State space model for an autoregressive process.

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_autoregressive_state_space_model(
  num_timesteps,
  coefficients,
  level_scale,
  initial_state_prior,
  observation_noise_scale = 0,
  initial_step = 0,
  validate_args = FALSE,
  name = NULL
)

Arguments

num_timesteps	Scalar integer tensor number of timesteps to model with this distribution.
coefficients	float tensor of shape tf$concat(batch_shape, list(order)) defining the autoregressive coefficients. The coefficients are defined backwards in time: coefficients[0] * level[t] + coefficients[1] * level[t-1] + ... + coefficients[order-1] * level[t-order+1].
level_scale	Scalar (any additional dimensions are treated as batch dimensions) float tensor indicating the standard deviation of the transition noise at each step.
initial_state_prior	instance of tfd_multivariate_normal representing the prior distribution on latent states. Must have event shape list(order).
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	Optional scalar int 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.
name	name prefixed to ops created by this class. Default value: "AutoregressiveStateSpaceModel".

Value

an instance of LinearGaussianStateSpaceModel.

Details

In an autoregressive process, the expected level at each timestep is a linear function of previous levels, with added Gaussian noise:

level[t+1] = (sum(coefficients * levels[t:t-order:-1]) + Normal(0., level_scale))

The process is characterized by a vector coefficients whose size determines the order of the process (how many previous values it looks at), and by level_scale, the standard deviation of the noise added at each step. This is formulated as a state space model by letting the latent state encode the most recent values; see 'Mathematical Details' below.

The parameters level_scale and observation_noise_scale are each (a batch of) scalars, and coefficients is a (batch) vector of size list(order). The batch shape of this Distribution is the broadcast batch shape of these parameters and of the initial_state_prior.

Mathematical Details

The autoregressive model implements a tfd_linear_gaussian_state_space_model with latent_size = order and observation_size = 1. The latent state vector encodes the recent history of the process, with the current value in the topmost dimension. At each timestep, the transition sums the previous values to produce the new expected value, shifts all other values down by a dimension, and adds noise to the current value. This is formally encoded by the transition model:

transition_matrix = [ coefs[0], coefs[1], ..., coefs[order]
                      1.,       0 ,       ..., 0.
                      0.,       1.,       ..., 0.
                      ...
                      0.,       0.,  ...,  1., 0.         ]

transition_noise ~ N(loc=0., scale=diag([level_scale, 0., 0., ..., 0.]))

The observation model simply extracts the current (topmost) value, and optionally adds independent noise at each step:

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

Models with observation_noise_scale = 0 are AR processes in the formal sense. Setting observation_noise_scale to a nonzero value corresponds to a latent AR process observed under an iid noise model.

See also

Other sts: sts_additive_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_state_space_model(), sts_smooth_seasonal(), sts_sparse_linear_regression(), sts_sum()