The local linear trend model 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)

sts_local_linear_trend(
observed_time_series = NULL,
level_scale_prior = NULL,
slope_scale_prior = NULL,
initial_level_prior = NULL,
initial_slope_prior = NULL,
name = NULL
)

## Arguments

observed_time_series optional float tensor of shape batch_shape + [T, 1] (omitting the trailing unit dimension is also supported when T > 1), specifying an observed time series. Any priors not explicitly set will be given default values according to the scale of the observed time series (or batch of time series). May optionally be an instance of sts_masked_time_series, which includes a mask tensor to specify timesteps with missing observations. Default value: NULL. optional tfp$distribution instance specifying a prior on the level_scale parameter. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL. optional tfd$Distribution instance specifying a prior on the slope_scale parameter. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL. optional tfp$distribution instance specifying a prior on the initial level. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL. optional tfd$Distribution instance specifying a prior on the initial slope. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL. the name of this model component. Default value: 'LocalLinearTrend'.

## Value

an instance of StructuralTimeSeries.

## Details

The latent state is the two-dimensional tuple [level, slope]. At each timestep we observe a noisy realization of the current level: f[t] = level[t] + Normal(0., observation_noise_scale). This model is appropriate for data where the trend direction and magnitude (latent slope) is consistent within short periods but may evolve over time.

Note that this model can produce very high uncertainty forecasts, as uncertainty over the slope compounds quickly. If you expect your data to have nonzero long-term trend, i.e. that slopes tend to revert to some mean, then the SemiLocalLinearTrend model may produce sharper forecasts.

For usage examples see sts_fit_with_hmc(), sts_forecast(), sts_decompose_by_component().
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_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()