State space model for a seasonal effect.

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.
num_seasons	Scalar `integer` number of seasons.
drift_scale	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.
initial_state_prior	instance of `tfd_multivariate_normal` representing the prior distribution on latent states; must have event shape `[num_seasons]`.
observation_noise_scale	Scalar (any additional dimensions are treated as batch dimensions) `float` `tensor` indicating the standard deviation of the observation noise.
num_steps_per_season	`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.
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: "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)

Arguments

Value

Details

See also