def test_prediction(self):
model = BayesSardModel(1,
self.ker_par_1d,
multi_ind=2,
point_str='gh',
point_par={'degree': 5})
xtest = np.linspace(-5, 5, 100)[na, :]
y = fcn(model.points)
f = fcn(xtest)
alpha = np.array([[0, 1, 2]])
mean, var = model.predict(xtest, y, mulind=alpha)
std = np.sqrt(var)
# plot training data, predictive mean and variance
fig_title = 'BSQ model predictions'
fig = plt.figure(fig_title)
xtest = np.squeeze(xtest)
plt.fill_between(xtest,
mean - 2 * std,
mean + 2 * std,
color='0.1',
alpha=0.15)
plt.plot(xtest, mean, color='k', lw=2)
plt.plot(model.points, y, 'ko', ms=8)
# true function values at test points if provided
if f is not None:
plt.plot(xtest, np.squeeze(f), lw=2, ls='--', color='tomato')
plt.show()
def test_weights_ut_5d(self):
model = BayesSardModel(5,
np.array([[1.0, 25, 25, 25, 25, 25]]),
point_str='ut')
alpha = np.hstack((np.zeros(
(5, 1)), np.eye(5), 2 * np.eye(5))).astype(np.int)
par = np.array([[1.0, 25, 25, 25, 25, 25]])
w, wc, wcc, emv, ivar = model.bq_weights(par, alpha)
# self.assertTrue(np.allclose(w, UnscentedTransform.weights(5, beta=0)[0]))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_weights_ut_2d(self):
# UT weights in 2D
par = np.array([[1.0, 1.0, 1]])
alpha = np.array([[0, 1, 0, 2, 0], [0, 0, 1, 0, 2]])
model = BayesSardModel(2,
par,
point_str='ut',
point_par=self.pt_par_ut)
w, wc, wcc, emv, ivar = model.bq_weights(par, alpha)
# UT weights reproduced in 2D?
self.assertTrue(np.allclose(w, UnscentedTransform.weights(2)[0]))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_weights_gh5_1d(self):
# GH-5 weights in 1D
model = BayesSardModel(1,
self.ker_par_1d,
point_str='gh',
point_par={'degree': 5})
alpha = np.array([[0, 1, 2, 3, 4]])
w, wc, wcc, emv, ivar = model.bq_weights(self.ker_par_1d, alpha)
# GH-5 weights in 1D reproduced?
self.assertTrue(
np.allclose(w, GaussHermiteTransform.weights(1, degree=5)))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_weights_gh3_2d(self):
# GH-3 weights in 2D
# there are 6 multivariate polynomials in 2D, UT has only 5 points in 2D
model = BayesSardModel(2,
self.ker_par_2d,
point_str='gh',
point_par={'degree': 3})
alpha = np.array([[0, 1, 0, 1, 2, 0, 1, 2, 2],
[0, 0, 1, 1, 0, 2, 2, 1, 2]])
par = np.array([[1.0, 1, 1]])
w, wc, wcc, emv, ivar = model.bq_weights(par, alpha)
self.assertTrue(np.allclose(w, GaussHermiteTransform.weights(2, 3)))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_weights_sr_1d(self):
# SR weights == UT weights for kappa=0 and alpha=1
model = BayesSardModel(1,
self.ker_par_1d,
point_str='ut',
point_par={
'kappa': 0,
'alpha': 1
})
alpha = np.array([[0, 1, 2]])
w, wc, wcc, emv, ivar = model.bq_weights(self.ker_par_1d, alpha)
# UT weights in 1D reproduced?
self.assertTrue(np.allclose(w[1:],
SphericalRadialTransform.weights(1)))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_exp_x_kxpx(self):
model = BayesSardModel(1,
self.ker_par_1d,
multi_ind=2,
point_str='ut',
point_par=self.pt_par_ut)
mi_1d = np.array([[0, 1, 2]])
par_1d = np.array([[1.0, 1.0]])
data = np.array([[0, 1, -1]], dtype=np.float)
ke = model._exp_x_kxpx(par_1d, mi_1d, data)
ke_true = np.array([[2**(-0.5), 0, 1 / (2 * (2**0.5))],
[
2**(-0.5) * np.exp(-0.25),
np.exp(-0.25) / (2 * (2**0.5)),
3 * np.exp(-0.25) / (4 * 2**0.5)
],
[
2**(-0.5) * np.exp(-0.25),
-np.exp(-0.25) / (2 * (2**0.5)),
3 * np.exp(-0.25) / (4 * 2**0.5)
]])
self.assertTrue(ke.shape == (data.shape[1], mi_1d.shape[1]))
self.assertTrue(np.allclose(ke, ke_true))
def test_mc_poly_verification(self):
dim = 1
alpha = np.array([[0, 1, 2]])
par = np.array([[1.0, 1]])
model = BayesSardModel(1,
par,
multi_ind=2,
point_str='ut',
point_par=self.pt_par_ut)
px = model._exp_x_px(alpha)
xpx = model._exp_x_xpx(alpha)
pxpx = model._exp_x_pxpx(alpha)
kxpx = model._exp_x_kxpx(par, alpha, self.data_1d)
# approximate expectations using cumulative moving average MC
def cma_mc(new_samples, old_avg, old_avg_size, axis=0):
b_size = new_samples.shape[axis]
return (new_samples.sum(axis=axis) +
old_avg_size * old_avg) / (old_avg_size + b_size)
batch_size = 100000
num_iter = 100
px_mc, xpx_mc, pxpx_mc, kxpx_mc = 0, 0, 0, 0
for i in range(num_iter):
# sample from standard Gaussian
x_samples = np.random.multivariate_normal(np.zeros((dim, )),
np.eye(dim),
size=batch_size).T
p = vandermonde(alpha, x_samples) # (N, Q)
k = model.kernel.eval(par, x_samples, self.data_1d,
scaling=False) # (N, M)
px_mc = cma_mc(p, px_mc, i * batch_size, axis=0)
xpx_mc = cma_mc(x_samples[..., na] * p[na, ...],
xpx_mc,
i * batch_size,
axis=1)
pxpx_mc = cma_mc(p[:, na, :] * p[..., na],
pxpx_mc,
i * batch_size,
axis=0)
kxpx_mc = cma_mc(k[..., na] * p[:, na, :],
kxpx_mc,
i * batch_size,
axis=0)
# compare MC approximates with analytic expressions
tol = 5e-3
print('Maximum absolute difference using {:d} samples.'.format(
batch_size * num_iter))
print('px {:.2e}'.format(np.abs(px - px_mc).max()))
print('xpx {:.2e}'.format(np.abs(xpx - xpx_mc).max()))
print('pxpx {:.2e}'.format(np.abs(pxpx - pxpx_mc).max()))
print('kxpx {:.2e}'.format(np.abs(kxpx - kxpx_mc).max()))
self.assertLessEqual(np.abs(px - px_mc).max(), tol)
self.assertLessEqual(np.abs(xpx - xpx_mc).max(), tol)
self.assertLessEqual(np.abs(pxpx - pxpx_mc).max(), tol)
self.assertLessEqual(np.abs(kxpx - kxpx_mc).max(), tol)
def test_weights_ut_1d(self):
# UT weights in 1D
model = BayesSardModel(1,
self.ker_par_1d,
point_str='ut',
point_par=self.pt_par_ut)
alpha = np.array([[0, 1, 2]])
w, wc, wcc, emv, ivar = model.bq_weights(self.ker_par_1d, alpha)
# UT weights in 1D reproduced?
self.assertTrue(np.allclose(w, UnscentedTransform.weights(1)[0]))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
# UT weights in 1D, different kappa and alpha
model = BayesSardModel(1,
self.ker_par_1d,
point_str='ut',
point_par={
'kappa': 2,
'alpha': 1
})
alpha = np.array([[0, 1, 2]])
w, wc, wcc, emv, ivar = model.bq_weights(self.ker_par_1d, alpha)
# UT weights in 1D reproduced?
self.assertTrue(
np.allclose(w,
UnscentedTransform.weights(1, kappa=2, alpha=1)[0]))
self.assertGreaterEqual(emv, 0)
self.assertGreaterEqual(ivar, 0)
# test positive definiteness
try:
la.cholesky(wc)
except la.LinAlgError:
self.fail("Weights not positive definite. Min eigval: {}".format(
la.eigvalsh(wc).min()))
def test_exp_x_xpx(self):
model = BayesSardModel(1,
self.ker_par_1d,
multi_ind=2,
point_str='ut',
point_par=self.pt_par_ut)
mi_1d = np.array([[0, 1, 2]])
ke = model._exp_x_xpx(mi_1d)
self.assertTrue(ke.shape == mi_1d.shape)
self.assertTrue(np.array_equal(ke, np.array([[0, 1, 0]])))
model = BayesSardModel(2,
self.ker_par_2d,
multi_ind=2,
point_str='ut',
point_par=self.pt_par_ut)
mi_2d = np.array([[0, 1, 0, 1, 0, 2], [0, 0, 1, 1, 2, 0]])
ke_true = np.array([[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]])
ke = model._exp_x_xpx(mi_2d)
self.assertTrue(ke.shape == mi_2d.shape)
self.assertTrue(np.array_equal(ke, ke_true))
def test_init(self):
BayesSardModel(1,
self.ker_par_1d,
multi_ind=2,
point_str='ut',
point_par=self.pt_par_ut)