tfp.substrates.jax.distributions.LinearGaussianStateSpaceModel

Observation distribution from a linear Gaussian state space model.

Inherits From: AutoCompositeTensorDistribution, Distribution

View aliases

Main aliases

tfp.experimental.substrates.jax.distributions.LinearGaussianStateSpaceModel

tfp.substrates.jax.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 matrix Q, 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], where observation_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 and observation_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 deviation step_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 of `tfd.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 Tensor `t` and returning a timestep-specific Tensor or LinearOperator.
`observation_noise`	An instance of `tfd.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 `MultivariateNormalLinearOperator` representing 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 `x` at 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 `Tensor`s 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`.
`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 instances `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.
`trainable_variables`
`transition_matrix`
`transition_noise`
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.
`variables`

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.

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 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.`.

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`.

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`.

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]` 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)`.

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 to `sample()`.
`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]` 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.

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.substrates.jax.distributions.LinearGaussianStateSpaceModel

View aliases

Shapes

Time-varying processes

Examples

Args

Attributes

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

__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`

`getitem`

`iter`