# tfp.substrates.numpy.distributions.LinearGaussianStateSpaceModel

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Observation distribution from a linear Gaussian state space model.

Inherits From: `Distribution`

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, :, :]))
``````

*

`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.

`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`

## Methods

### `backward_smoothing_pass`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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)`, (Shann