdf, r_cdf, atol=1e-5)
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# Also test bivariate cdf with some values precomputed
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# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
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# The values below were generated by the following R-script:
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# > library(mnormt)
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# > x <- seq(0, 2, length=5)
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# > y <- 3*x - 2
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# > mu <- c(1, 3)
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# > Sigma <- matrix(c(1,2,2,5), 2, 2)
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# > r_cdf2 <- pmnorm(cbind(x,y), mu, Sigma)
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r_cdf2 = np.array([0.01262147, 0.05838989, 0.18389571,
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0.40696599, 0.66470577])
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r2 = np.array([x, y]).T
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mean2 = np.array([1, 3], 'd')
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cov2 = np.array([[1, 2], [2, 5]], 'd')
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cdf2 = multivariate_normal.cdf(r2, mean2, cov2)
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assert_allclose(cdf2, r_cdf2, atol=1e-5)
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def test_multivariate_normal_rvs_zero_covariance(self):
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mean = np.zeros(2)
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covariance = np.zeros((2, 2))
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model = multivariate_normal(mean, covariance, allow_singular=True)
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sample = model.rvs()
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assert_equal(sample, [0, 0])
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def test_rvs_shape(self):
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# Check that rvs parses the mean and covariance correctly, and returns
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# an array of the right shape
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N = 300
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d = 4
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sample = multivariate_normal.rvs(mean=np.zeros(d), cov=1, size=N)
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assert_equal(sample.shape, (N, d))
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sample = multivariate_normal.rvs(mean=None,
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cov=np.array([[2, .1], [.1, 1]]),
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size=N)
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assert_equal(sample.shape, (N, 2))
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u = multivariate_normal(mean=0, cov=1)
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sample = u.rvs(N)
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assert_equal(sample.shape, (N, ))
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def test_large_sample(self):
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# Generate large sample and compare sample mean and sample covariance
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# with mean and covariance matrix.
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np.random.seed(2846)
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n = 3
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mean = np.random.randn(n)
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M = np.random.randn(n, n)
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cov = np.dot(M, M.T)
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size = 5000
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sample = multivariate_normal.rvs(mean, cov, size)
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assert_allclose(numpy.cov(sample.T), cov, rtol=1e-1)
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assert_allclose(sample.mean(0), mean, rtol=1e-1)
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def test_entropy(self):
|
|
np.random.seed(2846)
|
|
|
|
n = 3
|
|
mean = np.random.randn(n)
|
|
M = np.random.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)
|
|
|
|
class TestMatrixNormal(object):
|
|
|
|
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)
|
|
|
|
# Singular covariance
|
|
e = np.linalg.LinAlgError
|
|
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)
|
|
|
|
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)
|
|
|
|
assert_allclose(pdf1, pdf2, rtol=1E-10)
|
|
assert_allclose(logpdf1, logpdf2, rtol=1E-10)
|
|
|
|
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)
|
|
|
|
class TestDirichlet(object):
|
|
|
|
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_and_var(self):
|
|
alpha = np.array([1., 0.8, 0.2])
|
|
d = dirichlet(alpha)
|
|
|
|
expected_var = [1. / 12., 0.08, 0.03]
|
|
expected_mean = [0.5, 0.4, 0.1]
|
|
|
|
assert_array_almost_equal(d.var(), expected_var)
|
|
assert_array_almost_equal(d.mean(), expected_mean)
|
|
|
|
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(object):
|
|
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
|
|
assert_raises(ValueError, wishart, 1, np.eye(2))
|
|
|
|
# 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_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
|
|
np.random.seed(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)
|
|
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)`
|
|
np.random.seed(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)
|
|
args = (df,0,sigma_lamda)
|
|
alpha = 0.01
|
|
check_distribution_rvs('chi2', args, alpha, rvs)
|
|
|
|
class TestMultinomial(object):
|
|
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_allclose(vals2, np.NAN, rtol=1e-8)
|
|
|
|
vals3 = multinomial.logpmf([3, 4], 0, [-2, 3])
|
|
assert_allclose(vals3, 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)
|
|
|
|
def test_rvs_np(self):
|
|
# test that .rvs agrees w/numpy
|
|
sc_rvs = multinomial.rvs(3, [1/4.]*3, size=7, random_state=123)
|
|
rndm = np.random.RandomState(123)
|
|
np_rvs = rndm.multinomial(3, [1/4.]*3, 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)
|
|
|
|
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)
|
|
|
|
def test_cov(self):
|
|
cov1 = multinomial.cov(5, (.2, .3, .5))
|
|
cov2 = [[5*.2*.8, -5*.2*.3, -5*.2*.5],
|
|
[-5*.3*.2, 5*.3*.7, -5*.3*.5],
|
|
[-5*.5*.2, -5*.5*.3, 5*.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)
|
|
|
|
def test_entropy(self):
|
|
# this is equivalent to a binomial distribution with n=2, so the
|
|
# entropy .77899774929 is easily computed "by hand"
|
|
ent0 = multinomial.entropy(2, [.2, .8])
|
|
assert_allclose(ent0, binom.entropy(2, .2), rtol=1e-8)
|
|
|
|
def test_entropy_broadcasting(self):
|
|
ent0 = multinomial.entropy([2, 3], [.2, .3])
|
|
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)
|
|
|
|
def test_mean(self):
|
|
mean1 = multinomial.mean(5, [.2, .8])
|
|
assert_allclose(mean1, [5*.2, 5*.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))
|
|
|
|
class TestInvwishart(object):
|
|
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
|
|
# Kolgomorov-Smirnov test for rvs
|
|
np.random.seed(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)
|
|
args = (df/2, 0, 1./2)
|
|
alpha = 0.01
|
|
check_distribution_rvs('invgamma', args, alpha, rvs)
|
|
|
|
def test_wishart_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 Wishart and inverse Wishart random variables
|
|
w = wishart(df, scale)
|
|
iw = invwishart(df, scale)
|
|
|
|
# Get the generated random variables from a known seed
|
|
np.random.seed(248042)
|
|
w_rvs = wishart.rvs(df, scale)
|
|
np.random.seed(248042)
|
|
frozen_w_rvs = w.rvs()
|
|
np.random.seed(248042)
|
|
iw_rvs = invwishart.rvs(df, scale)
|
|
np.random.seed(248042)
|
|
frozen_iw_rvs = iw.rvs()
|
|
|
|
# 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.
|
|
np.random.seed(248042)
|
|
covariances = np.random.normal(size=3)
|
|
variances = np.r_[
|
|
np.random.chisquare(df),
|
|
np.random.chisquare(df-1),
|
|
np.random.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)
|
|
|
|
# inverse Wishart random variate
|
|
# Supposing that the inverse wishart has scale matrix `scale`, then the
|
|
# random variate is the inverse of a random variate drawn from a Wishart
|
|
# distribution with scale matrix `inv_scale = np.linalg.inv(scale)`
|
|
iD = np.linalg.cholesky(np.linalg.inv(scale))
|
|
iDA = iD.dot(A)
|
|
manual_iw_rvs = np.linalg.inv(np.dot(iDA, iDA.T))
|
|
|
|
# Test for equality
|
|
assert_allclose(w_rvs, manual_w_rvs)
|
|
assert_allclose(frozen_w_rvs, manual_w_rvs)
|
|
assert_allclose(iw_rvs, manual_iw_rvs)
|
|
assert_allclose(frozen_iw_rvs, manual_iw_rvs)
|
|
|
|
def test_cho_inv_batch(self):
|
|
"""Regression test for gh-8844."""
|
|
a0 = np.array([[2, 1, 0, 0.5],
|
|
[1, 2, 0.5, 0.5],
|
|
[0, 0.5, 3, 1],
|
|
[0.5, 0.5, 1, 2]])
|
|
a1 = np.array([[2, -1, 0, 0.5],
|
|
[-1, 2, 0.5, 0.5],
|
|
[0, 0.5, 3, 1],
|
|
[0.5, 0.5, 1, 4]])
|
|
a = np.array([a0, a1])
|
|
ainv = a.copy()
|
|
_cho_inv_batch(ainv)
|
|
ident = np.eye(4)
|
|
assert_allclose(a[0].dot(ainv[0]), ident, atol=1e-15)
|
|
assert_allclose(a[1].dot(ainv[1]), ident, atol=1e-15)
|
|
|
|
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(object):
|
|
def test_reproducibility(self):
|
|
np.random.seed(514)
|
|
x = special_ortho_group.rvs(3)
|
|
expected = np.array([[-0.99394515, -0.04527879, 0.10011432],
|
|
[0.04821555, -0.99846897, 0.02711042],
|
|
[0.09873351, 0.03177334, 0.99460653]])
|
|
assert_array_almost_equal(x, expected)
|
|
|
|
random_state = np.random.RandomState(seed=514)
|
|
x = special_ortho_group.rvs(3, random_state=random_state)
|
|
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
|
|
np.random.seed(514)
|
|
xs = special_ortho_group.rvs(dim, size=samples)
|
|
|
|
# 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 disribution,
|
|
# 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 = dict(((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)
|
|
|
|
class TestOrthoGroup(object):
|
|
def test_reproducibility(self):
|
|
np.random.seed(515)
|
|
x = ortho_group.rvs(3)
|
|
x2 = ortho_group.rvs(3, random_state=515)
|
|
# Note this matrix has det -1, distinguishing O(N) from SO(N)
|
|
assert_almost_equal(np.linalg.det(x), -1)
|
|
expected = np.array([[0.94449759, -0.21678569, -0.24683651],
|
|
[-0.13147569, -0.93800245, 0.3207266],
|
|
[0.30106219, 0.27047251, 0.9144431]])
|
|
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_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 we get both positive and negative determinants
|
|
# Check that we have at least one and less than 10 negative dets in a sample of 10. The rest are positive by the previous test.
|
|
# Test each dimension separately
|
|
assert_array_less([0]*10, [np.nonzero(d < 0)[0].shape[0] for d in dets])
|
|
assert_array_less([np.nonzero(d < 0)[0].shape[0] for d in dets], [10]*10)
|
|
|
|
# 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]))
|
|
|
|
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
|
|
np.random.seed(518) # Note that the test is sensitive to seed too
|
|
xs = ortho_group.rvs(dim, size=samples)
|
|
|
|
# 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 disribution,
|
|
# 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 = dict(((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)
|
|
|
|
@pytest.mark.slow
|
|
def test_pairwise_distances(self):
|
|
# Test that the distribution of pairwise distances is close to correct.
|
|
np.random.seed(514)
|
|
|
|
def random_ortho(dim):
|
|
u, _s, v = np.linalg.svd(np.random.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) - rvs(dim=dim))**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(object):
|
|
def test_reproducibility(self):
|
|
np.random.seed(514)
|
|
eigs = (.5, .8, 1.2, 1.5)
|
|
x = random_correlation.rvs(eigs)
|
|
x2 = random_correlation.rvs(eigs, random_state=514)
|
|
expected = np.array([[1., -0.20387311, 0.18366501, -0.04953711],
|
|
[-0.20387311, 1., -0.24351129, 0.06703474],
|
|
[0.18366501, -0.24351129, 1., 0.38530195],
|
|
[-0.04953711, 0.06703474, 0.38530195, 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_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)
|
|
|
|
np.random.seed(123)
|
|
|
|
eigs = [norm(i, np.random.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) 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 TestUnitaryGroup(object):
|
|
def test_reproducibility(self):
|
|
np.random.seed(514)
|
|
x = unitary_group.rvs(3)
|
|
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_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
|
|
dim = 5
|
|
samples = 1000 # Not too many, or the test takes too long
|
|
np.random.seed(514) # Note that the test is sensitive to seed too
|
|
xs = unitary_group.rvs(dim, size=samples)
|
|
|
|
# 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 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
|
|
|
|
|
|
def test_random_state_property():
|
|
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)
|