State space model for a dynamic linear regression from provided covariates.

A state space model (SSM) posits a set of latent (unobserved) variables that evolve over time with dynamics specified by a probabilistic transition model p(z[t+1] | z[t]). At each timestep, we observe a value sampled from an observation model conditioned on the current state, p(x[t] | z[t]). The special case where both the transition and observation models are Gaussians with mean specified as a linear function of the inputs, is known as a linear Gaussian state space model and supports tractable exact probabilistic calculations; see tfd_linear_gaussian_state_space_model for details.

sts_dynamic_linear_regression_state_space_model(
  num_timesteps,
  design_matrix,
  drift_scale,
  initial_state_prior,
  observation_noise_scale = 0,
  initial_step = 0,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = NULL
)

Arguments

num_timesteps	Scalar integer tensor, number of timesteps to model with this distribution.
design_matrix	float tensor of shape tf$concat(list(batch_shape, list(num_timesteps, num_features))). This may also optionally be an instance of tf$linalg$LinearOperator.
drift_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the latent state transitions.
initial_state_prior	instance of tfd_multivariate_normal representing the prior distribution on latent states. Must have event shape list(num_features).
observation_noise_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the observation noise. Default value: 0.
initial_step	scalar integer tensor specifying the starting timestep. Default value: 0.
validate_args	logical. Whether to validate input with asserts. If validate_args is FALSE, and the inputs are invalid, correct behavior is not guaranteed. Default value: FALSE.
allow_nan_stats	logical. If FALSE, raise an exception if a statistic (e.g. mean/mode/etc...) is undefined for any batch member. If TRUE, batch members with valid parameters leading to undefined statistics will return NaN for this statistic. Default value: TRUE.
name	name prefixed to ops created by this class. Default value: 'DynamicLinearRegressionStateSpaceModel'.

Value

an instance of LinearGaussianStateSpaceModel.

Details

The dynamic linear regression model is a special case of a linear Gaussian SSM and a generalization of typical (static) linear regression. The model represents regression weights with a latent state which evolves via a Gaussian random walk: weights[t] ~ Normal(weights[t-1], drift_scale)

The latent state (the weights) has dimension num_features, while the parameters drift_scale and observation_noise_scale are each (a batch of) scalars. The batch shape of this Distribution is the broadcast batch shape of these parameters, the initial_state_prior, and the design_matrix. num_features is determined from the last dimension of design_matrix (equivalent to the number of columns in the design matrix in linear regression).

Mathematical Details

The dynamic linear regression model implements a tfd_linear_gaussian_state_space_model with latent_size = num_features and observation_size = 1 following the transition model:

transition_matrix = eye(num_features)
transition_noise ~ Normal(0, diag([drift_scale]))

which implements the evolution of weights described above. The observation model is:

observation_matrix[t] = design_matrix[t]
observation_noise ~ Normal(0, observation_noise_scale)

See also

Other sts: sts_additive_state_space_model(), sts_autoregressive_state_space_model(), sts_autoregressive(), sts_constrained_seasonal_state_space_model(), sts_dynamic_linear_regression(), sts_linear_regression(), sts_local_level_state_space_model(), sts_local_level(), sts_local_linear_trend_state_space_model(), sts_local_linear_trend(), sts_seasonal_state_space_model(), sts_seasonal(), sts_semi_local_linear_trend_state_space_model(), sts_semi_local_linear_trend(), sts_smooth_seasonal_state_space_model(), sts_smooth_seasonal(), sts_sparse_linear_regression(), sts_sum()