tfp.distributions.LinearGaussianStateSpaceModel | TensorFlow Probability (original) (raw)

Observation distribution from a linear Gaussian state space model.

Inherits From: AutoCompositeTensorDistribution, Distribution, AutoCompositeTensor

tfp.distributions.LinearGaussianStateSpaceModel(
    num_timesteps,
    transition_matrix,
    transition_noise,
    observation_matrix,
    observation_noise,
    initial_state_prior,
    initial_step=0,
    mask=None,
    experimental_parallelize=False,
    validate_args=False,
    allow_nan_stats=True,
    name='LinearGaussianStateSpaceModel'
)

A linear Gaussian state space model, sometimes called a Kalman filter, posits a latent state vector z[t] of dimension latent_size that evolves over time following linear Gaussian transitions,

z[t+1] = F * z[t] + N(b; Q)  # latent state
x[t] = H * z[t] + N(c; R)    # observed series

for transition matrix F, transition bias b and covariance matrixQ, and observation matrix H, bias c and covariance matrix R. At each timestep, the model generates an observable vector x[t], a noisy projection of the latent state. The transition and observation models may be fixed or may vary between timesteps.

This Distribution represents the marginal distribution on observations, p(x). The marginal log_prob is implemented by Kalman filtering [1], and sample by an efficient forward recursion. Both operations require time linear in T, the total number of timesteps.

Shapes

The event shape is [num_timesteps, observation_size], whereobservation_size is the dimension of each observation x[t]. The observation and transition models must return consistent shapes.

This implementation supports vectorized computation over a batch of models. All of the parameters (prior distribution, transition and observation operators and noise models) must have a consistent batch shape.

Time-varying processes

Any of the model-defining parameters (prior distribution, transition and observation operators and noise models) may be specified as a callable taking an integer timestep t and returning a time-dependent value. The dimensionality (latent_size andobservation_size) must be the same at all timesteps.

Importantly, the timestep is passed as a Tensor, not a Python integer, so any conditional behavior must occur inside the TensorFlow graph. For example, suppose we want to use a different transition model on even days than odd days. It does not work to write

def transition_matrix(t):
  if t % 2 == 0:
    return even_day_matrix
  else:
    return odd_day_matrix

since the value of t is not fixed at graph-construction time. Instead we need to write

def transition_matrix(t):
  return tf.cond(tf.equal(tf.mod(t, 2), 0),
                 lambda : even_day_matrix,
                 lambda : odd_day_matrix)

so that TensorFlow can switch between operators appropriately at runtime.

Examples

Consider a simple tracking model, in which a two-dimensional latent state represents the position of a vehicle, and at each timestep we see a noisy observation of this position (e.g., a GPS reading). The vehicle is assumed to move by a random walk with standard deviationstep_std at each step, and observation noise level std. We build the marginal distribution over noisy observations as a state space model:

tfd = tfp.distributions
ndims = 2
step_std = 1.0
noise_std = 5.0
model = tfd.LinearGaussianStateSpaceModel(
  num_timesteps=100,
  transition_matrix=tf.linalg.LinearOperatorIdentity(ndims),
  transition_noise=tfd.MultivariateNormalDiag(
   scale_diag=step_std**2 * tf.ones([ndims])),
  observation_matrix=tf.linalg.LinearOperatorIdentity(ndims),
  observation_noise=tfd.MultivariateNormalDiag(
   scale_diag=noise_std**2 * tf.ones([ndims])),
  initial_state_prior=tfd.MultivariateNormalDiag(
   scale_diag=tf.ones([ndims])))

using the identity matrix for the transition and observation operators. We can then use this model to generate samples, compute marginal likelihood of observed sequences, and perform posterior inference.

x = model.sample(5) # Sample from the prior on sequences of observations.
lp = model.log_prob(x) # Marginal likelihood of a (batch of) observations.

# Compute the filtered posterior on latent states given observations,
# and extract the mean and covariance for the current (final) timestep.
_, filtered_means, filtered_covs, _, _, _, _ = model.forward_filter(x)
current_location_posterior = tfd.MultivariateNormalTriL(
              loc=filtered_means[..., -1, :],
              scale_tril=tf.linalg.cholesky(filtered_covs[..., -1, :, :]))

# Run a smoothing recursion to extract posterior marginals for locations
# at previous timesteps.
posterior_means, posterior_covs = model.posterior_marginals(x)
initial_location_posterior = tfd.MultivariateNormalTriL(
              loc=posterior_means[..., 0, :],
              scale_tril=tf.linalg.cholesky(posterior_covs[..., 0, :, :]))

Args
num_timesteps	Integer Tensor total number of timesteps.
transition_matrix	A transition operator, represented by a Tensor or LinearOperator of shape [latent_size, latent_size], or by a callable taking as argument a scalar integer Tensor t and returning a Tensor or LinearOperator representing the transition operator from latent state at time t to time t + 1.
transition_noise	An instance oftfd.MultivariateNormalLinearOperator with event shape[latent_size], representing the mean and covariance of the transition noise model, or a callable taking as argument a scalar integer Tensor t and returning such a distribution representing the noise in the transition from time t to time t + 1.
observation_matrix	An observation operator, represented by a Tensor or LinearOperator of shape [observation_size, latent_size], or by a callable taking as argument a scalar integer Tensort and returning a timestep-specific Tensor or LinearOperator.
observation_noise	An instance oftfd.MultivariateNormalLinearOperator with event shape[observation_size], representing the mean and covariance of the observation noise model, or a callable taking as argument a scalar integer Tensor t and returning a timestep-specific noise model.
initial_state_prior	An instance of MultivariateNormalLinearOperatorrepresenting the prior distribution on latent states; must have event shape [latent_size].
initial_step	optional int specifying the time of the first modeled timestep. This is added as an offset when passing timesteps t to (optional) callables specifying timestep-specific transition and observation models.
mask	Optional default missingness mask used for density and posterior inference calculations (any method that takes a mask argument). Bool-type Tensor with rightmost dimension[num_timesteps]; True values specify that the value of xat that timestep is masked, i.e., not conditioned on. Default value: None.
experimental_parallelize	If True, use parallel message passing algorithms from tfp.experimental.parallel_filter to perform operations in O(log num_timesteps) sequential steps. The overall FLOP and memory cost may be larger than for the sequential implementations, though only by a constant factor. Default value: False.
validate_args	Python bool, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
allow_nan_stats	Python bool, default True. If False, raise an exception if a statistic (e.g. mean/mode/etc...) is undefined for any batch member If True, batch members with valid parameters leading to undefined statistics will return NaN for this statistic.
name	The name to give Ops created by the initializer.

Attributes
allow_nan_stats	Python bool describing behavior when a stat is undefined.Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.
batch_shape	Shape of a single sample from a single event index as a TensorShape.May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
dtype	The DType of Tensors handled by this Distribution.
event_shape	Shape of a single sample from a single batch as a TensorShape.May be partially defined or unknown.
experimental_parallelize
experimental_shard_axis_names	The list or structure of lists of active shard axis names.
initial_state_prior
initial_step
mask
name	Name prepended to all ops created by this Distribution.
name_scope	Returns a tf.name_scope instance for this class.
non_trainable_variables	Sequence of non-trainable variables owned by this module and its submodules.
num_timesteps
observation_matrix
observation_noise
parameters	Dictionary of parameters used to instantiate this Distribution.
reparameterization_type	Describes how samples from the distribution are reparameterized.Currently this is one of the static instancestfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.
submodules	Sequence of all sub-modules.Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on). a = tf.Module() b = tf.Module() c = tf.Module() a.b = b b.c = c list(a.submodules) == [b, c] True list(b.submodules) == [c] True list(c.submodules) == [] True
trainable_variables	Sequence of trainable variables owned by this module and its submodules.
transition_matrix
transition_noise
validate_args	Python bool indicating possibly expensive checks are enabled.
variables	Sequence of variables owned by this module and its submodules.

Args
filtered_means	Means of the per-timestep filtered marginal distributions p(z[t] \| x[:t]), as a Tensor of shapesample_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 shapesample_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 shapesample_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 shapesample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]. which is of the same shape as filtered_covs.

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.

Args
other	tfp.distributions.Distribution instance.
name	Python str prepended to names of ops created by this function.

Args
*args	Passed to implementation _default_event_space_bijector.
**kwargs	Passed to implementation _default_event_space_bijector.

Args
value	a Tensor valid sample from this distribution family.
sample_ndims	Positive int Tensor number of leftmost dimensions ofvalue 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 tocls.__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..

Args
value	float or double Tensor.
backward_compat	bool specifying whether to fall back to returningFullSpace 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 shapesample_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.

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 Tensors, with prepended dimensionssample_shape.
log_prob	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

Returns
observation_means	float Tensor of shape[..., num_timesteps, observation_size]
observation_covs	float Tensor of shape[..., num_timesteps, observation_size, observation_size]

Args
x	a float-type Tensor with rightmost dimensions[num_timesteps, observation_size] matchingself.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 xat 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_timestepsdimension 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 Tensorof 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 shapesample_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 shapesample_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).

Args
latent_means	float Tensor of shape [..., num_timesteps, latent_size]
latent_covs	float Tensor of shape[..., num_timesteps, latent_size, latent_size].

Args
sample_shape	Tensor or python list/tuple. Desired shape of a call tosample().
name	name to prepend ops with.

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
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.

Args
x	a float-type Tensor with rightmost dimensions[num_timesteps, observation_size] matchingself.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 xat that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable with self.batch_shape andx.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.

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.

tfp.distributions.LinearGaussianStateSpaceModel | TensorFlow Probability (original) (raw)

Shapes

Time-varying processes

Examples

Methods

backward_smoothing_pass

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

experimental_default_event_space_bijector

experimental_fit

experimental_local_measure

experimental_sample_and_log_prob

forward_filter

is_scalar_batch

is_scalar_event

kl_divergence

latent_size_tensor

latents_to_observations

log_cdf

log_prob

kwargs:

log_survival_function

mean

mode

observation_size_tensor

param_shapes

param_static_shapes

parameter_properties

posterior_marginals

posterior_sample

prob

kwargs:

quantile

sample

stddev

survival_function

unnormalized_log_prob

variance

with_name_scope

__getitem__

__iter__

`backward_smoothing_pass`

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`experimental_default_event_space_bijector`

`experimental_fit`

`experimental_local_measure`

`experimental_sample_and_log_prob`

`forward_filter`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`latent_size_tensor`

`latents_to_observations`

`log_cdf`

`log_prob`

`kwargs`:

`log_survival_function`

`mean`

`mode`

`observation_size_tensor`

`param_shapes`

`param_static_shapes`

`parameter_properties`

`posterior_marginals`

`posterior_sample`

`prob`

`kwargs`:

`quantile`

`sample`

`stddev`

`survival_function`

`unnormalized_log_prob`

`variance`

`with_name_scope`

`getitem`

`iter`