tfp.sts.Seasonal

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Formal representation of a seasonal effect model.

Inherits From: StructuralTimeSeries

Used in the notebooks

Used in the tutorials

A seasonal effect model posits a fixed set of recurring, discrete 'seasons', each of which is active for a fixed number of timesteps and, while active, contributes a different effect to the time series. These are generally not meteorological seasons, but represent regular recurring patterns such as hour-of-day or day-of-week effects. Each season lasts for a fixed number of timesteps. 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 drift_scale parameter governs the standard deviation of the random walk; for example, in a day-of-week model it governs the change in effect from this Monday to next Monday.

Examples

A seasonal effect model representing day-of-week seasonality on hourly data:

day_of_week = tfp.sts.Seasonal(num_seasons=7,
                               num_steps_per_season=24,
                               observed_time_series=y,
                               name='day_of_week')

A seasonal effect model representing month-of-year seasonality on daily data, with explicit priors:

month_of_year = tfp.sts.Seasonal(
  num_seasons=12,
  num_steps_per_season=[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
  drift_scale_prior=tfd.LogNormal(loc=-1., scale=0.1),
  initial_effect_prior=tfd.Normal(loc=0., scale=5.),
  name='month_of_year')

Note that this version works over time periods not involving a leap year. A general implementation of month-of-year seasonality would require additional logic:

num_days_per_month = np.array(
  [[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
   [31, 29, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],  # year with leap day
   [31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
   [31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31]])

month_of_year = tfp.sts.Seasonal(
  num_seasons=12,
  num_steps_per_season=num_days_per_month,
  drift_scale_prior=tfd.LogNormal(loc=-1., scale=0.1),
  initial_effect_prior=tfd.Normal(loc=0., scale=5.),
  name='month_of_year')

A model representing both day-of-week and hour-of-day seasonality, on hourly data:

day_of_week = tfp.sts.Seasonal(num_seasons=7,
                               num_steps_per_season=24,
                               observed_time_series=y,
                               name='day_of_week')
hour_of_day = tfp.sts.Seasonal(num_seasons=24,
                               num_steps_per_season=1,
                               observed_time_series=y,
                               name='hour_of_day')
model = tfp.sts.Sum(components=[day_of_week, hour_of_day],
                    observed_time_series=y)

num_seasons Scalar Python int number of seasons.
num_steps_per_season Python int number of steps in each season. This may be either a scalar (shape []), in which case all seasons have the same length, or a NumPy array of shape [num_seasons], in which seasons have different length, but remain constant around different cycles, or a NumPy 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.
allow_drift optional Python bool specifying whether the seasonal effects can drift over time. Setting this to False removes the drift_scale parameter from the model. This is mathematically equivalent to drift_scale_prior = tfd.Deterministic(0.), but removing drift directly is preferred because it avoids the use of a degenerate prior. Default value: True.
drift_scale_prior optional tfd.Distribution instance specifying a prior on the drift_scale parameter. If None, a heuristic default prior is constructed based on the provided observed_time_series. Default value: None.
initial_effect_prior optional tfd.Distribution instance specifying a normal prior on the initial effect of each season. This may be either a scalar tfd.Normal prior, in which case it applies independently to every season, or it may be multivariate normal (e.g., tfd.MultivariateNormalDiag) with event shape [num_seasons], in which case it specifies a joint prior across all seasons. If None, a heuristic default prior is constructed based on the provided observed_time_series. Default value: None.
constrain_mean_effect_to_zero if True, use a model parameterization that constrains the mean effect across all seasons to be zero. This constraint is generally helpful in identifying the contributions of different model components and can lead to more interpretable posterior decompositions. It may be undesirable if you plan to directly examine the latent space of the underlying state space model. Default value: True.
observed_time_series optional float Tensor of shape batch_shape + [T, 1] (omitting the trailing unit dimension is also supported when T > 1), specifying an observed time series. Any priors not explicitly set will be given default values according to the scale of the observed time series (or batch of time series). May optionally be an instance of tfp.sts.MaskedTimeSeries, which includes a mask Tensor to specify timesteps with missing observations. Default value: None.
name the name of this model component. Default value: 'Seasonal'.

allow_drift Whether the seasonal effects are allowed to drift over time.
batch_shape Static batch shape of models represented by this component.
constrain_mean_effect_to_zero Whether to constrain the mean effect to zero.
initial_state_prior Prior distribution on the initial latent state (level and scale).
latent_size Python int dimensionality of the latent space in this model.
name Name of this model component.
num_seasons Number of seasons.
num_steps_per_season Number of steps per season.
parameters List of Parameter(name, prior, bijector) namedtuples for this model.

Methods

batch_shape_tensor

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Runtime batch shape of models represented by this component.

Returns
batch_shape int Tensor giving the broadcast batch shape of all model parameters. This should match the batch shape of derived state space models, i.e., self.make_state_space_model(...).batch_shape_tensor().

joint_log_prob

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Build the joint density log p(params) + log p(y|params) as a callable.

Args
observed_time_series Observed Tensor trajectories of shape sample_shape + batch_shape + [num_timesteps, 1] (the trailing 1 dimension is optional if num_timesteps > 1), where batch_shape should match self.batch_shape (the broadcast batch shape of all priors on parameters for this structural time series model). May optionally be an instance of tfp.sts.MaskedTimeSeries, which includes a mask Tensor to specify timesteps with missing observations.

Returns
log_joint_fn A function taking a Tensor argument for each model parameter, in canonical order, and returning a Tensor log probability of shape batch_shape. Note that, unlike tfp.Distributions log_prob methods, the log_joint sums over the sample_shape from y, so that sample_shape does not appear in the output log_prob. This corresponds to viewing multiple samples in y as iid observations from a single model, which is typically the desired behavior for parameter inference.

make_state_space_model

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Instantiate this model as a Distribution over specified num_timesteps.

Args
num_timesteps Python int number of timesteps to model.
param_vals a list of Tensor parameter values in order corresponding to self.parameters, or a dict mapping from parameter names to values.
initial_state_prior an optional Distribution instance overriding the default prior on the model's initial state. This is used in forecasting ("today's prior is yesterday's posterior").
initial_step optional int specifying the initial timestep to model. This is relevant when the model contains time-varying components, e.g., holidays or seasonality.

Returns
dist a LinearGaussianStateSpaceModel Distribution object.

prior_sample

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Sample from the joint prior over model parameters and trajectories.

Args
num_timesteps Scalar int Tensor number of timesteps to model.
initial_step Optional scalar int Tensor specifying the starting timestep. Default value: 0.
params_sample_shape Number of possible worlds to sample iid from the parameter prior, or more generally, Tensor int shape to fill with iid samples. Default value: [] (i.e., draw a single sample and don't expand the shape).
trajectories_sample_shape For each sampled set of parameters, number of trajectories to sample, or more generally, Tensor int shape to fill with iid samples. Default value: [] (i.e., draw a single sample and don't expand the shape).
seed Python int random seed.

Returns
trajectories float Tensor of shape trajectories_sample_shape + params_sample_shape + [num_timesteps, 1] containing all sampled trajectories.
param_samples list of sampled parameter value Tensors, in order corresponding to self.parameters, each of shape params_sample_shape + prior.batch_shape + prior.event_shape.