def test_update_mean_cov_L_lmbda_converges_to_weighted_mean_and_cov():
N_init = 10
N = 10000
D = 2
X = np.random.randn(N, D)
weights = np.random.rand(N)
old_mean = np.average(X[:N_init], axis=0, weights=weights[:N_init])
old_cov_L = np.linalg.cholesky(np.cov(X[:N_init].T, ddof=0))
sum_old_weights = np.sum(weights[:N_init])
lmbdas = weights_to_lmbdas(sum_old_weights, weights[N_init:])
mean, cov_L = update_mean_cov_L_lmbda(X[N_init:], old_mean, old_cov_L, lmbdas)
full_mean = np.average(X, axis=0, weights=weights)
# the above method uses N rather than N-1 to normalise covariance (biased)
try:
full_cov = np.cov(X.T, ddof=0, aweights=weights)
except TypeError:
raise SkipTest("Numpy's cov method does not support aweights keyword.")
cov = np.dot(cov_L, cov_L.T)
assert_allclose(full_mean, mean)
assert_allclose(full_cov, cov, atol=1e-2)
def test_update_mean_cov_L_lmbda_converges_to_weighted_mean_and_cov():
N_init = 10
N = 10000
D = 2
X = np.random.randn(N, D)
weights = np.random.rand(N)
old_mean = np.average(X[:N_init], axis=0, weights=weights[:N_init])
old_cov_L = np.linalg.cholesky(np.cov(X[:N_init].T, ddof=0))
sum_old_weights = np.sum(weights[:N_init])
lmbdas = weights_to_lmbdas(sum_old_weights, weights[N_init:])
mean, cov_L = update_mean_cov_L_lmbda(X[N_init:], old_mean, old_cov_L, lmbdas)
full_mean = np.average(X, axis=0, weights=weights)
# the above method uses N rather than N-1 to normalise covariance (biased)
try:
full_cov = np.cov(X.T, ddof=0, aweights=weights)
except TypeError:
raise SkipTest("Numpy's cov method does not support aweights keyword.")
cov = np.dot(cov_L, cov_L.T)
assert_allclose(full_mean, mean, atol=1e-2)
assert_allclose(full_cov, cov, atol=1e-2)
def test_weights_to_lmbdas_single():
N = 1
weights = np.ones(N)
sum_old_weights = np.sum(weights)
lmbdas = weights_to_lmbdas(sum_old_weights, weights)
# if there are only two elements, the mean is produced by using lmbda = 0.5
desired = np.array([0.5])
assert_allclose(lmbdas, desired)
def test_weights_to_lmbdas_single():
N = 1
weights = np.ones(N)
sum_old_weights = np.sum(weights)
lmbdas = weights_to_lmbdas(sum_old_weights, weights)
# if there are only two elements, the mean is produced by using lmbda = 0.5
desired = np.array([0.5])
assert_allclose(lmbdas, desired)
def test_weights_to_lmbdas_equals_log_version():
N = 30
sum_old_weights = 1
new_weights = np.ones(N - 1)
lmbdas = weights_to_lmbdas(sum_old_weights, new_weights)
log_sum_old_weights = np.log(sum_old_weights)
log_new_weights = np.log(new_weights)
lmbdas2 = log_weights_to_lmbdas(log_sum_old_weights, log_new_weights)
assert_allclose(lmbdas, lmbdas2)
def test_weights_to_lmbdas_equals_log_version():
N = 30
sum_old_weights = 1
new_weights = np.ones(N - 1)
lmbdas = weights_to_lmbdas(sum_old_weights, new_weights)
log_sum_old_weights = np.log(sum_old_weights)
log_new_weights = np.log(new_weights)
lmbdas2 = log_weights_to_lmbdas(log_sum_old_weights, log_new_weights)
assert_allclose(lmbdas, lmbdas2)
def test_weights_to_lmbdas_produces_mean_weighted():
N = 20
X = np.random.randn(N)
weights = np.random.rand(N)
sum_old_weights = weights[0]
lmbdas = weights_to_lmbdas(sum_old_weights, weights[1:])
old_mean = X[0]
full_mean = update_mean_lmbda(X[1:], old_mean, lmbdas)
X_weighted = np.array([X[i] * weights[i] for i in range(N)])
full_mean_batch = np.sum(X_weighted, axis=0) / np.sum(weights)
assert_allclose(full_mean_batch, full_mean)
def test_weights_to_lmbdas_produces_mean_weighted():
N = 20
X = np.random.randn(N)
weights = np.random.rand(N)
sum_old_weights = weights[0]
lmbdas = weights_to_lmbdas(sum_old_weights, weights[1:])
old_mean = X[0]
full_mean = update_mean_lmbda(X[1:], old_mean, lmbdas)
X_weighted = np.array([X[i] * weights[i] for i in range(N)])
full_mean_batch = np.sum(X_weighted, axis=0) / np.sum(weights)
assert_allclose(full_mean_batch, full_mean)
def test_weights_to_lmbdas_produces_mean():
N = 30
D = 2
X = np.random.randn(N, D)
full_mean_batch = np.mean(X, axis=0)
sum_old_weights = 1
new_weights = np.ones(N - 1)
lmbdas = weights_to_lmbdas(sum_old_weights, new_weights)
old_mean = X[0]
full_mean = update_mean_lmbda(X[1:], old_mean, lmbdas)
assert_allclose(full_mean_batch, full_mean)
def test_weights_to_lmbdas_produces_mean():
N = 30
D = 2
X = np.random.randn(N, D)
full_mean_batch = np.mean(X, axis=0)
sum_old_weights = 1
new_weights = np.ones(N - 1)
lmbdas = weights_to_lmbdas(sum_old_weights, new_weights)
old_mean = X[0]
full_mean = update_mean_lmbda(X[1:], old_mean, lmbdas)
assert_allclose(full_mean_batch, full_mean)
def test_update_mean_cov_L_lmbda_converges_to_mean_and_cov():
N_init = 10
N = 10000
D = 2
X = np.random.randn(N, D)
weights = np.ones(N)
old_mean = np.mean(X[:N_init], axis=0)
old_cov_L = np.linalg.cholesky(np.cov(X[:N_init].T, ddof=0))
sum_old_weights = np.sum(weights[:N_init])
lmbdas = weights_to_lmbdas(sum_old_weights, weights[N_init:])
mean, cov_L = update_mean_cov_L_lmbda(X[N_init:], old_mean, old_cov_L, lmbdas)
full_mean = np.mean(X, axis=0)
# the above method uses N rather than N-1 to normalise covariance (biased)
full_cov = np.cov(X.T, ddof=0)
cov = np.dot(cov_L, cov_L.T)
assert_allclose(full_mean, mean)
assert_allclose(full_cov, cov, atol=1e-3)
def test_update_mean_cov_L_lmbda_converges_to_mean_and_cov():
N_init = 10
N = 10000
D = 2
X = np.random.randn(N, D)
weights = np.ones(N)
old_mean = np.mean(X[:N_init], axis=0)
old_cov_L = np.linalg.cholesky(np.cov(X[:N_init].T, ddof=0))
sum_old_weights = np.sum(weights[:N_init])
lmbdas = weights_to_lmbdas(sum_old_weights, weights[N_init:])
mean, cov_L = update_mean_cov_L_lmbda(X[N_init:], old_mean, old_cov_L, lmbdas)
full_mean = np.mean(X, axis=0)
# the above method uses N rather than N-1 to normalise covariance (biased)
full_cov = np.cov(X.T, ddof=0)
cov = np.dot(cov_L, cov_L.T)
assert_allclose(full_mean, mean)
assert_allclose(full_cov, cov, atol=1e-3)
