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_seasonal_state_space_model(
num_timesteps,
num_seasons,
drift_scale,
initial_state_prior,
observation_noise_scale = 0,
num_steps_per_season = 1,
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. Scalar integer number of seasons. Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the change in effect between consecutive occurrences of a given season. This is assumed to be the same for all seasons. instance of tfd_multivariate_normal representing the prior distribution on latent states; must have event shape [num_seasons]. Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the observation noise. integer number of steps in each season. This may be either a scalar (shape []), in which case all seasons have the same length, or an array of shape [num_seasons], in which seasons have different length, but remain constant around different cycles, or an array of shape [num_cycles, num_seasons], in which num_steps_per_season for each season also varies in different cycle (e.g., a 4 years cycle with leap day). Default value: 1. Optional scalar integer tensor specifying the starting timestep. Default value: 0. 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. 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. string prefixed to ops created by this class. Default value: "SeasonalStateSpaceModel".

## Value

an instance of LinearGaussianStateSpaceModel.

## Details

A seasonal effect model is a special case of a linear Gaussian SSM. The latent states represent an unknown effect from each of several 'seasons'; these are generally not meteorological seasons, but represent regular recurring patterns such as hour-of-day or day-of-week effects. The effect of each season drifts from one occurrence to the next, following a Gaussian random walk:

effects[season, occurrence[i]] = (effects[season, occurrence[i-1]] + Normal(loc=0., scale=drift_scale))


The latent state has dimension num_seasons, containing one effect for each seasonal component. The parameters drift_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. Note: there is no requirement that the effects sum to zero.

Mathematical Details

The seasonal effect model implements a tfd_linear_gaussian_state_space_model with latent_size = num_seasons and observation_size = 1. The latent state is organized so that the current seasonal effect is always in the first (zeroth) dimension. The transition model rotates the latent state to shift to a new effect at the end of each season:

transition_matrix[t] = (permutation_matrix([1, 2, ..., num_seasons-1, 0])
if season_is_changing(t)
else eye(num_seasons)
transition_noise[t] ~ Normal(loc=0., scale_diag=(
[drift_scale, 0, ..., 0]
if season_is_changing(t)
else [0, 0, ..., 0]))


where season_is_changing(t) is True if t mod sum(num_steps_per_season) is in the set of final days for each season, given by cumsum(num_steps_per_season) - 1. The observation model always picks out the effect for the current season, i.e., the first element of the latent state:

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


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(), 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()