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Formal representation of a linear regression from provided covariates.

Inherits From: StructuralTimeSeries

    design_matrix, weights_prior=None, name=None

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. Alternately, event_shape may be scalar ([]), in which case the prior is internally broadcast as TransformedDistribution(weights_prior, tfb.Identity(), event_shape=[num_features], batch_shape=design_matrix.batch_shape). If None, defaults to StudentT(df=5, loc=0., scale=10.), 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.

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


  • 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.


  • 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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    num_timesteps, param_vals=None, initial_state_prior=None, initial_step=0

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.


  • dist: a LinearGaussianStateSpaceModel Distribution object.


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

Sample from the joint prior over model parameters and trajectories.


  • 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.


  • 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.