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Formal representation of a linear regression from provided covariates.

Inherits From: StructuralTimeSeries

Used in the notebooks

Used in the tutorials

This model defines a time series given by a linear combination of covariate time series provided in a design matrix:

observed_time_series = matmul(design_matrix, weights)

The design matrix has shape [num_timesteps, num_features]. The weights are treated as an unknown random variable of size [num_features] (both components also support batch shape), and are integrated over using the same approximate inference tools as other model parameters, i.e., generally HMC or variational inference.

This component does not itself include observation noise; it defines a deterministic distribution with mass at the point matmul(design_matrix, weights). In practice, it should be combined with observation noise from another component such as tfp.sts.Sum, as demonstrated below.


Given series1, series2 as Tensors each of shape [num_timesteps] representing covariate time series, we create a regression model that conditions on these covariates:

regression = tfp.sts.LinearRegression(
  design_matrix=tf.stack([series1, series2], axis=-1),
  weights_prior=tfd.Normal(loc=0., scale=1.))

Here we've also demonstrated specifying a custom prior, using an informative Normal(0., 1.) prior instead of the default weakly-informative prior.

As a more advanced application, we might use the design matrix to encode holiday effects. For example, suppose we are modeling data from the month of December. We can combine day-of-week seasonality with special effects for Christmas Eve (Dec 24), Christmas (Dec 25), and New Year's Eve (Dec 31), by constructing a design matrix with indicators for those dates.

holiday_indicators = np.zeros([31, 3])
holiday_indicators[23, 0] = 1  # Christmas Eve
holiday_indicators[24, 1] = 1  # Christmas Day
holiday_indicators[30, 2] = 1  # New Year's Eve

holidays = tfp.sts.LinearRegression(design_matrix=holiday_indicators,
day_of_week = tfp.sts.Seasonal(num_seasons=7,
model = tfp.sts.Sum(components=[holidays, seasonal],

Note that the Sum component in the above model also incorporates observation noise, with prior scale heuristically inferred from observed_time_series.

In these examples, we've used a single design matrix, but batching is also supported. If the design matrix has batch shape, the default behavior constructs weights with matching batch shape, which will fit a separate regression for each design matrix. This can be overridden by passing an explicit weights prior with appropriate batch shape. For example, if each design matrix in a batch contains features with the same semantics (e.g., if they represent per-group or per-observation covariates), we might choose to share statistical strength by fitting a single weight vector that broadcasts across all design matrices:

design_matrix = get_batch_of_inputs()
design_matrix.shape  # => concat([batch_shape, [num_timesteps, num_features]])

# Construct a prior with batch shape `[]` and event shape `[num_features]`,
# so that it describes a single vector of weights.
weights_prior = tfd.Independent(
linear_regression = LinearRegression(design_matrix=design_matrix,

design_matrix float Tensor of shape concat([batch_shape, [num_timesteps, num_features]]). This may also optionally be an instance of tf.linalg.LinearOperator.
weights_prior tfd.Distribution representing a prior over the regression weights. Must have event shape [num_features] and batch shape broadcastable to the design matrix's batch_shape. If None, defaults to Sample(StudentT(df=5, loc=0., scale=10.), num_features]), a weakly-informative prior loosely inspired by the Stan prior choice recommendations. Default value: None.
name the name of this model component. Default value: 'LinearRegression'.

batch_shape Static batch shape of models represented by this component.
design_matrix LinearOperator representing the design matrix.
init_parameters Parameters used to instantiate this StructuralTimeSeries.
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.



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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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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.