Formal representation of an autoregressive model.
Inherits From: StructuralTimeSeries
tfp.sts.Autoregressive(
order, coefficients_prior=None, level_scale_prior=None,
initial_state_prior=None, coefficient_constraining_bijector=None,
observed_time_series=None, name=None
)
Used in the notebooks
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 for details on
stationarity and other mathematical properties of autoregressive processes.
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'.
|
Attributes |
|---|
batch_shape
|
Static batch shape of models represented by this component.
|
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
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().
|
joint_log_prob
View source
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
View source
make_state_space_model(
num_timesteps, param_vals, 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
View source
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: [] (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.
|
View source on GitHub