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Formal representation of a smooth seasonal effect model.

Inherits From: StructuralTimeSeries

The smooth seasonal model uses a set of trigonometric terms in order to capture a recurring pattern whereby adjacent (in time) effects are similar. The model uses frequencies calculated via:

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

and then posits two latent states for each frequency. The two latent states associated with frequency j drift over time via:

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

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

where effect is the smooth seasonal effect and auxiliary only appears as a matter of construction. The interpretation of auxiliary is thus not particularly important.


A smooth seasonal effect model representing smooth weekly seasonality on daily data:

component = SmoothSeasonal(
    frequency_multipliers=[1, 2, 3],

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.
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_state_prior instance of tfd.MultivariateNormal representing the prior distribution on the latent states. Must have event shape [2 * len(frequency_multipliers)]. If None, a heuristic default prior is constructed based on the provided observed_time_series.
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 NaNs are interpreted as missing observations; missingness may be also be explicitly specified by passing a tfp.sts.MaskedTimeSeries instance. Any priors not explicitly set will be given default values according to the scale of the observed time series (or batch of time series). Default value: None.
name the name of this model component. Default value: 'SmoothSeasonal'.

allow_drift Whether the seasonal effects are allowed to drift over time.
batch_shape Static batch shape of models represented by this component.
frequency_multipliers Multipliers of the fundamental frequency.
init_parameters Parameters used to instantiate this StructuralTimeSeries.
initial_state_prior Prior distribution on the initial latent states.
latent_size Python int dimensionality of the latent space in this model.
name Name of this model component.
parameters List of Parameter(name, prior, bijector) namedtuples for this model.
period The seasonal period.



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Runtime batch shape of models represented by this component.

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().


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Creates a deep copy.

**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

copy A new instance of type(self) initialized from the union of self.init_parameters and override_parameters_kwargs, i.e., dict(self.init_parameters, **override_parameters_kwargs).


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Returns the parameter with the given name, or a KeyError.


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Constructs the joint distribution over parameters and observed values.

observed_time_series Optional observed time series to model, as a Tensor or tfp.sts.MaskedTimeSeries instance having shape concat([batch_shape, trajectories_shape, num_timesteps, 1]). If an observed time series is provided, the num_timesteps, trajectories_shape, and mask arguments are ignored, and an unnormalized (pinned) distribution over parameter values is returned. Default value: None.
num_timesteps scalar int Tensor number of timesteps to model. This must be specified either directly or by passing an observed_time_series. Default value: 0.
trajectories_shape int Tensor shape of sampled trajectories for each set of parameter values. Ignored if an observed_time_series is passed. Default value: ().
initial_step Optional scalar int Tensor specifying the starting timestep. Default value: 0.
mask Optional bool Tensor having shape concat([batch_shape, trajectories_shape, num_timesteps]), in which True entries indicate that the series value at the corresponding step is missing and should be ignored. This argument should be passed only if observed_time_series is not specified or does not already contain a missingness mask; it is an error to pass both this argument and an observed_time_series value containing a missingness mask. Default value: None.
experimental_parallelize If True, use parallel message passing algorithms from tfp.experimental.parallel_filter to perform time series operations in O(log num_timesteps) sequential steps. The overall FLOP and memory cost may be larger than for the sequential implementations by a constant factor. Default value: False.

joint_distribution joint distribution of model parameters and observed trajectories. If no observed_time_series was specified, this is an instance of tfd.JointDistributionNamedAutoBatched with a random variable for each model parameter (with names and order matching self.parameters), plus a final random variable observed_time_series representing a trajectory(ies) conditioned on the parameters. If observed_time_series was specified, the return value is given by joint_distribution.experimental_pin( observed_time_series=observed_time_series) where joint_distribution is as just described, so it defines an unnormalized posterior distribution over the parameters.


The joint distribution can generate prior samples of parameters and trajectories:

from matplotlib import pylab as plt
import tensorflow_probability as tfp

# Sample and plot 100 trajectories from the prior.
model = tfp.sts.LocalLinearTrend()
prior_samples = model.joint_distribution(num_timesteps=200).sample([100])
  tf.linalg.matrix_transpose(prior_samples['observed_time_series'][..., 0]))

It also integrates with TFP inference APIs, providing a more flexible alternative to the STS-specific fitting utilities.

jd = model.joint_distribution(observed_time_series)

# Variational inference.
surrogate_posterior = (
losses =
parameter_samples = surrogate_posterior.sample(50)

# No U-Turn Sampler.
samples, kernel_results = tfp.experimental.mcmc.windowed_adaptive_nuts(
  n_draws=500, joint_dist=dist)


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

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). Any NaNs are interpreted as missing observations; missingness may be also be explicitly specified by passing a tfp.sts.MaskedTimeSeries instance.

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.


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Instantiate this model as a Distribution over specified num_timesteps.

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.
**linear_gaussian_ssm_kwargs Optional additional keyword arguments to to the base tfd.LinearGaussianStateSpaceModel constructor.

dist a LinearGaussianStateSpaceModel Distribution object.


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Sample from the joint prior over model parameters and trajectories. (deprecated)

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 PRNG seed; see tfp.random.sanitize_seed for details. Default value: None.

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.


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Models the sum of the series from the two components.