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Formal representation of a linear regression from provided covariates.
Inherits From: StructuralTimeSeries
tfp.substrates.numpy.sts.LinearRegression(
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.
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.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,
name='holidays')
day_of_week = tfp.sts.Seasonal(num_seasons=7,
observed_time_series=observed_time_series,
name='day_of_week')
model = tfp.sts.Sum(components=[holidays, seasonal],
observed_time_series=observed_time_series)
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(
tfd.StudentT(df=5,
loc=tf.zeros([num_features]),
scale=tf.ones([num_features])),
reinterpreted_batch_ndims=1)
linear_regression = LinearRegression(design_matrix=design_matrix,
weights_prior=weights_prior)
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
|
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'. |
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().
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copy
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
get_parameter(
parameter_name
)
Returns the parameter with the given name, or a KeyError.
joint_distribution
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.
|
Example:
The joint distribution can generate prior samples of parameters and trajectories:
from matplotlib import pylab as plt
import tensorflow_probability as tfp; tfp = tfp.substrates.numpy
# 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
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). Any NaNs 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.
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make_state_space_model
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").
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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.
|
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: [] (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.
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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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__add__
__add__(
other
)
Models the sum of the series from the two components.