tfp.sts.SparseLinearRegression

Formal representation of a sparse linear regression.

Inherits From: StructuralTimeSeries

tfp.sts.SparseLinearRegression(
    design_matrix, weights_prior_scale=0.1, weights_batch_shape=None, name=None
)

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

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'.

Attributes
`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.
`weights_prior_scale`

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()`.

`copy`

View source

copy(
    **override_parameters_kwargs
)

Creates a deep copy.

Args
`**override_parameters_kwargs`	String/value dictionary of initialization arguments to override with new values.

Returns
`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)`.

`get_parameter`

View source

get_parameter(
    parameter_name
)

Returns the parameter with the given name, or a KeyError.

`joint_distribution`

View source

joint_distribution(
    observed_time_series=None,
    num_timesteps=None,
    trajectories_shape=(),
    initial_step=0,
    mask=None,
    experimental_parallelize=False
)

Constructs the joint distribution over parameters and observed values.

Args
`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`.

Returns

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.

Returns
`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.

Example:

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])
plt.plot(
  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 = (
  tfp.experimental.vi.build_factored_surrogate_posterior(
    event_shape=jd.event_shape,
    bijector=jd.experimental_default_event_space_bijector()))
losses = tfp.vi.fit_surrogate_posterior(
  target_log_prob_fn=jd.unnormalized_log_prob,
  surrogate_posterior=surrogate_posterior,
  optimizer=tf.optimizers.Adam(0.1),
  num_steps=200)
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)

`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. (deprecated)

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). Any NaNs are interpreted as missing observations; missingness may be also be explicitly specified by passing a tfp.sts.MaskedTimeSeries instance.

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). Any `NaN`s are interpreted as missing observations; missingness may be also be explicitly specified by passing a `tfp.sts.MaskedTimeSeries` instance.

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.

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,
    **linear_gaussian_ssm_kwargs
)

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.
`**linear_gaussian_ssm_kwargs`	Optional additional keyword arguments to to the base `tfd.LinearGaussianStateSpaceModel` constructor.

Returns
`dist`	a `LinearGaussianStateSpaceModel` Distribution object.

`params_to_weights`

View source

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`

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. (deprecated)

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`	PRNG seed; see `tfp.random.sanitize_seed` for details. Default value: `None`.

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 `Tensor`s, in order corresponding to `self.parameters`, each of shape `params_sample_shape + prior.batch_shape + prior.event_shape`.

`add`

View source

__add__(
    other
)

Models the sum of the series from the two components.

tfp.sts.SparseLinearRegression

Examples

Mathematical Details

References

Args

Attributes

Methods

batch_shape_tensor

copy

get_parameter

joint_distribution

Example:

joint_log_prob

make_state_space_model

params_to_weights

prior_sample

__add__

`batch_shape_tensor`

`copy`

`get_parameter`

`joint_distribution`

`joint_log_prob`

`make_state_space_model`

`params_to_weights`

`prior_sample`

`add`