Observation distribution from a linear Gaussian state space model

The 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) for transition matrix F, bias b and covariance matrix Q. At each timestep, we observe a noisy projection of the latent state x_t = H * z_t + N(c; R). The transition and observation models may be fixed or may vary between timesteps.

tfd_linear_gaussian_state_space_model(
  num_timesteps,
  transition_matrix,
  transition_noise,
  observation_matrix,
  observation_noise,
  initial_state_prior,
  initial_step = 0L,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "LinearGaussianStateSpaceModel"
)

Arguments

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 integer 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.
validate_args	Logical, 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.
allow_nan_stats	Logical, default TRUE. When TRUE, statistics (e.g., mean, mode, variance) use the value NaN to indicate the result is undefined. When FALSE, an exception is raised if one or more of the statistic's batch members are undefined.
name	name prefixed to Ops created by this class.

Value

a distribution instance.

Details

This Distribution represents the marginal distribution on observations, p(x). The marginal log_prob is computed by Kalman filtering, 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

transition_matrix <- function(t) {
if(t %% 2 == 0) even_day_matrix else odd_day_matrix
}

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

transition_matrix <- function(t) {
tf$cond(tf$equal(tf$mod(t, 2), 0), function() even_day_matrix, function() odd_day_matrix)
}

so that TensorFlow can switch between operators appropriately at runtime.

See also

For usage examples see e.g. tfd_sample(), tfd_log_prob(), tfd_mean().

Other distributions: tfd_autoregressive(), tfd_batch_reshape(), tfd_bates(), tfd_bernoulli(), tfd_beta_binomial(), tfd_beta(), tfd_binomial(), tfd_categorical(), tfd_cauchy(), tfd_chi2(), tfd_chi(), tfd_cholesky_lkj(), tfd_continuous_bernoulli(), tfd_deterministic(), tfd_dirichlet_multinomial(), tfd_dirichlet(), tfd_empirical(), tfd_exp_gamma(), tfd_exp_inverse_gamma(), tfd_exponential(), tfd_gamma_gamma(), tfd_gamma(), tfd_gaussian_process_regression_model(), tfd_gaussian_process(), tfd_generalized_normal(), tfd_geometric(), tfd_gumbel(), tfd_half_cauchy(), tfd_half_normal(), tfd_hidden_markov_model(), tfd_horseshoe(), tfd_independent(), tfd_inverse_gamma(), tfd_inverse_gaussian(), tfd_johnson_s_u(), tfd_joint_distribution_named_auto_batched(), tfd_joint_distribution_named(), tfd_joint_distribution_sequential_auto_batched(), tfd_joint_distribution_sequential(), tfd_kumaraswamy(), tfd_laplace(), tfd_lkj(), tfd_log_logistic(), tfd_log_normal(), tfd_logistic(), tfd_mixture_same_family(), tfd_mixture(), tfd_multinomial(), tfd_multivariate_normal_diag_plus_low_rank(), tfd_multivariate_normal_diag(), tfd_multivariate_normal_full_covariance(), tfd_multivariate_normal_linear_operator(), tfd_multivariate_normal_tri_l(), tfd_multivariate_student_t_linear_operator(), tfd_negative_binomial(), tfd_normal(), tfd_one_hot_categorical(), tfd_pareto(), tfd_pixel_cnn(), tfd_poisson_log_normal_quadrature_compound(), tfd_poisson(), tfd_power_spherical(), tfd_probit_bernoulli(), tfd_quantized(), tfd_relaxed_bernoulli(), tfd_relaxed_one_hot_categorical(), tfd_sample_distribution(), tfd_sinh_arcsinh(), tfd_skellam(), tfd_spherical_uniform(), tfd_student_t_process(), tfd_student_t(), tfd_transformed_distribution(), tfd_triangular(), tfd_truncated_cauchy(), tfd_truncated_normal(), tfd_uniform(), tfd_variational_gaussian_process(), tfd_vector_diffeomixture(), tfd_vector_exponential_diag(), tfd_vector_exponential_linear_operator(), tfd_vector_laplace_diag(), tfd_vector_laplace_linear_operator(), tfd_vector_sinh_arcsinh_diag(), tfd_von_mises_fisher(), tfd_von_mises(), tfd_weibull(), tfd_wishart_linear_operator(), tfd_wishart_tri_l(), tfd_wishart(), tfd_zipf()