## Class `SeasonalStateSpaceModel`

Inherits From: `LinearGaussianStateSpaceModel`

State space model for a seasonal effect.

A state space model (SSM) posits a set of latent (unobserved) variables that
evolve over time with dynamics specified by a probabilistic transition model
`p(z[t+1] | z[t])`

. At each timestep, we observe a value sampled from an
observation model conditioned on the current state, `p(x[t] | z[t])`

. The
special case where both the transition and observation models are Gaussians
with mean specified as a linear function of the inputs, is known as a linear
Gaussian state space model and supports tractable exact probabilistic
calculations; see `tfp.distributions.LinearGaussianStateSpaceModel`

for
details.

A seasonal effect model is a special case of a linear Gaussian SSM. The latent states represent an unknown effect from each of several 'seasons'; these are generally not meteorological seasons, but represent regular recurring patterns such as hour-of-day or day-of-week effects. The effect of each season drifts from one occurrence to the next, following a Gaussian random walk:

```
effects[season, occurrence[i]] = (
effects[season, occurrence[i-1]] + Normal(loc=0., scale=drift_scale))
```

The latent state has dimension `num_seasons`

, containing one effect for each
seasonal component. The parameters `drift_scale`

and
`observation_noise_scale`

are each (a batch of) scalars. The batch shape of
this `Distribution`

is the broadcast batch shape of these parameters and of
the `initial_state_prior`

.

#### Mathematical Details

The seasonal effect model implements a
`tfp.distributions.LinearGaussianStateSpaceModel`

with
`latent_size = num_seasons`

and `observation_size = 1`

. The latent state
is organized so that the *current* seasonal effect is always in the first
(zeroth) dimension. The transition model rotates the latent state to shift
to a new effect at the end of each season:

```
transition_matrix[t] = (permutation_matrix([1, 2, ..., num_seasons-1, 0])
if season_is_changing(t)
else eye(num_seasons)
transition_noise[t] ~ Normal(loc=0., scale_diag=(
[drift_scale, 0, ..., 0]
if season_is_changing(t)
else [0, 0, ..., 0]))
```

where `season_is_changing(t)`

is `True`

if ```
t `mod`
sum(num_steps_per_season)
```

is in the set of final days for each season,
given by `cumsum(num_steps_per_season) - 1`

. The observation model always
picks out the effect for the current season, i.e., the first element of
the latent state:

```
observation_matrix = [[1., 0., ..., 0.]]
observation_noise ~ Normal(loc=0, scale=observation_noise_scale)
```

#### Examples

A state-space model with day-of-week seasonality on hourly data:

```
day_of_week = SeasonalStateSpaceModel(
num_timesteps=30,
num_seasons=7,
drift_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([7], 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 = SeasonalStateSpaceModel(
num_timesteps=2 * 365, # 2 years
num_seasons=12,
drift_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([12], 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). A general implementation of
month-of-year seasonality would require additional logic; this
version works over time periods not involving a leap year.

`__init__`

```
__init__(
num_timesteps,
num_seasons,
drift_scale,
initial_state_prior,
observation_noise_scale=0.0,
num_steps_per_season=1,
initial_step=0,
validate_args=False,
allow_nan_stats=True,
name=None
)
```

Build a state space model implementing seasonal effects.

#### Args:

: Scalar`num_timesteps`

`int`

`Tensor`

number of timesteps to model with this distribution.: Scalar Python`num_seasons`

`int`

number of seasons.: Scalar (any additional dimensions are treated as batch dimensions)`drift_scale`

`float`

`Tensor`

indicating the standard deviation of the change in effect between consecutive occurrences of a given season. This is assumed to be the same for all seasons.: instance of`initial_state_prior`

`tfd.MultivariateNormal`

representing the prior distribution on latent states; must have event shape`[num_seasons]`

.: Scalar (any additional dimensions are treated as batch dimensions)`observation_noise_scale`

`float`

`Tensor`

indicating the standard deviation of the observation noise. Default value: 0.: Python`num_steps_per_season`

`int`

number of steps in each season. This may be either a scalar (shape`[]`

), in which case all seasons have the same length, or a NumPy array of shape`[num_seasons]`

. Default value: 1.: Optional scalar`initial_step`

`int`

`Tensor`

specifying the starting timestep. Default value: 0.: Python`validate_args`

`bool`

. Whether to validate input with asserts. If`validate_args`

is`False`

, and the inputs are invalid, correct behavior is not guaranteed. Default value:`False`

.: Python`allow_nan_stats`

`bool`

. 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. Default value:`True`

.: Python`name`

`str`

name prefixed to ops created by this class. Default value: "SeasonalStateSpaceModel".

#### Raises:

: if`ValueError`

`num_steps_per_season`

has invalid shape (neither scalar nor`[num_seasons]`

).

## Properties

`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.

#### Returns:

: Python`allow_nan_stats`

`bool`

.

`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.

#### Returns:

:`batch_shape`

`TensorShape`

, possibly unknown.

`drift_scale`

Standard deviation of the drift in effects between seasonal cycles.

`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.

#### Returns:

:`event_shape`

`TensorShape`

, possibly unknown.

`name`

Name prepended to all ops created by this `Distribution`

.

`num_seasons`

Number of seasons.

`num_steps_per_season`

Number of steps in each season.

`observation_noise_scale`

Standard deviation of the 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`

.

#### Returns:

An instance of `ReparameterizationType`

.

`validate_args`

Python `bool`

indicating possibly expensive checks are enabled.

## Methods

`backward_smoothing_pass`

```
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.

#### Args:

: Means of the per-timestep filtered marginal distributions p(z`filtered_means`

*t | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep filtered marginal distributions p(z`filtered_covs`

*t | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.: Means of the per-timestep predictive distributions over latent states, p(z`predicted_means`

*{t+1} | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep predictive distributions over latent states, p(z`predicted_covs`

*{t+1} | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.

#### Returns:

: Means of the smoothed marginal distributions p(z`posterior_means`

*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.: Covariances of the smoothed marginal distributions p(z`posterior_covs`

*t | x*{1:T}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

. which is of the same shape as filtered_covs.

`batch_shape_tensor`

```
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.

#### Args:

: name to give to the op`name`

#### Returns:

:`batch_shape`

`Tensor`

.

`cdf`

```
cdf(
value,
name='cdf'
)
```

Cumulative distribution function.

Given random variable `X`

, the cumulative distribution function `cdf`

is:

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

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`cdf`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`copy`

```
copy(**override_parameters_kwargs)
```

Creates a deep copy of the distribution.

#### Args:

: String/value dictionary of initialization arguments to override with new values.`**override_parameters_kwargs`

#### Returns:

: A new instance of`distribution`

`type(self)`

initialized from the union of self.parameters and override_parameters_kwargs, i.e.,`dict(self.parameters, **override_parameters_kwargs)`

.

`covariance`

```
covariance(name='covariance')
```

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:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`covariance`

`Tensor`

with shape`[B1, ..., Bn, k', k']`

where the first`n`

dimensions are batch coordinates and`k' = reduce_prod(self.event_shape)`

.

`cross_entropy`

```
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`

.

#### Args:

:`other`

`tfp.distributions.Distribution`

instance.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`cross_entropy`

`self.dtype`

`Tensor`

with shape`[B1, ..., Bn]`

representing`n`

different calculations of (Shannon) cross entropy.

`entropy`

```
entropy(name='entropy')
```

Shannon entropy in nats.

`event_shape_tensor`

```
event_shape_tensor(name='event_shape_tensor')
```

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

.

#### Args:

: name to give to the op`name`

#### Returns:

:`event_shape`

`Tensor`

.

`forward_filter`

```
forward_filter(x)
```

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.

#### Args:

: a float-type`x`

`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.

#### Returns:

: Per-timestep log marginal likelihoods`log_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].`

: Means of the per-timestep filtered marginal distributions p(z`filtered_means`

*t | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep filtered marginal distributions p(z`filtered_covs`

*t | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.: Means of the per-timestep predictive distributions over latent states, p(z`predicted_means`

*{t+1} | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep predictive distributions over latent states, p(z`predicted_covs`

*{t+1} | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.: Means of the per-timestep predictive distributions over observations, p(x`observation_means`

*{t} | x*{:t-1}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, observation_size]`

.: Covariances of the per-timestep predictive distributions over observations, p(x`observation_covs`

*{t} | x*{:t-1}), as a Tensor of shape`batch_shape + [num_timesteps, observation_size, observation_size]`

.

`is_scalar_batch`

```
is_scalar_batch(name='is_scalar_batch')
```

Indicates that `batch_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_batch`

`bool`

scalar`Tensor`

.

`is_scalar_event`

```
is_scalar_event(name='is_scalar_event')
```

Indicates that `event_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_event`

`bool`

scalar`Tensor`

.

`kl_divergence`

```
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.

#### Args:

:`other`

`tfp.distributions.Distribution`

instance.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`kl_divergence`

`self.dtype`

`Tensor`

with shape`[B1, ..., Bn]`

representing`n`

different calculations of the Kullback-Leibler divergence.

`log_cdf`

```
log_cdf(
value,
name='log_cdf'
)
```

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`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`logcdf`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_prob`

```
log_prob(
value,
name='log_prob'
)
```

Log probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`log_prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_survival_function`

```
log_survival_function(
value,
name='log_survival_function'
)
```

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`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`mean`

```
mean(name='mean')
```

Mean.

`mode`

```
mode(name='mode')
```

Mode.

`param_shapes`

```
param_shapes(
cls,
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`

.

#### Args:

:`sample_shape`

`Tensor`

or python list/tuple. Desired shape of a call to`sample()`

.: name to prepend ops with.`name`

#### Returns:

`dict`

of parameter name to `Tensor`

shapes.

`param_static_shapes`

```
param_static_shapes(
cls,
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.

#### Args:

:`sample_shape`

`TensorShape`

or python list/tuple. Desired shape of a call to`sample()`

.

#### Returns:

`dict`

of parameter name to `TensorShape`

.

#### Raises:

: if`ValueError`

`sample_shape`

is a`TensorShape`

and is not fully defined.

`posterior_marginals`

```
posterior_marginals(x)
```

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(x)
```

where `x`

is an observation sequence.

#### Args:

: a float-type`x`

`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.

#### Returns:

: Means of the per-timestep smoothed distributions over latent states, p(x`smoothed_means`

*{t} | x*{:T}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, observation_size]`

.: Covariances of the per-timestep smoothed distributions over latent states, p(x`smoothed_covs`

*{t} | x*{:T}), as a Tensor of shape`batch_shape + [num_timesteps, observation_size, observation_size]`

.

`prob`

```
prob(
value,
name='prob'
)
```

Probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`quantile`

```
quantile(
value,
name='quantile'
)
```

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
```

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: a`quantile`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`sample`

```
sample(
sample_shape=(),
seed=None,
name='sample'
)
```

Generate samples of the specified shape.

Note that a call to `sample()`

without arguments will generate a single
sample.

#### Args:

: 0D or 1D`sample_shape`

`int32`

`Tensor`

. Shape of the generated samples.: Python integer seed for RNG`seed`

: name to give to the op.`name`

#### Returns:

: a`samples`

`Tensor`

with prepended dimensions`sample_shape`

.

`stddev`

```
stddev(name='stddev')
```

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`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`stddev`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.

`survival_function`

```
survival_function(
value,
name='survival_function'
)
```

Survival function.

Given random variable `X`

, the survival function is defined:

```
survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
```

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`variance`

```
variance(name='variance')
```

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`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

: Floating-point`variance`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.