def positive(self, init=None, shape=(), dtype=None, name=None):
def generate_init(shape, dtype):
return B.rand(dtype, *shape)
return self._get_var(transform=lambda x: B.exp(x),
inverse_transform=lambda x: B.log(x),
init=init,
generate_init=generate_init,
shape=shape,
dtype=dtype,
name=name)
def normalise_logdet(self, y):
"""Compute the log-determinant of the Jacobian of the normalisation.
Accepts multiple arguments.
Args:
y (matrix): Data that was transformed.
Returns:
scalar: Log-determinant.
"""
return -B.shape(y)[0] * B.sum(B.log(self.scale))
def normal1d_logpdf(x, var, mean=0):
"""Broadcast the one-dimensional normal logpdf.
Args:
x (tensor): Point to evaluate at.
var (tensor): Variances.
mean (tensor): Means.
Returns:
tensor: Logpdf.
"""
return -(B.log_2_pi + B.log(var) + (x - mean)**2 / var) / 2
def _project_pattern(self, x, y, pattern):
# Check whether all data is available.
no_missing = all(pattern)
if no_missing:
# All data is available. Nothing to be done.
u = self.u
else:
# Data is missing. Pick the available entries.
y = B.take(y, pattern, axis=1)
# Ensure that `u` remains a structured matrix.
u = Dense(B.take(self.u, pattern))
# Get number of data points and outputs in this part of the data.
n = B.shape(x)[0]
p = sum(pattern)
# Perform projection.
proj_y_partial = B.matmul(y, B.pinv(u), tr_b=True)
proj_y = B.matmul(proj_y_partial, B.inv(self.s_sqrt), tr_b=True)
# Compute projected noise.
u_square = B.matmul(u, u, tr_a=True)
proj_noise = (
self.noise_obs / B.diag(self.s_sqrt) ** 2 * B.diag(B.pd_inv(u_square))
)
# Convert projected noise to weights.
noises = self.model.noises
weights = noises / (noises + proj_noise)
proj_w = B.ones(B.dtype(weights), n, self.m) * weights[None, :]
# Compute Frobenius norm.
frob = B.sum(y ** 2)
frob = frob - B.sum(proj_y_partial * B.matmul(proj_y_partial, u_square))
# Compute regularising term.
reg = 0.5 * (
n * (p - self.m) * B.log(2 * B.pi * self.noise_obs)
+ frob / self.noise_obs
+ n * B.logdet(B.matmul(u, u, tr_a=True))
+ n * 2 * B.logdet(self.s_sqrt)
)
return x, proj_y, proj_w, reg
def positive(self, init=None, shape=None, dtype=None, name=None):
# If nothing is specific, generate a scalar.
if init is None and shape is None:
shape = ()
def generate_init(shape, dtype):
return B.rand(dtype, *shape)
return self._get_var(
transform=lambda x: B.exp(x),
inverse_transform=lambda x: B.log(x),
init=init,
generate_init=generate_init,
shape=shape,
shape_latent=shape,
dtype=dtype,
name=name,
)
def apply_to_psd(f):
raw = 10**(psds / 10)
return 10 * B.log(f(raw)) / B.log(10)
pred_k = wd.load("pred_k.pickle")
# Unpack prediction for the PDF and cut off a frequency 0.5.
freqs, mean, lower, upper, samps = pred_psd
upper_freq = 0.5
samps = samps[freqs <= upper_freq, :]
mean = mean[freqs <= upper_freq]
lower = lower[freqs <= upper_freq]
upper = upper[freqs <= upper_freq]
freqs = freqs[freqs <= upper_freq]
# Compute the spectrum of the excitation process.
instance = model()
spec_x = (2 * instance.lam) / (instance.lam**2 + (2 * B.pi * freqs)**2)
spec_x *= instance.alpha_t**2 / (2 * instance.alpha)
spec_x = 10 * B.log(spec_x) / B.log(10)
plt.figure(figsize=(12, 3.5))
# Plot the prediction for the PSD.
plt.subplot(1, 2, 1)
plt.title("PSD")
plt.plot(
freqs,
mean,
label="$|\\mathcal{F}h(f)|^2\\mathcal{F}k_{x}(f)$",
style="pred",
zorder=1,
)
plt.fill_between(freqs, lower, upper, style="pred", zorder=1)
plt.plot(freqs, lower, style="pred", lw=1, zorder=1)
posterior = model.condition(t_train, y_train)
pred_f = (t_pred, ) + normaliser.untransform(posterior.predict(t_pred))
pred_f_test = (t_test, ) + normaliser.untransform(
posterior.predict(t_test))
pred_k = posterior.predict_kernel()
# Carefully untransform kernel prediction.
pred_k = (
pred_k.x,
pred_k.mean * normaliser._scale,
pred_k.var * normaliser._scale**2,
)
pred_psd = posterior.predict_psd()
# Carefully untransform PSD prediction.
pred_psd = (
pred_psd.x,
pred_psd.mean + 20 * B.log(normaliser._scale),
pred_psd.err_95_lower + 20 * B.log(normaliser._scale),
pred_psd.err_95_upper + 20 * B.log(normaliser._scale),
)
# Save predictions.
wd.save(pred_f, model.name.lower(), "pred_f.pickle")
wd.save(pred_f_test, model.name.lower(), "pred_f_test.pickle")
wd.save(pred_k, model.name.lower(), "pred_k.pickle")
wd.save(pred_psd, model.name.lower(), "pred_psd.pickle")
# Load predictions.
preds_f = {}
preds_f_test = {}
preds_k = {}
preds_psd = {}
for model in models:
def logdet(a: Union[Diagonal, LowerTriangular, UpperTriangular]):
return B.sum(B.log(B.diag(a)))
def inverse_transform(x):
return B.log(upper - x) - B.log(x - lower)
def inverse_transform(x):
chol = B.cholesky(B.reg(x))
return B.concat(B.log(B.diag(chol)), B.tril_to_vec(chol,
offset=-1))