def test_lstsq_with_constraints_is_solution_for_matrices():
for m, n, k in itertools.permutations([10, 20, 30]):
# Solution
A = np.random.randint(-100, 101, size=(m, n))
X = np.random.randint(-100, 101, size=(n, k))
Y = np.matmul(A, X)
# Constraints
constraints = np.empty((m, n))
constraints[:] = np.nan
size = 5
i = np.random.randint(m - size + 1)
j = np.random.randint(n - size + 1)
constraints[i:i + size, j:j + size] = A[i:i + size, j:j + size]
# Result
r = lstsq(X.T, Y.T, constraints).T
# Test result is a solution
assert np.allclose(np.matmul(r, X), Y)
# Test result has constrained elements
assert np.allclose(r[i:i + size, j:j + size], constraints[i:i + size,
j:j + size])
def test_lstsq_none_and_nan_constraints_coincide_for_matrices():
for m, n, k in itertools.permutations([10, 20, 30]):
# Solution
A = np.random.randint(-100, 101, size=(m, n))
X = np.random.randint(-100, 101, size=(n, k))
Y = np.matmul(A, X)
# Constraints
constraints = np.empty((m, n))
constraints[:] = np.nan
# Results
results = [lstsq(X.T, Y.T).T, lstsq(X.T, Y.T, constraints).T]
# Test they are all close to each other
assert np.alltrue([
np.allclose(ri, rj)
for ri, rj in itertools.combinations(results, r=2)
])
def test_lstsq_none_and_nan_constraints_coincide_for_vectors():
for m, n in itertools.permutations([10, 20]):
# Solution
x = np.random.randint(-100, 101, size=n)
A = np.random.randint(-100, 101, size=(m, n))
b = np.matmul(A, x)
# Constraints
constraints = np.empty(n)
constraints[:] = np.nan
# Results
results = [lstsq(A, b), lstsq(A, b, constraints)]
# Test they are all close to each other
assert np.alltrue([
np.allclose(ri, rj)
for ri, rj in itertools.combinations(results, r=2)
])
def test_lstsq_none_constraints_is_solution_for_vectors():
for m, n in itertools.permutations([10, 20]):
# Solution
x = np.random.randint(-100, 101, size=n)
A = np.random.randint(-100, 101, size=(m, n))
b = np.matmul(A, x)
# Result
r = lstsq(A, b)
# Test result is a solution
assert np.allclose(np.matmul(A, r), b)
def test_lstsq_none_constraints_is_solution_for_matrices():
for m, n, k in itertools.permutations([10, 20, 30]):
# Solution
A = np.random.randint(-100, 101, size=(m, n))
X = np.random.randint(-100, 101, size=(n, k))
Y = np.matmul(A, X)
# Result
r = lstsq(X.T, Y.T).T
# Test result is a solution
assert np.allclose(np.matmul(r, X), Y)
def test_lstsq_ransac_is_robust_solution():
k = 50
size = 10
# k random points in the square [-size/2, size/2) x [-size/2, size/2)
# Each column is a 2D point
X = size * np.random.random(size=(2, k)) - size / 2
# X plus a last row of ones
X = np.concatenate((X, np.ones((1, X.shape[1]))))
sx = 2
sy = 3
tx = -4
ty = 1
# Unobservable 2D affine transformation
A = np.array([[sx, 0, tx], [0, sy, ty]])
# Transformed points
Y = np.matmul(A, X)
p = 0.5 # Ratio of inliers
r = int(p * k) # Number of inliers
# Last k - r points are not from y = A*x model but random
Y[:, r:] = size * np.random.random(size=(Y.shape[0], k - r)) - size / 2
constraints = np.empty((2, 3))
constraints[:] = np.nan
constraints[0, 1] = constraints[1, 0] = 0
# This yields a bad model!
x1 = lstsq(X.T, Y.T, constraints).T
assert not np.allclose(x1, A)
num_iter = 100
sample_size = 2
min_num_inliers = 8
tol = 0.05
constraints = np.empty((2, 3))
constraints[:] = np.nan
constraints[0, 1] = constraints[1, 0] = 0
# This yields a good model
x2 = lstsq_ransac(X.T, Y.T, num_iter, sample_size, min_num_inliers, tol,
constraints).T
assert np.allclose(x2, A)
def test_lstsq_with_constraints_is_solution_for_vectors():
for m, n in itertools.permutations([10, 20]):
# Solution
x = np.random.randint(-100, 101, size=n)
A = np.random.randint(-100, 101, size=(m, n))
b = np.matmul(A, x)
# Constraints
constraints = np.empty(n)
constraints[:] = np.nan
size = 5
i = np.random.randint(n - size + 1)
constraints[i:i + size] = x[i:i + size]
# Result
r = lstsq(A, b, constraints)
# Test result is a solution
assert np.allclose(np.matmul(A, r), b)
# Test result has constrained elements
assert np.allclose(r[i:i + size], constraints[i:i + size])