def objective(vs, m, x_data, y_data, locs):
"""NLML objective.
Args:
vs (:class:`varz.Vars`): Variable container.
m (int): Number of latent processes.
x_data (tensor): Time stamps of the observations.
y_data (tensor): Observations.
locs (tensor): Spatial locations of observations.
Returns:
scalar: Negative log-marginal likelihood.
"""
y_proj, _, S, noises_obs = project(vs, m, y_data, locs)
xs, noise_obs, noises_latent = model(vs, m)
# Add contribution of latent processes.
lml = 0
for i, (x, y) in enumerate(zip(xs, y_proj)):
e_signal = GP((noise_obs / S[i] + noises_latent[i]) * Delta(),
graph=x.graph)
lml += (x + e_signal)(x_data).logpdf(y)
e_noise = GP(noise_obs / S[i] * Delta(), graph=x.graph)
lml -= e_noise(x_data).logpdf(y)
# Add regularisation contribution.
lml += B.sum(Normal(Diagonal(noises_obs)).logpdf(B.transpose(y_data)))
# Return negative the evidence, normalised by the number of data points.
n, p = B.shape(y_data)
return -lml / (n * p)
def project(vs, m, y_data, locs):
"""Project the data.
Args:
vs (:class:`varz.Vars`): Variable container.
m (int): Number of latent processes.
y_data (tensor): Observations.
locs (tensor): Spatial locations of observations.
Returns:
tuple: Tuple containing the projected outputs, the mixing matrix,
S from the mixing matrix, and the observation noises.
"""
_, noise_obs, noises_latent = model(vs, m)
# Construct mixing matrix and projection.
scales = vs.bnd(B.ones(2), name='scales')
K = dense(Matern52().stretch(scales)(locs))
U, S, _ = B.svd(K)
S = S[:m]
H = U[:, :m] * S[None, :]**.5
T = B.transpose(U[:, :m]) / S[:, None]**.5
# Project data and unstack over latent processes.
y_proj = B.unstack(B.matmul(T, y_data, tr_b=True))
# Observation noises:
noises_obs = noise_obs * B.ones(B.dtype(noise_obs), B.shape(y_data)[1])
return y_proj, H, S, noises_obs