tfp.sts.SmoothSeasonal

Formal representation of a smooth seasonal effect model.

Inherits From: StructuralTimeSeries

tfp.sts.SmoothSeasonal(
    period, frequency_multipliers, allow_drift=True, drift_scale_prior=None,
    initial_state_prior=None, observed_time_series=None, name=None
)

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.

Examples

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

component = SmoothSeasonal(
    period=7,
    frequency_multipliers=[1, 2, 3],
    initial_state_prior=tfd.MultivariateNormalDiag(scale_diag=tf.ones([6])),
)

Args
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'.

Attributes
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.

Methods

batch_shape_tensor

View source

batch_shape_tensor()

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().

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy.

Note: the copy distribution may continue to depend on the original initialization arguments.

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

Returns
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).

joint_distribution

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joint_distribution(
    observed_time_series=None, num_timesteps=None, trajectories_shape=None,
    initial_step=0, mask=None, experimental_parallelize=False
)

Constructs the joint distribution over parameters and observed values.

Args
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. If not specified (either directly or by passing an observed_time_series), defaults to a one-to-one correspondence between trajectories and parameter settings (implicitly trajectories_shape=()). Default value: None.
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.

Returns

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.

Example:

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.LocalLinearTrendModel()
prior_samples = model.joint_distribution().sample([100])
plt.plot(
  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 = (
  tfp.experimental.vi.build_factored_surrogate_posterior(
    event_shape=jd.event_shape,
    bijector=jd.experimental_default_event_space_bijector()))
losses = tfp.vi.fit_surrogate_posterior(
  target_log_prob_fn=jd.unnormalized_log_prob,
  surrogate_posterior=surrogate_posterior,
  optimizer=tf.optimizers.Adam(0.1),
  num_steps=200)
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)

joint_log_prob

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joint_log_prob(
    observed_time_series
)

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

Warning: THIS FUNCTION IS DEPRECATED. It will be removed after 2022-03-01. Instructions for updating: Please use StructuralTimeSeries.joint_distribution(observed_time_series).log_prob

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

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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make_state_space_model(
    num_timesteps, param_vals, initial_state_prior=None, initial_step=0,
    **linear_gaussian_ssm_kwargs
)

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

Returns
dist	a LinearGaussianStateSpaceModel Distribution object.

prior_sample

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prior_sample(
    num_timesteps, initial_step=0, params_sample_shape=(),
    trajectories_sample_shape=(), seed=None
)

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

Warning: THIS FUNCTION IS DEPRECATED. It will be removed after 2022-03-01. Instructions for updating: Please use StructuralTimeSeries.joint_distribution(...).sample()

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

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.

__add__

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__add__(
    other
)

Models the sum of the series from the two components.