class GenericLGSSMWithGPNoiseModel(TimeSeriesModel):
"""
A generic Linear Gaussian State Space Model parameterized with arbitrary time invariant
transition and observation dynamics together with separate Gaussian Process noise models
for each output dimension. In more detail, the generative process is:
:math:`y_i(t) = \\sum_j A_{ij} z_j(t) + f_i(t) + \\epsilon_i(t)`
where the latent variables :math:`{\\bf z}(t)` follow generic time invariant Linear Gaussian dynamics
and the :math:`f_i(t)` are Gaussian Processes with Matern kernels.
The targets are (implicitly) assumed to be evenly spaced in time. In particular a timestep of
:math:`dt=1.0` for the continuous-time GP dynamics corresponds to a single discrete step of
the :math:`{\\bf z}`-space dynamics. Training and inference are logarithmic in the length of
the time series T.
:param int obs_dim: The dimension of the targets at each time step.
:param int state_dim: The dimension of the :math:`{\\bf z}` latent state at each time step.
:param float nu: The order of the Matern kernel; one of 0.5, 1.5 or 2.5.
:param torch.Tensor length_scale_init: optional initial values for the kernel length scale
given as a ``obs_dim``-dimensional tensor
:param torch.Tensor kernel_scale_init: optional initial values for the kernel scale
given as a ``obs_dim``-dimensional tensor
:param torch.Tensor obs_noise_scale_init: optional initial values for the observation noise scale
given as a ``obs_dim``-dimensional tensor
:param bool learnable_observation_loc: whether the mean of the observation model should be learned or not;
defaults to False.
"""
def __init__(self,
obs_dim=1,
state_dim=2,
nu=1.5,
obs_noise_scale_init=None,
length_scale_init=None,
kernel_scale_init=None,
learnable_observation_loc=False):
self.obs_dim = obs_dim
self.state_dim = state_dim
self.nu = nu
if obs_noise_scale_init is None:
obs_noise_scale_init = 0.2 * torch.ones(obs_dim)
assert obs_noise_scale_init.shape == (obs_dim, )
super().__init__()
self.kernel = MaternKernel(nu=nu,
num_gps=obs_dim,
length_scale_init=length_scale_init,
kernel_scale_init=kernel_scale_init)
self.dt = 1.0
self.full_state_dim = self.kernel.state_dim * obs_dim + state_dim
self.full_gp_state_dim = self.kernel.state_dim * obs_dim
self.obs_noise_scale = PyroParam(obs_noise_scale_init,
constraint=constraints.positive)
self.trans_noise_scale_sq = PyroParam(torch.ones(state_dim),
constraint=constraints.positive)
self.z_trans_matrix = nn.Parameter(
torch.eye(state_dim) + 0.03 * torch.randn(state_dim, state_dim))
self.z_obs_matrix = nn.Parameter(0.3 * torch.randn(state_dim, obs_dim))
self.init_noise_scale_sq = PyroParam(torch.ones(state_dim),
constraint=constraints.positive)
gp_obs_matrix = torch.zeros(self.kernel.state_dim * obs_dim, obs_dim)
for i in range(obs_dim):
gp_obs_matrix[self.kernel.state_dim * i, i] = 1.0
self.register_buffer("gp_obs_matrix", gp_obs_matrix)
self.obs_selector = torch.tensor(
[self.kernel.state_dim * d for d in range(obs_dim)],
dtype=torch.long)
if learnable_observation_loc:
self.obs_loc = nn.Parameter(torch.zeros(obs_dim))
else:
self.register_buffer('obs_loc', torch.zeros(obs_dim))
def _get_obs_matrix(self):
# (obs_dim + state_dim, obs_dim) => (gp_state_dim * obs_dim + state_dim, obs_dim)
return torch.cat([self.gp_obs_matrix, self.z_obs_matrix], dim=0)
def _get_init_dist(self):
loc = self.z_trans_matrix.new_zeros(self.full_state_dim)
covar = self.z_trans_matrix.new_zeros(self.full_state_dim,
self.full_state_dim)
covar[:self.full_gp_state_dim, :self.
full_gp_state_dim] = block_diag_embed(
self.kernel.stationary_covariance())
covar[self.full_gp_state_dim:,
self.full_gp_state_dim:] = self.init_noise_scale_sq.diag_embed()
return MultivariateNormal(loc, covar)
def _get_obs_dist(self):
return dist.Normal(self.obs_loc, self.obs_noise_scale).to_event(1)
def _get_dist(self):
"""
Get the :class:`~pyro.distributions.GaussianHMM` distribution that corresponds
to :class:`GenericLGSSMWithGPNoiseModel`.
"""
gp_trans_matrix, gp_process_covar = self.kernel.transition_matrix_and_covariance(
dt=self.dt)
trans_covar = self.z_trans_matrix.new_zeros(self.full_state_dim,
self.full_state_dim)
trans_covar[:self.full_gp_state_dim, :self.
full_gp_state_dim] = block_diag_embed(gp_process_covar)
trans_covar[
self.full_gp_state_dim:,
self.full_gp_state_dim:] = self.trans_noise_scale_sq.diag_embed()
trans_dist = MultivariateNormal(
trans_covar.new_zeros(self.full_state_dim), trans_covar)
full_trans_mat = trans_covar.new_zeros(self.full_state_dim,
self.full_state_dim)
full_trans_mat[:self.full_gp_state_dim, :self.
full_gp_state_dim] = block_diag_embed(gp_trans_matrix)
full_trans_mat[self.full_gp_state_dim:,
self.full_gp_state_dim:] = self.z_trans_matrix
return dist.GaussianHMM(self._get_init_dist(), full_trans_mat,
trans_dist, self._get_obs_matrix(),
self._get_obs_dist())
@pyro_method
def log_prob(self, targets):
"""
:param torch.Tensor targets: A 2-dimensional tensor of real-valued targets
of shape ``(T, obs_dim)``, where ``T`` is the length of the time series and ``obs_dim``
is the dimension of the real-valued ``targets`` at each time step
:returns torch.Tensor: A (scalar) log probability.
"""
assert targets.dim() == 2 and targets.size(-1) == self.obs_dim
return self._get_dist().log_prob(targets)
@torch.no_grad()
def _filter(self, targets):
"""
Return the filtering state for the associated state space model.
"""
assert targets.dim() == 2 and targets.size(-1) == self.obs_dim
return self._get_dist().filter(targets)
@torch.no_grad()
def _forecast(self,
N_timesteps,
filtering_state,
include_observation_noise=True):
"""
Internal helper for forecasting.
"""
dts = torch.arange(N_timesteps,
dtype=self.z_trans_matrix.dtype,
device=self.z_trans_matrix.device) + 1.0
dts = dts.unsqueeze(-1).unsqueeze(-1).unsqueeze(-1)
gp_trans_matrix, gp_process_covar = self.kernel.transition_matrix_and_covariance(
dt=dts)
gp_trans_matrix = block_diag_embed(gp_trans_matrix)
gp_process_covar = block_diag_embed(gp_process_covar[..., 0:1, 0:1])
N_trans_matrix = repeated_matmul(self.z_trans_matrix, N_timesteps)
N_trans_obs = torch.matmul(N_trans_matrix, self.z_obs_matrix)
# z-state contribution + gp contribution
predicted_mean1 = torch.matmul(
filtering_state.loc[-self.state_dim:].unsqueeze(-2),
N_trans_obs).squeeze(-2)
predicted_mean2 = torch.matmul(
filtering_state.loc[:self.full_gp_state_dim].unsqueeze(-2),
gp_trans_matrix[..., self.obs_selector]).squeeze(-2)
predicted_mean = predicted_mean1 + predicted_mean2
# first compute the contributions from filtering_state.covariance_matrix: z-space and gp
fs_cov = filtering_state.covariance_matrix
predicted_covar1z = torch.matmul(N_trans_obs.transpose(-1, -2),
torch.matmul(
fs_cov[self.full_gp_state_dim:,
self.full_gp_state_dim:],
N_trans_obs)) # N O O
gp_trans = gp_trans_matrix[..., self.obs_selector]
predicted_covar1gp = torch.matmul(
gp_trans.transpose(-1, -2),
torch.matmul(
fs_cov[:self.full_gp_state_dim:, :self.full_gp_state_dim],
gp_trans))
# next compute the contribution from process noise that is injected at each timestep.
# (we need to do a cumulative sum to integrate across time for the z-state contribution)
z_process_covar = self.trans_noise_scale_sq.diag_embed()
N_trans_obs_shift = torch.cat(
[self.z_obs_matrix.unsqueeze(0), N_trans_obs[0:-1]])
predicted_covar2z = torch.matmul(N_trans_obs_shift.transpose(
-1, -2), torch.matmul(z_process_covar, N_trans_obs_shift)) # N O O
predicted_covar = predicted_covar1z + predicted_covar1gp + gp_process_covar + \
torch.cumsum(predicted_covar2z, dim=0)
if include_observation_noise:
predicted_covar = predicted_covar + self.obs_noise_scale.pow(
2.0).diag_embed()
return predicted_mean, predicted_covar
@pyro_method
def forecast(self, targets, N_timesteps):
"""
:param torch.Tensor targets: A 2-dimensional tensor of real-valued targets
of shape ``(T, obs_dim)``, where ``T`` is the length of the time series and ``obs_dim``
is the dimension of the real-valued targets at each time step. These
represent the training data that are conditioned on for the purpose of making
forecasts.
:param int N_timesteps: The number of timesteps to forecast into the future from
the final target ``targets[-1]``.
:returns torch.distributions.MultivariateNormal: Returns a predictive MultivariateNormal distribution
with batch shape ``(N_timesteps,)`` and event shape ``(obs_dim,)``
"""
filtering_state = self._filter(targets)
predicted_mean, predicted_covar = self._forecast(
N_timesteps, filtering_state)
return MultivariateNormal(predicted_mean, predicted_covar)
class GenericLGSSM(TimeSeriesModel):
"""
A generic Linear Gaussian State Space Model parameterized with arbitrary time invariant
transition and observation dynamics. The targets are (implicitly) assumed to be evenly
spaced in time. Training and inference are logarithmic in the length of the time series T.
:param int obs_dim: The dimension of the targets at each time step.
:param int state_dim: The dimension of latent state at each time step.
:param bool learnable_observation_loc: whether the mean of the observation model should be learned or not;
defaults to False.
"""
def __init__(self,
obs_dim=1,
state_dim=2,
obs_noise_scale_init=None,
learnable_observation_loc=False):
self.obs_dim = obs_dim
self.state_dim = state_dim
if obs_noise_scale_init is None:
obs_noise_scale_init = 0.2 * torch.ones(obs_dim)
assert obs_noise_scale_init.shape == (obs_dim, )
super().__init__()
self.obs_noise_scale = PyroParam(obs_noise_scale_init,
constraint=constraints.positive)
self.trans_noise_scale_sq = PyroParam(torch.ones(state_dim),
constraint=constraints.positive)
self.trans_matrix = nn.Parameter(
torch.eye(state_dim) + 0.03 * torch.randn(state_dim, state_dim))
self.obs_matrix = nn.Parameter(0.3 * torch.randn(state_dim, obs_dim))
self.init_noise_scale_sq = PyroParam(torch.ones(state_dim),
constraint=constraints.positive)
if learnable_observation_loc:
self.obs_loc = nn.Parameter(torch.zeros(obs_dim))
else:
self.register_buffer('obs_loc', torch.zeros(obs_dim))
def _get_init_dist(self):
loc = self.obs_matrix.new_zeros(self.state_dim)
return MultivariateNormal(loc, self.init_noise_scale_sq.diag_embed())
def _get_obs_dist(self):
return dist.Normal(self.obs_loc, self.obs_noise_scale).to_event(1)
def _get_trans_dist(self):
loc = self.obs_matrix.new_zeros(self.state_dim)
return MultivariateNormal(loc, self.trans_noise_scale_sq.diag_embed())
def _get_dist(self):
"""
Get the :class:`~pyro.distributions.GaussianHMM` distribution that corresponds to :class:`GenericLGSSM`.
"""
return dist.GaussianHMM(self._get_init_dist(), self.trans_matrix,
self._get_trans_dist(), self.obs_matrix,
self._get_obs_dist())
@pyro_method
def log_prob(self, targets):
"""
:param torch.Tensor targets: A 2-dimensional tensor of real-valued targets
of shape ``(T, obs_dim)``, where ``T`` is the length of the time series and ``obs_dim``
is the dimension of the real-valued ``targets`` at each time step
:returns torch.Tensor: A (scalar) log probability.
"""
assert targets.dim() == 2 and targets.size(-1) == self.obs_dim
return self._get_dist().log_prob(targets)
@torch.no_grad()
def _filter(self, targets):
"""
Return the filtering state for the associated state space model.
"""
assert targets.dim() == 2 and targets.size(-1) == self.obs_dim
return self._get_dist().filter(targets)
@torch.no_grad()
def _forecast(self,
N_timesteps,
filtering_state,
include_observation_noise=True):
"""
Internal helper for forecasting.
"""
N_trans_matrix = repeated_matmul(self.trans_matrix, N_timesteps)
N_trans_obs = torch.matmul(N_trans_matrix, self.obs_matrix)
predicted_mean = torch.matmul(filtering_state.loc, N_trans_obs)
# first compute the contribution from filtering_state.covariance_matrix
predicted_covar1 = torch.matmul(N_trans_obs.transpose(-1, -2),
torch.matmul(
filtering_state.covariance_matrix,
N_trans_obs)) # N O O
# next compute the contribution from process noise that is injected at each timestep.
# (we need to do a cumulative sum to integrate across time)
process_covar = self._get_trans_dist().covariance_matrix
N_trans_obs_shift = torch.cat(
[self.obs_matrix.unsqueeze(0), N_trans_obs[:-1]])
predicted_covar2 = torch.matmul(N_trans_obs_shift.transpose(
-1, -2), torch.matmul(process_covar, N_trans_obs_shift)) # N O O
predicted_covar = predicted_covar1 + torch.cumsum(predicted_covar2,
dim=0)
if include_observation_noise:
predicted_covar = predicted_covar + self.obs_noise_scale.pow(
2.0).diag_embed()
return predicted_mean, predicted_covar
@pyro_method
def forecast(self, targets, N_timesteps):
"""
:param torch.Tensor targets: A 2-dimensional tensor of real-valued targets
of shape ``(T, obs_dim)``, where ``T`` is the length of the time series and ``obs_dim``
is the dimension of the real-valued targets at each time step. These
represent the training data that are conditioned on for the purpose of making
forecasts.
:param int N_timesteps: The number of timesteps to forecast into the future from
the final target ``targets[-1]``.
:returns torch.distributions.MultivariateNormal: Returns a predictive MultivariateNormal distribution
with batch shape ``(N_timesteps,)`` and event shape ``(obs_dim,)``
"""
filtering_state = self._filter(targets)
predicted_mean, predicted_covar = self._forecast(
N_timesteps, filtering_state)
return torch.distributions.MultivariateNormal(predicted_mean,
predicted_covar)