State space model for a local linear trend

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_local_linear_trend_state_space_model(
  num_timesteps,
  level_scale,
  slope_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.
slope_scale	Scalar (any additional dimensions are treated as batch dimensions) `float` `tensor` indicating the standard deviation of the slope 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 prefixed to ops created by this class. Default value: "LocalLinearTrendStateSpaceModel".

Value

an instance of LinearGaussianStateSpaceModel.

Details

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

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

The latent state is the two-dimensional tuple [level, slope]. The level is observed at each timestep.

The parameters level_scale, slope_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 linear trend model implements a tfd_linear_gaussian_state_space_model with latent_size = 2 and observation_size = 1, following the transition model:

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

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

observation_matrix = [[1., 0.]]
observation_noise ~ N(loc= 0 , scale = observation_noise_scale)

which picks out the first latent component, i.e., the level, as the observation at each timestep.

Arguments

Value

Details

See also