State space model for a local level

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. The local level model is a special case of a linear Gaussian SSM, in which the latent state posits a level evolving via a Gaussian random walk:

level[t] = level[t-1] + Normal(0., level_scale)

sts_local_level_state_space_model(
  num_timesteps,
  level_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.
level_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the level transitions.
initial_state_prior	instance of tfd_multivariate_normal representing the prior distribution on latent states. Must have event shape [1] (as tfd_linear_gaussian_state_space_model requires a rank-1 event shape).
observation_noise_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the observation noise.
initial_step	Optional 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	string name prefixed to ops created by this class. Default value: "LocalLevelStateSpaceModel".

Value

an instance of LinearGaussianStateSpaceModel.

Details

The latent state is [level] and [level] is observed (with noise) at each timestep.

The parameters level_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 and of the initial_state_prior.

Mathematical Details

The local level model implements a tfp$distributions$LinearGaussianStateSpaceModel with latent_size = 1 and observation_size = 1, following the transition model:

transition_matrix = [[1]]
transition_noise ~ N(loc = 0, scale = diag([level_scale]))

which implements the evolution of level described above, and the observation model:

observation_matrix = [[1]]
observation_noise ~ N(loc = 0, scale = 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_state_space_model(), sts_dynamic_linear_regression(), sts_linear_regression(), 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()