Seasonal state space model with effects constrained to sum to zero.
sts_constrained_seasonal_state_space_model(
num_timesteps,
num_seasons,
drift_scale,
initial_state_prior,
observation_noise_scale = 1e-04,
num_steps_per_season = 1,
initial_step = 0,
validate_args = FALSE,
allow_nan_stats = TRUE,
name = NULL
)an instance of LinearGaussianStateSpaceModel.
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".
sts_seasonal_state_space_model().
Mathematical details
The constrained model implements a reparameterization of the
naive SeasonalStateSpaceModel. Instead of directly representing the
seasonal effects in the latent space, the latent space of the constrained
model represents the difference between each effect and the mean effect.
The following discussion assumes familiarity with the mathematical details
of SeasonalStateSpaceModel.
Reparameterization and constraints: let the seasonal effects at a given
timestep be E = [e_1, ..., e_N]. The difference between each effect e_i
and the mean effect is z_i = e_i - sum_i(e_i)/N. By itself, this
transformation is not invertible because recovering the absolute effects
requires that we know the mean as well. To fix this, we'll define
z_N = sum_i(e_i)/N as the mean effect. It's easy to see that this is
invertible: given the mean effect and the differences of the first N - 1
effects from the mean, it's easy to solve for all N effects. Formally,
we've defined the invertible linear reparameterization Z = R E, where
R = [1 - 1/N, -1/N, ..., -1/N
-1/N, 1 - 1/N, ..., -1/N,
...
1/N, 1/N, ..., 1/N]
represents the change of basis from 'effect coordinates' E to
'residual coordinates' Z. The Zs form the latent space of the
ConstrainedSeasonalStateSpaceModel.
To constrain the mean effect z_N to zero, we fix the prior to zero,
p(z_N) ~ N(0., 0), and after the transition at each timestep we project
z_N back to zero. Note that this projection is linear: to set the Nth
dimension to zero, we simply multiply by the identity matrix with a missing
element in the bottom right, i.e., Z_constrained = P Z,
where P = eye(N) - scatter((N-1, N-1), 1).
Model: concretely, suppose a naive seasonal effect model has initial state
prior N(m, S), transition matrix F and noise covariance
Q, and observation matrix H. Then the corresponding constrained seasonal
effect model has initial state prior N(P R m, P R S R' P'),
transition matrix P R F R^-1 and noise covariance F R Q R' F', and
observation matrix H R^-1, where the change-of-basis matrix R and
constraint projection matrix P are as defined above. This follows
directly from applying the reparameterization Z = R E, and then enforcing
the zero-sum constraint on the prior and transition noise covariances.
In practice, because the sum of effects z_N is constrained to be zero, it
will never contribute a term to any linear operation on the latent space,
so we can drop that dimension from the model entirely.
ConstrainedSeasonalStateSpaceModel does this, so that it implements the
N - 1 dimension latent space z_1, ..., z_[N-1].
Note that since we constrained the mean effect to be zero, the latent
z_i's now recover their interpretation as the actual effects,
z_i = e_i for i = 1, ..., N - 1, even though they were originally defined as residuals. The Nth effect is represented only implicitly, as the nonzero mean of the first N - 1effects. Although the computational represention is not symmetric across allNeffects, we derived theConstrainedSeasonalStateSpaceModelby starting with a symmetric representation and imposing only a symmetric constraint (the zero-sum constraint), so the probability model remains symmetric over allN`
seasonal effects.
Other sts:
sts_additive_state_space_model(),
sts_autoregressive_state_space_model(),
sts_autoregressive(),
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()