tfp.sts.SparseLinearRegression

Class SparseLinearRegression

Formal representation of a sparse linear regression.

Inherits From: StructuralTimeSeries

Defined in python/sts/regression.py.

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

observed_time_series = matmul(design_matrix, weights)

This is identical to tfp.sts.LinearRegression, except that SparseLinearRegression uses a parameterization of a Horseshoe prior [1][2] to encode the assumption that many of the weights are zero, i.e., many of the covariate time series are irrelevant. See the mathematical details section below for further discussion. The prior parameterization used by SparseLinearRegression is more suitable for inference than that obtained by simply passing the equivalent tfd.Horseshoe prior to LinearRegression; when sparsity is desired, SparseLinearRegression will likely yield better results.

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.

Examples

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.SparseLinearRegression(
  design_matrix=tf.stack([series1, series2], axis=-1),
  weights_prior_scale=0.1)

The weights_prior_scale determines the level of sparsity; small scales encourage the weights to be sparse. In some cases, such as when the likelihood is iid Gaussian with known scale, the prior scale can be analytically related to the expected number of nonzero weights [2]; however, this is not the case in general for STS models.

If the design matrix has batch dimensions, by default the model will create a matching batch of weights. For example, if design_matrix.shape == [ num_users, num_timesteps, num_features], by default the model will fit separate weights for each user, i.e., it will internally represent weights.shape == [num_users, num_features]. To share weights across some or all batch dimensions, you can manually specify the batch shape for the weights:

# design_matrix.shape == [num_users, num_timesteps, num_features]
regression = tfp.sts.SparseLinearRegression(
  design_matrix=design_matrix,
  weights_batch_shape=[])  # weights.shape -> [num_features]

Mathematical Details

The basic horseshoe prior [1] is defined as a Cauchy-normal scale mixture:

scales[i] ~ HalfCauchy(loc=0, scale=1)
weights[i] ~ Normal(loc=0., scale=scales[i] * global_scale)`

The Cauchy scale parameters puts substantial mass near zero, encouraging weights to be sparse, but their heavy tails allow weights far from zero to be estimated without excessive shrinkage. The horseshoe can be thought of as a continuous relaxation of a traditional 'spike-and-slab' discrete sparsity prior, in which the latent Cauchy scale mixes between 'spike' (scales[i] ~= 0) and 'slab' (scales[i] >> 0) regimes.

Following the recommendations in [2], SparseLinearRegression implements a horseshoe with the following adaptations:

• The Cauchy prior on scales[i] is represented as an InverseGamma-Normal compound.
• The global_scale parameter is integrated out following a Cauchy(0., scale=weights_prior_scale) hyperprior, which is also represented as an InverseGamma-Normal compound.
• All compound distributions are implemented using a non-centered parameterization.

The compound, non-centered representation defines the same marginal prior as the original horseshoe (up to integrating out the global scale), but allows samplers to mix more efficiently through the heavy tails; for variational inference, the compound representation implicity expands the representational power of the variational model.

Note that we do not yet implement the regularized ('Finnish') horseshoe, proposed in [2] for models with weak likelihoods, because the likelihood in STS models is typically Gaussian, where it's not clear that additional regularization is appropriate. If you need this functionality, please email tfprobability@tensorflow.org.

The full prior parameterization implemented in SparseLinearRegression is as follows:

# Sample global_scale from Cauchy(0, scale=weights_prior_scale).
global_scale_variance ~ InverseGamma(alpha=0.5, beta=0.5)
global_scale_noncentered ~ HalfNormal(loc=0, scale=1)
global_scale = (global_scale_noncentered *
                sqrt(global_scale_variance) *
                weights_prior_scale)

# Sample local_scales from Cauchy(0, 1).
local_scale_variances[i] ~ InverseGamma(alpha=0.5, beta=0.5)
local_scales_noncentered[i] ~ HalfNormal(loc=0, scale=1)
local_scales[i] = local_scales_noncentered[i] * sqrt(local_scale_variances[i])

weights[i] ~ Normal(loc=0., scale=local_scales[i] * global_scale)

References

[1]: Carvalho, C., Polson, N. and Scott, J. Handling Sparsity via the Horseshoe. AISTATS (2009). http://proceedings.mlr.press/v5/carvalho09a/carvalho09a.pdf [2]: Juho Piironen, Aki Vehtari. Sparsity information and regularization in the horseshoe and other shrinkage priors (2017). https://arxiv.org/abs/1707.01694

__init__

__init__(
    design_matrix,
    weights_prior_scale=0.1,
    weights_batch_shape=None,
    name=None
)

Specify a sparse linear regression model.

Args:

• 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_scale: float Tensor defining the scale of the Horseshoe prior on regression weights. Small values encourage the weights to be sparse. The shape must broadcast with weights_batch_shape. Default value: 0.1.
• weights_batch_shape: if None, defaults to design_matrix.batch_shape_tensor(). Must broadcast with the batch shape of design_matrix. Default value: None.
• name: the name of this model component. Default value: 'SparseLinearRegression'.

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.

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.

weights_prior_scale

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.

params_to_weights

params_to_weights(
    global_scale_variance,
    global_scale_noncentered,
    local_scale_variances,
    local_scales_noncentered,
    weights_noncentered
)

Build regression weights from model parameters.

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.