# Time Series¶

The pyro.contrib.timeseries module provides a collection of Bayesian time series models useful for forecasting applications.

See the GP example for example usage.

## Abstract Models¶

class TimeSeriesModel(name='')[source]

Base class for univariate and multivariate time series models.

log_prob(targets)[source]

Log probability function.

Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 A 0-dimensional log probability for the case of properly multivariate time series models in which the output dimensions are correlated; otherwise returns a 1-dimensional tensor of log probabilities for batched univariate time series models.
forecast(targets, dts)[source]
Parameters: Returns torch.distributions.Distribution: targets (torch.Tensor) – 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. dts (torch.Tensor) – A 1-dimensional tensor of times to forecast into the future, with zero corresponding to the time of the final target targets[-1]. Returns a predictive distribution with batch shape (S,) and event shape (obs_dim,), where S is the size of dts. That is, the resulting predictive distributions do not encode correlations between distinct times in dts.
get_dist()[source]

Get a Distribution object corresponding to this time series model. Often this is a GaussianHMM.

## Gaussian Processes¶

class IndependentMaternGP(nu=1.5, dt=1.0, obs_dim=1, length_scale_init=None, kernel_scale_init=None, obs_noise_scale_init=None)[source]

A time series model in which each output dimension is modeled independently with a univariate Gaussian Process with a Matern kernel. The targets are assumed to be evenly spaced in time. Training and inference are logarithmic in the length of the time series T.

Parameters: nu (float) – The order of the Matern kernel; one of 0.5, 1.5 or 2.5. dt (float) – The time spacing between neighboring observations of the time series. obs_dim (int) – The dimension of the targets at each time step. length_scale_init (torch.Tensor) – optional initial values for the kernel length scale given as a obs_dim-dimensional tensor kernel_scale_init (torch.Tensor) – optional initial values for the kernel scale given as a obs_dim-dimensional tensor obs_noise_scale_init (torch.Tensor) – optional initial values for the observation noise scale given as a obs_dim-dimensional tensor
get_dist(duration=None)[source]

Get the GaussianHMM distribution that corresponds to obs_dim-many independent Matern GPs.

Parameters: duration (int) – Optional size of the time axis event_shape[0]. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.
log_prob(targets)[source]
Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 A 1-dimensional tensor of log probabilities of shape (obs_dim,)
forecast(targets, dts)[source]
Parameters: Returns torch.distributions.Normal: targets (torch.Tensor) – 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. dts (torch.Tensor) – A 1-dimensional tensor of times to forecast into the future, with zero corresponding to the time of the final target targets[-1]. Returns a predictive Normal distribution with batch shape (S,) and event shape (obs_dim,), where S is the size of dts.
class LinearlyCoupledMaternGP(nu=1.5, dt=1.0, obs_dim=2, num_gps=1, length_scale_init=None, kernel_scale_init=None, obs_noise_scale_init=None)[source]

A time series model in which each output dimension is modeled as a linear combination of shared univariate Gaussian Processes with Matern kernels.

In more detail, the generative process is:

$$y_i(t) = \sum_j A_{ij} f_j(t) + \epsilon_i(t)$$

The targets $$y_i$$ are assumed to be evenly spaced in time. Training and inference are logarithmic in the length of the time series T.

Parameters: nu (float) – The order of the Matern kernel; one of 0.5, 1.5 or 2.5. dt (float) – The time spacing between neighboring observations of the time series. obs_dim (int) – The dimension of the targets at each time step. num_gps (int) – The number of independent GPs that are mixed to model the time series. Typical values might be $$\N_{\rm gp} \in [\D_{\rm obs} / 2, \D_{\rm obs}]$$ length_scale_init (torch.Tensor) – optional initial values for the kernel length scale given as a num_gps-dimensional tensor kernel_scale_init (torch.Tensor) – optional initial values for the kernel scale given as a num_gps-dimensional tensor obs_noise_scale_init (torch.Tensor) – optional initial values for the observation noise scale given as a obs_dim-dimensional tensor
get_dist(duration=None)[source]

Get the GaussianHMM distribution that corresponds to a LinearlyCoupledMaternGP.

Parameters: duration (int) – Optional size of the time axis event_shape[0]. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.
log_prob(targets)[source]
Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 a (scalar) log probability
forecast(targets, dts)[source]
Parameters: Returns torch.distributions.MultivariateNormal: targets (torch.Tensor) – 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. dts (torch.Tensor) – A 1-dimensional tensor of times to forecast into the future, with zero corresponding to the time of the final target targets[-1]. Returns a predictive MultivariateNormal distribution with batch shape (S,) and event shape (obs_dim,), where S is the size of dts.
class DependentMaternGP(nu=1.5, dt=1.0, obs_dim=1, linearly_coupled=False, length_scale_init=None, obs_noise_scale_init=None)[source]

A time series model in which each output dimension is modeled as a univariate Gaussian Process with a Matern kernel. The different output dimensions become correlated because the Gaussian Processes are driven by a correlated Wiener process; see reference [1] for details. If, in addition, linearly_coupled is True, additional correlation is achieved through linear mixing as in LinearlyCoupledMaternGP. The targets are assumed to be evenly spaced in time. Training and inference are logarithmic in the length of the time series T.

Parameters: nu (float) – The order of the Matern kernel; must be 1.5. dt (float) – The time spacing between neighboring observations of the time series. obs_dim (int) – The dimension of the targets at each time step. linearly_coupled (bool) – Whether to linearly mix the various gaussian processes in the likelihood. Defaults to False. length_scale_init (torch.Tensor) – optional initial values for the kernel length scale given as a obs_dim-dimensional tensor obs_noise_scale_init (torch.Tensor) – optional initial values for the observation noise scale given as a obs_dim-dimensional tensor

References [1] “Dependent Matern Processes for Multivariate Time Series,” Alexander Vandenberg-Rodes, Babak Shahbaba.

get_dist(duration=None)[source]

Get the GaussianHMM distribution that corresponds to a DependentMaternGP

Parameters: duration (int) – Optional size of the time axis event_shape[0]. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.
log_prob(targets)[source]
Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 A (scalar) log probability
forecast(targets, dts)[source]
Parameters: Returns torch.distributions.MultivariateNormal: targets (torch.Tensor) – 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. dts (torch.Tensor) – A 1-dimensional tensor of times to forecast into the future, with zero corresponding to the time of the final target targets[-1]. Returns a predictive MultivariateNormal distribution with batch shape (S,) and event shape (obs_dim,), where S is the size of dts.

## Linear Gaussian State Space Models¶

class GenericLGSSM(obs_dim=1, state_dim=2, obs_noise_scale_init=None, learnable_observation_loc=False)[source]

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.

Parameters: obs_dim (int) – The dimension of the targets at each time step. state_dim (int) – The dimension of latent state at each time step. learnable_observation_loc (bool) – whether the mean of the observation model should be learned or not; defaults to False.
get_dist(duration=None)[source]

Get the GaussianHMM distribution that corresponds to GenericLGSSM.

Parameters: duration (int) – Optional size of the time axis event_shape[0]. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.
log_prob(targets)[source]
Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 A (scalar) log probability.
forecast(targets, N_timesteps)[source]
Parameters: Returns torch.distributions.MultivariateNormal: targets (torch.Tensor) – 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. N_timesteps (int) – The number of timesteps to forecast into the future from the final target targets[-1]. Returns a predictive MultivariateNormal distribution with batch shape (N_timesteps,) and event shape (obs_dim,)
class GenericLGSSMWithGPNoiseModel(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)[source]

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:

$$y_i(t) = \sum_j A_{ij} z_j(t) + f_i(t) + \epsilon_i(t)$$

where the latent variables $${\bf z}(t)$$ follow generic time invariant Linear Gaussian dynamics and the $$f_i(t)$$ are Gaussian Processes with Matern kernels.

The targets are (implicitly) assumed to be evenly spaced in time. In particular a timestep of $$dt=1.0$$ for the continuous-time GP dynamics corresponds to a single discrete step of the $${\bf z}$$-space dynamics. Training and inference are logarithmic in the length of the time series T.

Parameters: obs_dim (int) – The dimension of the targets at each time step. state_dim (int) – The dimension of the $${\bf z}$$ latent state at each time step. nu (float) – The order of the Matern kernel; one of 0.5, 1.5 or 2.5. length_scale_init (torch.Tensor) – optional initial values for the kernel length scale given as a obs_dim-dimensional tensor kernel_scale_init (torch.Tensor) – optional initial values for the kernel scale given as a obs_dim-dimensional tensor obs_noise_scale_init (torch.Tensor) – optional initial values for the observation noise scale given as a obs_dim-dimensional tensor learnable_observation_loc (bool) – whether the mean of the observation model should be learned or not; defaults to False.
get_dist(duration=None)[source]

Get the GaussianHMM distribution that corresponds to GenericLGSSMWithGPNoiseModel.

Parameters: duration (int) – Optional size of the time axis event_shape[0]. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.
log_prob(targets)[source]
Parameters: Returns torch.Tensor: targets (torch.Tensor) – 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 A (scalar) log probability.
forecast(targets, N_timesteps)[source]
Parameters: Returns torch.distributions.MultivariateNormal: targets (torch.Tensor) – 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. N_timesteps (int) – The number of timesteps to forecast into the future from the final target targets[-1]. Returns a predictive MultivariateNormal distribution with batch shape (N_timesteps,) and event shape (obs_dim,)