State space model for a semi-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_semi_local_linear_trend_state_space_model(
  num_timesteps,
  level_scale,
  slope_mean,
  slope_scale,
  autoregressive_coef,
  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_mean	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the expected long-term mean of the latent slope.
slope_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the slope transitions.
autoregressive_coef	Scalar (any additional dimensions are treated as batch dimensions) float tensor defining the AR1 process on the latent slope.
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: "SemiLocalLinearTrendStateSpaceModel".

Value

an instance of LinearGaussianStateSpaceModel.

Details

The semi-local linear trend model is a special case of a linear Gaussian SSM, in which the latent state posits a level and slope. The level evolves via a Gaussian random walk centered at the current slope, while the slope follows a first-order autoregressive (AR1) process with mean slope_mean:

level[t] = level[t-1] + slope[t-1] + Normal(0, level_scale)
slope[t] = (slope_mean + autoregressive_coef * (slope[t-1] - slope_mean) +
           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_mean, slope_scale, autoregressive_coef, 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 semi-local linear trend model implements a tfp.distributions.LinearGaussianStateSpaceModel with latent_size = 2 and observation_size = 1, following the transition model:

transition_matrix = [[1., 1.]
                     [0., autoregressive_coef]]
transition_noise ~ N(loc=slope_mean - autoregressive_coef * slope_mean,
                     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.

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_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(), sts_smooth_seasonal_state_space_model(), sts_smooth_seasonal(), sts_sparse_linear_regression(), sts_sum()