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Seasonal state space model with effects constrained to sum to zero.
Inherits From: LinearGaussianStateSpaceModel
, AutoCompositeTensorDistribution
, Distribution
tfp.substrates.numpy.sts.ConstrainedSeasonalStateSpaceModel(
num_timesteps,
num_seasons,
drift_scale,
initial_state_prior,
observation_noise_scale=0.0001,
num_steps_per_season=1,
name=None,
**linear_gaussian_ssm_kwargs
)
See SeasonalStateSpaceModel
for background.
Mathematical details
The constrained model implements a reparameterization of the
naive SeasonalStateSpaceModel
. Instead of directly representing the
seasonal effects in the latent space, the latent space of the constrained
model represents the difference between each effect and the mean effect.
The following discussion assumes familiarity with the mathematical details
of SeasonalStateSpaceModel
.
Reparameterization and constraints: let the seasonal effects at a given
timestep be E = [e_1, ..., e_N]
. The difference between each effect e_i
and the mean effect is z_i = e_i - sum_i(e_i)/N
. By itself, this
transformation is not invertible because recovering the absolute effects
requires that we know the mean as well. To fix this, we'll define
z_N = sum_i(e_i)/N
as the mean effect. It's easy to see that this is
invertible: given the mean effect and the differences of the first N - 1
effects from the mean, it's easy to solve for all N
effects. Formally,
we've defined the invertible linear reparameterization Z = R E
, where
R = [1 - 1/N, -1/N, ..., -1/N, -1/N
-1/N, 1 - 1/N, ..., -1/N, -1/N
...
-1/N, -1/N, ..., 1/ - 1/N, -1/N
1/N, 1/N, ..., 1/N, 1/N]
E = [e_1, ..., e_(N-1), z_N]
represents the change of basis from 'effect coordinates' E to
'residual coordinates' Z. The Z
s form the latent space of the
ConstrainedSeasonalStateSpaceModel
.
To constrain the mean effect z_N
to zero, we fix the prior to zero,
p(z_N) ~ N(0., 0)
, and after the transition at each timestep we project
z_N
back to zero. Note that this projection is linear: to set the Nth
dimension to zero, we simply multiply by the identity matrix with a missing
element in the bottom right, i.e., Z_constrained = P Z
,
where P = eye(N) - scatter((N-1, N-1), 1)
.
Model: concretely, suppose a naive seasonal effect model has initial state
prior N(m, S)
, transition matrix F
and noise covariance
Q
, and observation matrix H
. Then the corresponding constrained seasonal
effect model has initial state prior N(P R m, P R S R' P')
,
transition matrix P R F R^-1
and noise covariance F R Q R' F'
, and
observation matrix H R^-1
, where the change-of-basis matrix R
and
constraint projection matrix P
are as defined above. This follows
directly from applying the reparameterization Z = R E
, and then enforcing
the zero-sum constraint on the prior and transition noise covariances.
In practice, because the sum of effects z_N
is constrained to be zero, it
will never contribute a term to any linear operation on the latent space,
so we can drop that dimension from the model entirely.
ConstrainedSeasonalStateSpaceModel
does this, so that it implements the
N - 1
dimension latent space z_1, ..., z_[N-1]
.
Note that since we constrained the mean effect to be zero, the latent
z_i
's now recover their interpretation as the actual effects,
z_i = e_i
for i =
1, ..., N - 1, even though they were originally
defined as residuals. The
Nth effect is represented only implicitly, as
the nonzero mean of the first
N - 1effects. Although the computational
represention is not symmetric across all
Neffects, we derived the
ConstrainedSeasonalStateSpaceModelby starting with a symmetric
representation and imposing only a symmetric constraint (the zero-sum
constraint), so the probability model remains symmetric over all
N`
seasonal effects.
Examples
A constrained state-space model with day-of-week seasonality on hourly data:
day_of_week = ConstrainedSeasonalStateSpaceModel(
num_timesteps=30,
num_seasons=7,
drift_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([7-1], dtype=tf.float32)),
num_steps_per_season=24)
A model with basic month-of-year seasonality on daily data, demonstrating seasons of varying length:
month_of_year = ConstrainedSeasonalStateSpaceModel(
num_timesteps=2 * 365, # 2 years
num_seasons=12,
drift_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([12-1], dtype=tf.float32)),
num_steps_per_season=[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
initial_step=22)
Note that we've used initial_step=22
to denote that the model begins
on January 23 (steps are zero-indexed). This version works over time periods
not involving a leap year. A general implementation of month-of-year
seasonality would require additional logic:
num_days_per_month = np.array(
[[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
[31, 29, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31], # year with leap day
[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31]])
month_of_year = ConstrainedSeasonalStateSpaceModel(
num_timesteps=4 * 365 + 2, # 8 years with leap days
num_seasons=12,
drift_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([12-1], dtype=tf.float32)),
num_steps_per_season=num_days_per_month,
initial_step=22)
Methods
backward_smoothing_pass
backward_smoothing_pass(
filtered_means, filtered_covs, predicted_means, predicted_covs
)
Run the backward pass in Kalman smoother.
The backward smoothing is using Rauch, Tung and Striebel smoother as
as discussed in section 18.3.2 of Kevin P. Murphy, 2012, Machine Learning:
A Probabilistic Perspective, The MIT Press. The inputs are returned by
forward_filter
function.
Args | |
---|---|
filtered_means
|
Means of the per-timestep filtered marginal
distributions p(z[t] | x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] .
|
filtered_covs
|
Covariances of the per-timestep filtered marginal
distributions p(z[t] | x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] .
|
predicted_means
|
Means of the per-timestep predictive
distributions over latent states, p(z[t+1] | x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size] .
|
predicted_covs
|
Covariances of the per-timestep predictive
distributions over latent states, p(z[t+1] | x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size, latent_size] .
|
Returns | |
---|---|
posterior_means
|
Means of the smoothed marginal distributions
p(z[t] | x[1:T]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] ,
which is of the same shape as filtered_means.
|
posterior_covs
|
Covariances of the smoothed marginal distributions
p(z[t] | x[1:T]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] . which is of the same shape as filtered_covs.
|
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1-D Tensor
.
The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
Args | |
---|---|
name
|
name to give to the op |
Returns | |
---|---|
batch_shape
|
Tensor .
|
cdf
cdf(
value, name='cdf', **kwargs
)
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
cdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
Args | |
---|---|
**override_parameters_kwargs
|
String/value dictionary of initialization arguments to override with new values. |
Returns | |
---|---|
distribution
|
A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .
|
covariance
covariance(
name='covariance', **kwargs
)
Covariance.
Covariance is (possibly) defined only for non-scalar-event distributions.
For example, for a length-k
, vector-valued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
where Cov
is a (batch of) k x k
matrix, 0 <= (i, j) < k
, and E
denotes expectation.
Alternatively, for non-vector, multivariate distributions (e.g.,
matrix-valued, Wishart), Covariance
shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov
is a (batch of) k' x k'
matrices,
0 <= (i, j) < k' = reduce_prod(event_shape)
, and Vec
is some function
mapping indices of this distribution's event dimensions to indices of a
length-k'
vector.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
covariance
|
Floating-point Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape) .
|
cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self
) by P
and the other
distribution by
Q
. Assuming P, Q
are absolutely continuous with respect to
one another and permit densities p(x) dr(x)
and q(x) dr(x)
, (Shannon)
cross entropy is defined as:
H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
where F
denotes the support of the random variable X ~ P
.
Args | |
---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
cross_entropy
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shannon) cross entropy.
|
entropy
entropy(
name='entropy', **kwargs
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1-D int32 Tensor
.
Args | |
---|---|
name
|
name to give to the op |
Returns | |
---|---|
event_shape
|
Tensor .
|
experimental_default_event_space_bijector
experimental_default_event_space_bijector(
*args, **kwargs
)
Bijector mapping the reals (R**n) to the event space of the distribution.
Distributions with continuous support may implement
_default_event_space_bijector
which returns a subclass of
tfp.bijectors.Bijector
that maps R**n to the distribution's event space.
For example, the default bijector for the Beta
distribution
is tfp.bijectors.Sigmoid()
, which maps the real line to [0, 1]
, the
support of the Beta
distribution. The default bijector for the
CholeskyLKJ
distribution is tfp.bijectors.CorrelationCholesky
, which
maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular
matrices with ones along the diagonal.
The purpose of experimental_default_event_space_bijector
is
to enable gradient descent in an unconstrained space for Variational
Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
choose bijectors such that the tails of the distribution in the
unconstrained space are between Gaussian and Exponential.
For distributions with discrete event space, or for which TFP currently
lacks a suitable bijector, this function returns None
.
Args | |
---|---|
*args
|
Passed to implementation _default_event_space_bijector .
|
**kwargs
|
Passed to implementation _default_event_space_bijector .
|
Returns | |
---|---|
event_space_bijector
|
Bijector instance or None .
|
experimental_fit
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
Instantiates a distribution that maximizes the likelihood of x
.
Args | |
---|---|
value
|
a Tensor valid sample from this distribution family.
|
sample_ndims
|
Positive int Tensor number of leftmost dimensions of
value that index i.i.d. samples.
Default value: 1 .
|
validate_args
|
Python bool , default False . When True , distribution
parameters are checked for validity despite possibly degrading runtime
performance. When False , invalid inputs may silently render incorrect
outputs.
Default value: False .
|
**init_kwargs
|
Additional keyword arguments passed through to
cls.__init__ . These take precedence in case of collision with the
fitted parameters; for example,
tfd.Normal.experimental_fit([1., 1.], scale=20.) returns a Normal
distribution with scale=20. rather than the maximum likelihood
parameter scale=0. .
|
Returns | |
---|---|
maximum_likelihood_instance
|
instance of cls with parameters that
maximize the likelihood of value .
|
experimental_local_measure
experimental_local_measure(
value, backward_compat=False, **kwargs
)
Returns a log probability density together with a TangentSpace
.
A TangentSpace
allows us to calculate the correct push-forward
density when we apply a transformation to a Distribution
on
a strict submanifold of R^n (typically via a Bijector
in the
TransformedDistribution
subclass). The density correction uses
the basis of the tangent space.
Args | |
---|---|
value
|
float or double Tensor .
|
backward_compat
|
bool specifying whether to fall back to returning
FullSpace as the tangent space, and representing R^n with the standard
basis.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
log_prob
|
a Tensor representing the log probability density, of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .
|
tangent_space
|
a TangentSpace object (by default FullSpace )
representing the tangent space to the manifold at value .
|
Raises | |
---|---|
UnspecifiedTangentSpaceError if backward_compat is False and
the _experimental_tangent_space attribute has not been defined.
|
experimental_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
Samples from this distribution and returns the log density of the sample.
The default implementation simply calls sample
and log_prob
:
def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
However, some subclasses may provide more efficient and/or numerically stable implementations.
Args | |
---|---|
sample_shape
|
integer Tensor desired shape of samples to draw.
Default value: () .
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
Default value: None .
|
name
|
name to give to the op.
Default value: 'sample_and_log_prob' .
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
samples
|
a Tensor , or structure of Tensor s, with prepended dimensions
sample_shape .
|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
forward_filter
forward_filter(
x, mask=None, final_step_only=False
)
Run a Kalman filter over a provided sequence of outputs.
Note that the returned values filtered_means
, predicted_means
, and
observation_means
depend on the observed time series x
, while the
corresponding covariances are independent of the observed series; i.e., they
depend only on the model itself. This means that the mean values have shape
concat([sample_shape(x), batch_shape, [num_timesteps,
{latent/observation}_size]])
, while the covariances have shape
concat[(batch_shape, [num_timesteps, {latent/observation}_size,
{latent/observation}_size]])
, which does not depend on the sample shape.
Args | |
---|---|
x
|
a float-type Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable to self.batch_shape ; any further dimensions
are interpreted as a sample shape.
|
mask
|
optional bool-type Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable to self.batch_shape ; any
further dimensions must match or be broadcastable to the sample
shape of x .
Default value: None (falls back to self.mask ).
|
final_step_only
|
optional Python bool . If True , the num_timesteps
dimension is omitted from all return values and only the value from the
final timestep is returned (in this case, log_likelihoods will
be the cumulative log marginal likelihood). This may be significantly
more efficient than returning all values (although note that no
efficiency gain is expected when self.experimental_parallelize=True ).
Default value: False .
|
Returns | |
---|---|
log_likelihoods
|
Per-timestep log marginal likelihoods log
p(x[t] | x[:t-1]) evaluated at the input x , as a Tensor
of shape sample_shape(x) + batch_shape + [num_timesteps].
If final_step_only is True , this will instead be the
cumulative log marginal likelihood at the final step.
|
filtered_means
|
Means of the per-timestep filtered marginal
distributions p(z[t] | x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] .
|
filtered_covs
|
Covariances of the per-timestep filtered marginal
distributions p(z[t] | x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] . Since posterior covariances do not depend on observed
data, some implementations may return a Tensor whose shape omits the
initial sample_shape(x) .
|
predicted_means
|
Means of the per-timestep predictive
distributions over latent states, p(z[t+1] | x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size] .
|
predicted_covs
|
Covariances of the per-timestep predictive
distributions over latent states, p(z[t+1] | x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size, latent_size] . Since posterior covariances
do not depend on observed data, some implementations may return a
Tensor whose shape omits the initial sample_shape(x) .
|
observation_means
|
Means of the per-timestep predictive
distributions over observations, p(x[t] | x[:t-1]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, observation_size] .
|
observation_covs
|
Covariances of the per-timestep predictive
distributions over observations, p(x[t] | x[:t-1]), as a
Tensor of shape sample_shape(x) + batch_shape + [num_timesteps,
observation_size, observation_size] . Since posterior covariances
do not depend on observed data, some implementations may return a
Tensor whose shape omits the initial sample_shape(x) .
|
is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == []
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
is_scalar_batch
|
bool scalar Tensor .
|
is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == []
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
is_scalar_event
|
bool scalar Tensor .
|
kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the Kullback--Leibler divergence.
Denote this distribution (self
) by p
and the other
distribution by
q
. Assuming p, q
are absolutely continuous with respect to reference
measure r
, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
where F
denotes the support of the random variable X ~ p
, H[., .]
denotes (Shannon) cross entropy, and H[.]
denotes (Shannon) entropy.
Args | |
---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
kl_divergence
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the Kullback-Leibler
divergence.
|
latent_size_tensor
latent_size_tensor()
latents_to_observations
latents_to_observations(
latent_means, latent_covs
)
Push latent means and covariances forward through the observation model.
Args | |
---|---|
latent_means
|
float Tensor of shape [..., num_timesteps, latent_size]
|
latent_covs
|
float Tensor of shape
[..., num_timesteps, latent_size, latent_size] .
|
Returns | |
---|---|
observation_means
|
float Tensor of shape
[..., num_timesteps, observation_size]
|
observation_covs
|
float Tensor of shape
[..., num_timesteps, observation_size, observation_size]
|
log_cdf
log_cdf(
value, name='log_cdf', **kwargs
)
Log cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x)
that yields
a more accurate answer than simply taking the logarithm of the cdf
when
x << -1
.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
logcdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_prob
log_prob(
value, name='log_prob', **kwargs
)
Log probability density/mass function.
Additional documentation from LinearGaussianStateSpaceModel
:
kwargs
:
mask
: optional bool-typeTensor
with rightmost dimension[num_timesteps]
;True
values specify that the value ofx
at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable toself.batch_shape
; any further dimensions must match or be broadcastable to the sample shape ofx
. Default value:None
(falls back toself.mask
).
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_survival_function
log_survival_function(
value, name='log_survival_function', **kwargs
)
Log survival function.
Given random variable X
, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1 - cdf(x)
when x >> 1
.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
mean
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
observation_size_tensor
observation_size_tensor()
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
.
Subclasses should override class method _param_shapes
.
Args | |
---|---|
sample_shape
|
Tensor or python list/tuple. Desired shape of a call to
sample() .
|
name
|
name to prepend ops with. |
Returns | |
---|---|
dict of parameter name to Tensor shapes.
|
param_static_shapes
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
. Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes
to return
constant-valued tensors when constant values are fed.
Args | |
---|---|
sample_shape
|
TensorShape or python list/tuple. Desired shape of a call
to sample() .
|
Returns | |
---|---|
dict of parameter name to TensorShape .
|
Raises | |
---|---|
ValueError
|
if sample_shape is a TensorShape and is not fully defined.
|
parameter_properties
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
Returns a dict mapping constructor arg names to property annotations.
This dict should include an entry for each of the distribution's
Tensor
-valued constructor arguments.
Distribution subclasses are not required to implement
_parameter_properties
, so this method may raise NotImplementedError
.
Providing a _parameter_properties
implementation enables several advanced
features, including:
- Distribution batch slicing (
sliced_distribution = distribution[i:j]
). - Automatic inference of
_batch_shape
and_batch_shape_tensor
, which must otherwise be computed explicitly. - Automatic instantiation of the distribution within TFP's internal property tests.
- Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.
In the future, parameter property annotations may enable additional
functionality; for example, returning Distribution instances from
tf.vectorized_map
.
Args | |
---|---|
dtype
|
Optional float dtype to assume for continuous-valued parameters.
Some constraining bijectors require advance knowledge of the dtype
because certain constants (e.g., tfb.Softplus.low ) must be
instantiated with the same dtype as the values to be transformed.
|
num_classes
|
Optional int Tensor number of classes to assume when
inferring the shape of parameters for categorical-like distributions.
Otherwise ignored.
|
Returns | |
---|---|
parameter_properties
|
A
str -> tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.
|
Raises | |
---|---|
NotImplementedError
|
if the distribution class does not implement
_parameter_properties .
|
posterior_marginals
posterior_marginals(
x, mask=None
)
Run a Kalman smoother to return posterior mean and cov.
Note that the returned values smoothed_means
depend on the observed
time series x
, while the smoothed_covs
are independent
of the observed series; i.e., they depend only on the model itself.
This means that the mean values have shape concat([sample_shape(x),
batch_shape, [num_timesteps, {latent/observation}_size]])
,
while the covariances have shape concat[(batch_shape, [num_timesteps,
{latent/observation}_size, {latent/observation}_size]])
, which
does not depend on the sample shape.
This function only performs smoothing. If the user wants the
intermediate values, which are returned by filtering pass forward_filter
,
one could get it by:
(log_likelihoods,
filtered_means, filtered_covs,
predicted_means, predicted_covs,
observation_means, observation_covs) = model.forward_filter(x)
smoothed_means, smoothed_covs = model.backward_smoothing_pass(
filtered_means, filtered_covs,
predicted_means, predicted_covs)
where x
is an observation sequence.
Args | |
---|---|
x
|
a float-type Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable to self.batch_shape ; any further dimensions
are interpreted as a sample shape.
|
mask
|
optional bool-type Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable to self.batch_shape ; any
further dimensions must match or be broadcastable to the sample
shape of x .
Default value: None (falls back to self.mask ).
|
Returns | |
---|---|
smoothed_means
|
Means of the per-timestep smoothed
distributions over latent states, p(z[t] | x[:T]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, observation_size] .
|
smoothed_covs
|
Covariances of the per-timestep smoothed
distributions over latent states, p(z[t] | x[:T]), as a
Tensor of shape sample_shape(mask) + batch_shape + [num_timesteps,
observation_size, observation_size] . Note that the covariances depend
only on the model and the mask, not on the data, so this may have fewer
dimensions than filtered_means .
|
posterior_sample
posterior_sample(
x, sample_shape=(), mask=None, seed=None, name=None
)
Draws samples from the posterior over latent trajectories.
This method uses Durbin-Koopman sampling [1], an efficient algorithm to
sample from the posterior latents of a linear Gaussian state space model.
The cost of drawing a sample is equal to the cost of drawing a prior
sample (.sample(sample_shape)
), plus the cost of Kalman smoothing (
.posterior_marginals(...)
on both the observed time series and the
prior sample. This method is significantly more efficient in graph mode,
because it uses only the posterior means and can elide the unneeded
calculation of marginal covariances.
[1] Durbin, J. and Koopman, S.J. A simple and efficient simulation smoother for state space time series analysis. Biometrika 89(3):603-615, 2002. https://www.jstor.org/stable/4140605
Args | |
---|---|
x
|
a float-type Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable with self.batch_shape .
|
sample_shape
|
int Tensor shape of samples to draw.
Default value: () .
|
mask
|
optional bool-type Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable with self.batch_shape and
x.shape[:-2] .
Default value: None (falls back to self.mask ).
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
|
name
|
Python str name for ops generated by this method.
|
Returns | |
---|---|
latent_posterior_sample
|
Float Tensor of shape
concat([sample_shape, batch_shape, [num_timesteps, latent_size]]) ,
where batch_shape is the broadcast shape of self.batch_shape ,
x.shape[:-2] , and mask.shape[:-1] , representing n samples from
the posterior over latent states given the observed value x .
|
prob
prob(
value, name='prob', **kwargs
)
Probability density/mass function.
Additional documentation from LinearGaussianStateSpaceModel
:
kwargs
:
mask
: optional bool-typeTensor
with rightmost dimension[num_timesteps]
;True
values specify that the value ofx
at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable toself.batch_shape
; any further dimensions must match or be broadcastable to the sample shape ofx
. Default value:None
(falls back toself.mask
).
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
quantile
quantile(
value, name='quantile', **kwargs
)
Quantile function. Aka 'inverse cdf' or 'percent point function'.
Given random variable X
and p in [0, 1]
, the quantile
is:
quantile(p) := x such that P[X <= x] == p
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
quantile
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
sample
sample(
sample_shape=(), seed=None, name='sample', **kwargs
)
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single
sample.
Args | |
---|---|
sample_shape
|
0D or 1D int32 Tensor . Shape of the generated samples.
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
|
name
|
name to give to the op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
samples
|
a Tensor with prepended dimensions sample_shape .
|
stddev
stddev(
name='stddev', **kwargs
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X - E[X])**2]**0.5
where X
is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
stddev
|
Floating-point Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .
|
survival_function
survival_function(
value, name='survival_function', **kwargs
)
Survival function.
Given random variable X
, the survival function is defined:
survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
unnormalized_log_prob
unnormalized_log_prob(
value, name='unnormalized_log_prob', **kwargs
)
Potentially unnormalized log probability density/mass function.
This function is similar to log_prob
, but does not require that the
return value be normalized. (Normalization here refers to the total
integral of probability being one, as it should be by definition for any
probability distribution.) This is useful, for example, for distributions
where the normalization constant is difficult or expensive to compute. By
default, this simply calls log_prob
.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
unnormalized_log_prob
|
a Tensor of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .
|
variance
variance(
name='variance', **kwargs
)
Variance.
Variance is defined as,
Var = E[(X - E[X])**2]
where X
is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
variance
|
Floating-point Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .
|
__getitem__
__getitem__(
slices
)
Slices the batch axes of this distribution, returning a new instance.
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args | |
---|---|
slices
|
slices from the [] operator |
Returns | |
---|---|
dist
|
A new tfd.Distribution instance with sliced parameters.
|
__iter__
__iter__()