tfp.sts.Autoregressive

Class Autoregressive

Formal representation of an autoregressive model.

Inherits From: StructuralTimeSeries

Defined in python/sts/autoregressive.py.

An autoregressive (AR) model posits a latent level whose value at each step is a noisy linear combination of previous steps:

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

The latent state is `levels[t:t-order:-1]`. We observe a noisy realization of
the current level: `f[t] = level[t] + Normal(0., observation_noise_scale)` at
each timestep.

If `coefficients=[1.]`, the AR process is a simple random walk, equivalent to
a `LocalLevel` model. However, a random walk's variance increases with time,
while many AR processes (in particular, any first-order process with
`abs(coefficient) < 1`) are *stationary*, i.e., they maintain a constant
variance over time. This makes AR processes useful models of uncertainty.

See the [Wikipedia article](
https://en.wikipedia.org/wiki/Autoregressive_model#Definition) for details on
stationarity and other mathematical properties of autoregressive processes.

<h2 id="__init__"><code>__init__</code></h2>

``` python
__init__(
    order,
    coefficients_prior=None,
    level_scale_prior=None,
    initial_state_prior=None,
    coefficient_constraining_bijector=None,
    observed_time_series=None,
    name=None
)

Specify an autoregressive model.

Args:

• order: scalar Python positive int specifying the number of past timesteps to regress on.
• coefficients_prior: optional tfd.Distribution instance specifying a prior on the coefficients parameter. If None, a default standard normal (tfd.MultivariateNormalDiag(scale_diag=tf.ones([order]))) prior is used. Default value: None.
• level_scale_prior: optional tfd.Distribution instance specifying a prior on the level_scale parameter. If None, a heuristic default prior is constructed based on the provided observed_time_series. Default value: None.
• initial_state_prior: optional tfd.Distribution instance specifying a prior on the initial state, corresponding to the values of the process at a set of size order of imagined timesteps before the initial step. If None, a heuristic default prior is constructed based on the provided observed_time_series. Default value: None.
• coefficient_constraining_bijector: optional tfb.Bijector instance representing a constraining mapping for the autoregressive coefficients. For example, tfb.Tanh() constrains the coefficients to lie in (-1, 1), while tfb.Softplus() constrains them to be positive, and tfb.Identity() implies no constraint. If None, the default behavior constrains the coefficients to lie in (-1, 1) using a Tanh bijector. Default value: None.
• 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: 'Autoregressive'.

Properties

batch_shape

Static batch shape of models represented by this component.

Returns:

• batch_shape: A tf.TensorShape 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. It may be partially defined or unknown.

initial_state_prior

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.

Methods

batch_shape_tensor

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().

joint_log_prob

joint_log_prob(observed_time_series)

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

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

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

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.

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: .
• 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: .
• 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.