State space model for a smooth 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 tfp$distributions$LinearGaussianStateSpaceModel for details. A smooth seasonal effect model is a special case of a linear Gaussian SSM. It is the sum of a set of "cyclic" components, with one component for each frequency:

frequencies[j] = 2. * pi * frequency_multipliers[j] / period

Each cyclic component contains two latent states which we denote effect and auxiliary. The two latent states for component j drift over time via:

effect[t] = (effect[t-1] * cos(frequencies[j]) +
             auxiliary[t-] * sin(frequencies[j]) +
             Normal(0., drift_scale))
auxiliary[t] = (-effect[t-1] * sin(frequencies[j]) +
                auxiliary[t-] * cos(frequencies[j]) +
                Normal(0., drift_scale))

sts_smooth_seasonal_state_space_model(
  num_timesteps,
  period,
  frequency_multipliers,
  drift_scale,
  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.
period	positive scalar float Tensor giving the number of timesteps required for the longest cyclic effect to repeat.
frequency_multipliers	One-dimensional float Tensor listing the frequencies (cyclic components) included in the model, as multipliers of the base/fundamental frequency 2. * pi / period. Each component is specified by the number of times it repeats per period, and adds two latent dimensions to the model. A smooth seasonal model that can represent any periodic function is given by frequency_multipliers = [1,2, ..., floor(period / 2)]. However, it is often desirable to enforce a smoothness assumption (and reduce the computational burden) by dropping some of the higher frequencies.
drift_scale	Scalar (any additional dimensions are treated as batch dimensions) float Tensor indicating the standard deviation of the latent state transitions.
initial_state_prior	instance of tfd$MultivariateNormal representing the prior distribution on latent states. Must have event shape [num_features].
observation_noise_scale	Scalar (any additional dimensions are treated as batch dimensions) float Tensor indicating the standard deviation of the observation noise. Default value: 0..
initial_step	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: "LocalLinearTrendStateSpaceModel".

Value

an instance of LinearGaussianStateSpaceModel.

Details

The auxiliary latent state only appears as a matter of construction and thus its interpretation is not particularly important. The total smooth seasonal effect is the sum of the effect values from each of the cyclic components. 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.

Mathematical Details

The smooth seasonal effect model implements a tfp$distributions$LinearGaussianStateSpaceModel with latent_size = 2 * len(frequency_multipliers) and observation_size = 1. The latent state is the concatenation of the cyclic latent states which themselves comprise an effect and an auxiliary state. The transition matrix is a block diagonal matrix where block j is:

transition_matrix[j] =  [[cos(frequencies[j]), sin(frequencies[j])],
                         [-sin(frequencies[j]), cos(frequencies[j])]]

The observation model picks out the cyclic effect values from the latent state:

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

For further mathematical details please see Harvey (1990).

references

• Harvey, A. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press, 1990.

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_state_space_model(), sts_semi_local_linear_trend(), sts_smooth_seasonal(), sts_sparse_linear_regression(), sts_sum()