EVOLUTION-MANAGER

Edit File: test_procrustes.py

from itertools import product, permutations import numpy as np from numpy.testing import assert_array_less, assert_allclose from pytest import raises as assert_raises from scipy.linalg import inv, eigh, norm from scipy.linalg import orthogonal_procrustes from scipy.sparse.sputils import matrix def test_orthogonal_procrustes_ndim_too_large(): np.random.seed(1234) A = np.random.randn(3, 4, 5) B = np.random.randn(3, 4, 5) assert_raises(ValueError, orthogonal_procrustes, A, B) def test_orthogonal_procrustes_ndim_too_small(): np.random.seed(1234) A = np.random.randn(3) B = np.random.randn(3) assert_raises(ValueError, orthogonal_procrustes, A, B) def test_orthogonal_procrustes_shape_mismatch(): np.random.seed(1234) shapes = ((3, 3), (3, 4), (4, 3), (4, 4)) for a, b in permutations(shapes, 2): A = np.random.randn(*a) B = np.random.randn(*b) assert_raises(ValueError, orthogonal_procrustes, A, B) def test_orthogonal_procrustes_checkfinite_exception(): np.random.seed(1234) m, n = 2, 3 A_good = np.random.randn(m, n) B_good = np.random.randn(m, n) for bad_value in np.inf, -np.inf, np.nan: A_bad = A_good.copy() A_bad[1, 2] = bad_value B_bad = B_good.copy() B_bad[1, 2] = bad_value for A, B in ((A_good, B_bad), (A_bad, B_good), (A_bad, B_bad)): assert_raises(ValueError, orthogonal_procrustes, A, B) def test_orthogonal_procrustes_scale_invariance(): np.random.seed(1234) m, n = 4, 3 for i in range(3): A_orig = np.random.randn(m, n) B_orig = np.random.randn(m, n) R_orig, s = orthogonal_procrustes(A_orig, B_orig) for A_scale in np.square(np.random.randn(3)): for B_scale in np.square(np.random.randn(3)): R, s = orthogonal_procrustes(A_orig * A_scale, B_orig * B_scale) assert_allclose(R, R_orig) def test_orthogonal_procrustes_array_conversion(): np.random.seed(1234) for m, n in ((6, 4), (4, 4), (4, 6)): A_arr = np.random.randn(m, n) B_arr = np.random.randn(m, n) As = (A_arr, A_arr.tolist(), matrix(A_arr)) Bs = (B_arr, B_arr.tolist(), matrix(B_arr)) R_arr, s = orthogonal_procrustes(A_arr, B_arr) AR_arr = A_arr.dot(R_arr) for A, B in product(As, Bs): R, s = orthogonal_procrustes(A, B) AR = A_arr.dot(R) assert_allclose(AR, AR_arr) def test_orthogonal_procrustes(): np.random.seed(1234) for m, n in ((6, 4), (4, 4), (4, 6)): # Sample a random target matrix. B = np.random.randn(m, n) # Sample a random orthogonal matrix # by computing eigh of a sampled symmetric matrix. X = np.random.randn(n, n) w, V = eigh(X.T + X) assert_allclose(inv(V), V.T) # Compute a matrix with a known orthogonal transformation that gives B. A = np.dot(B, V.T) # Check that an orthogonal transformation from A to B can be recovered. R, s = orthogonal_procrustes(A, B) assert_allclose(inv(R), R.T) assert_allclose(A.dot(R), B) # Create a perturbed input matrix. A_perturbed = A + 1e-2 * np.random.randn(m, n) # Check that the orthogonal procrustes function can find an orthogonal # transformation that is better than the orthogonal transformation # computed from the original input matrix. R_prime, s = orthogonal_procrustes(A_perturbed, B) assert_allclose(inv(R_prime), R_prime.T) # Compute the naive and optimal transformations of the perturbed input. naive_approx = A_perturbed.dot(R) optim_approx = A_perturbed.dot(R_prime) # Compute the Frobenius norm errors of the matrix approximations. naive_approx_error = norm(naive_approx - B, ord='fro') optim_approx_error = norm(optim_approx - B, ord='fro') # Check that the orthogonal Procrustes approximation is better. assert_array_less(optim_approx_error, naive_approx_error) def _centered(A): mu = A.mean(axis=0) return A - mu, mu def test_orthogonal_procrustes_exact_example(): # Check a small application. # It uses translation, scaling, reflection, and rotation. # # | # a b | # | # d c | w # | # --------+--- x ----- z --- # | # | y # | # A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float) B_orig = np.array([[3, 2], [1, 0], [3, -2], [5, 0]], dtype=float) A, A_mu = _centered(A_orig) B, B_mu = _centered(B_orig) R, s = orthogonal_procrustes(A, B) scale = s / np.square(norm(A)) B_approx = scale * np.dot(A, R) + B_mu assert_allclose(B_approx, B_orig, atol=1e-8) def test_orthogonal_procrustes_stretched_example(): # Try again with a target with a stretched y axis. A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float) B_orig = np.array([[3, 40], [1, 0], [3, -40], [5, 0]], dtype=float) A, A_mu = _centered(A_orig) B, B_mu = _centered(B_orig) R, s = orthogonal_procrustes(A, B) scale = s / np.square(norm(A)) B_approx = scale * np.dot(A, R) + B_mu expected = np.array([[3, 21], [-18, 0], [3, -21], [24, 0]], dtype=float) assert_allclose(B_approx, expected, atol=1e-8) # Check disparity symmetry. expected_disparity = 0.4501246882793018 AB_disparity = np.square(norm(B_approx - B_orig) / norm(B)) assert_allclose(AB_disparity, expected_disparity) R, s = orthogonal_procrustes(B, A) scale = s / np.square(norm(B)) A_approx = scale * np.dot(B, R) + A_mu BA_disparity = np.square(norm(A_approx - A_orig) / norm(A)) assert_allclose(BA_disparity, expected_disparity) def test_orthogonal_procrustes_skbio_example(): # This transformation is also exact. # It uses translation, scaling, and reflection. # # | # | a # | b # | c d # --+--------- # | # | w # | # | x # | # | z y # | # A_orig = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], dtype=float) B_orig = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], dtype=float) B_standardized = np.array([ [-0.13363062, 0.6681531], [-0.13363062, 0.13363062], [-0.13363062, -0.40089186], [0.40089186, -0.40089186]]) A, A_mu = _centered(A_orig) B, B_mu = _centered(B_orig) R, s = orthogonal_procrustes(A, B) scale = s / np.square(norm(A)) B_approx = scale * np.dot(A, R) + B_mu assert_allclose(B_approx, B_orig) assert_allclose(B / norm(B), B_standardized)