tfp.substrates.numpy.sts.ConstrainedSeasonalStateSpaceModel

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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 Zs 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. TheNth effect is represented only implicitly, as the nonzero mean of the firstN - 1effects. Although the computational represention is not symmetric across allNeffects, we derived theConstrainedSeasonalStateSpaceModelby starting with a symmetric representation and imposing only a symmetric constraint (the zero-sum constraint), so the probability model remains symmetric over allN` 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

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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.

batch_shape_tensor

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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.

cdf

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cdf(
    value, name='cdf', **kwargs
)

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

covariance

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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.

cross_entropy

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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.

entropy

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entropy(
    name='entropy', **kwargs
)

Shannon entropy in nats.

event_shape_tensor

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event_shape_tensor(
    name='event_shape_tensor'
)

Shape of a single sample from a single batch as a 1-D int32 Tensor.

experimental_default_event_space_bijector

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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.

experimental_fit

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@classmethod
experimental_fit(
    value, sample_ndims=1, validate_args=False, **init_kwargs
)

Instantiates a distribution that maximizes the likelihood of x.

experimental_local_measure

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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.

experimental_sample_and_log_prob

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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.

forward_filter

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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.

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

kl_divergence

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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.

latent_size_tensor

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latent_size_tensor()

latents_to_observations

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latents_to_observations(
    latent_means, latent_covs
)

Push latent means and covariances forward through the observation model.

log_cdf

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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.

log_prob

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log_prob(
    value, name='log_prob', **kwargs
)

Log probability density/mass function.

Additional documentation from LinearGaussianStateSpaceModel:

kwargs:

• 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).

log_survival_function

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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.

mean

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mean(
    name='mean', **kwargs
)

Mean.

mode

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mode(
    name='mode', **kwargs
)

Mode.

observation_size_tensor

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observation_size_tensor()

param_shapes

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@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.

param_static_shapes

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@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.

parameter_properties

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@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.

posterior_marginals

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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.

posterior_sample

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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

prob

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prob(
    value, name='prob', **kwargs
)

Probability density/mass function.

Additional documentation from LinearGaussianStateSpaceModel:

kwargs:

• 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).

quantile

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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

sample

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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.

stddev

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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.

survival_function

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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).

unnormalized_log_prob

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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.

variance

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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.

__getitem__

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__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]

__iter__

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__iter__()