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 )
num_timesteps |
Scalar |
level_scale |
Scalar (any additional dimensions are treated as batch
dimensions) |
slope_scale |
Scalar (any additional dimensions are treated as batch
dimensions) |
initial_state_prior |
instance of |
observation_noise_scale |
Scalar (any additional dimensions are
treated as batch dimensions) |
initial_step |
Optional scalar |
validate_args |
|
allow_nan_stats |
|
name |
string prefixed to ops created by this class. Default value: "LocalLinearTrendStateSpaceModel". |
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.
an instance of LinearGaussianStateSpaceModel.
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_state_space_model(),
sts_local_level(),
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()
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