<br />

|---|
| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/components/local_linear_trend.py#L297-L428) |

Formal representation of a local linear trend model.

Inherits From: [`StructuralTimeSeries`](https://www.tensorflow.org/probability/api_docs/python/tfp/sts/StructuralTimeSeries)

    tfp.sts.LocalLinearTrend(
        level_scale_prior=None,
        slope_scale_prior=None,
        initial_level_prior=None,
        initial_slope_prior=None,
        observed_time_series=None,
        name=None
    )

### Used in the notebooks

| Used in the tutorials |
|---|
| - [Approximate inference for STS models with non-Gaussian observations](https://www.tensorflow.org/probability/examples/STS_approximate_inference_for_models_with_non_Gaussian_observations) - [Structural Time Series Modeling Case Studies: Atmospheric CO2 and Electricity Demand](https://www.tensorflow.org/probability/examples/Structural_Time_Series_Modeling_Case_Studies_Atmospheric_CO2_and_Electricity_Demand) |

The local linear trend model posits a `level` and `slope`, each
evolving via a Gaussian random walk:

    level[t] = level[t-1] + slope[t-1] + Normal(0., level_scale)
    slope[t] = slope[t-1] + Normal(0., slope_scale)

The latent state is the two-dimensional tuple `[level, slope]`. At each
timestep we observe a noisy realization of the current level:
`f[t] = level[t] + Normal(0., observation_noise_scale)`. This model
is appropriate for data where the trend direction and magnitude (latent
`slope`) is consistent within short periods but may evolve over time.

Note that this model can produce very high uncertainty forecasts, as
uncertainty over the slope compounds quickly. If you expect your data to
have nonzero long-term trend, i.e. that slopes tend to revert to some mean,
then the `SemiLocalLinearTrend` model may produce sharper forecasts.

<br />

<br />

<br />

| ## Args ||
|---|---|
| `level_scale_prior` | optional `tfd.Distribution` instance specifying a prior on the `level_scale` parameter. If `None`, a heuristic default prior is constructed based on the provided `observed_time_series`. Default value: `None`. |
| `slope_scale_prior` | optional `tfd.Distribution` instance specifying a prior on the `slope_scale` parameter. If `None`, a heuristic default prior is constructed based on the provided `observed_time_series`. Default value: `None`. |
| `initial_level_prior` | optional `tfd.Distribution` instance specifying a prior on the initial level. If `None`, a heuristic default prior is constructed based on the provided `observed_time_series`. Default value: `None`. |
| `initial_slope_prior` | optional `tfd.Distribution` instance specifying a prior on the initial slope. If `None`, a heuristic default prior is constructed based on the provided `observed_time_series`. Default value: `None`. |
| `observed_time_series` | optional `float` `Tensor` of shape `batch_shape + [T, 1]` (omitting the trailing unit dimension is also supported when `T > 1`), specifying an observed time series. Any `NaN`s are interpreted as missing observations; missingness may be also be explicitly specified by passing a [`tfp.sts.MaskedTimeSeries`](https://www.tensorflow.org/probability/api_docs/python/tfp/sts/MaskedTimeSeries) instance. Any priors not explicitly set will be given default values according to the scale of the observed time series (or batch of time series). Default value: `None`. |
| `name` | the name of this model component. Default value: 'LocalLinearTrend'. |

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<br />

<br />

<br />

| ## Attributes ||
|---|---|
| `batch_shape` | Static batch shape of models represented by this component. |
| `init_parameters` | Parameters used to instantiate this `StructuralTimeSeries`. |
| `initial_state_prior` | Prior distribution on the initial latent state (level and scale). |
| `latent_size` | Python `int` dimensionality of the latent space in this model. |
| `name` | Name of this model component. |
| `parameters` | List of Parameter(name, prior, bijector) namedtuples for this model. |

<br />

## Methods

### `batch_shape_tensor`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L121-L134)

    batch_shape_tensor()

Runtime batch shape of models represented by this component.

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<br />

<br />

| Returns ||
|---|---|
| `batch_shape` | `int` `Tensor` giving the broadcast batch shape of all model parameters. This should match the batch shape of derived state space models, i.e., `self.make_state_space_model(...).batch_shape_tensor()`. |

<br />

### `copy`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L179-L194)

    copy(
        **override_parameters_kwargs
    )

Creates a deep copy.

> [!NOTE]
> **Note:** the copy distribution may continue to depend on the original initialization arguments.

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<br />

<br />

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

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| Returns ||
|---|---|
| `copy` | A new instance of `type(self)` initialized from the union of self.init_parameters and override_parameters_kwargs, i.e., `dict(self.init_parameters, **override_parameters_kwargs)`. |

<br />

### `get_parameter`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L86-L92)

    get_parameter(
        parameter_name
    )

Returns the parameter with the given name, or a KeyError.

### `joint_distribution`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L263-L406)

    joint_distribution(
        observed_time_series=None,
        num_timesteps=None,
        trajectories_shape=(),
        initial_step=0,
        mask=None,
        experimental_parallelize=False
    )

Constructs the joint distribution over parameters and observed values.

<br />

<br />

<br />

| Args ||
|---|---|
| `observed_time_series` | Optional observed time series to model, as a `Tensor` or [`tfp.sts.MaskedTimeSeries`](https://www.tensorflow.org/probability/api_docs/python/tfp/sts/MaskedTimeSeries) instance having shape `concat([batch_shape, trajectories_shape, num_timesteps, 1])`. If an observed time series is provided, the `num_timesteps`, `trajectories_shape`, and `mask` arguments are ignored, and an unnormalized (pinned) distribution over parameter values is returned. Default value: `None`. |
| `num_timesteps` | scalar `int` `Tensor` number of timesteps to model. This must be specified either directly or by passing an `observed_time_series`. Default value: `0`. |
| `trajectories_shape` | `int` `Tensor` shape of sampled trajectories for each set of parameter values. Ignored if an `observed_time_series` is passed. Default value: `()`. |
| `initial_step` | Optional scalar `int` `Tensor` specifying the starting timestep. Default value: `0`. |
| `mask` | Optional `bool` `Tensor` having shape `concat([batch_shape, trajectories_shape, num_timesteps])`, in which `True` entries indicate that the series value at the corresponding step is missing and should be ignored. This argument should be passed only if `observed_time_series` is not specified or does not already contain a missingness mask; it is an error to pass both this argument and an `observed_time_series` value containing a missingness mask. Default value: `None`. |
| `experimental_parallelize` | If `True`, use parallel message passing algorithms from [`tfp.experimental.parallel_filter`](https://www.tensorflow.org/probability/api_docs/python/tfp/experimental/parallel_filter) to perform time series operations in `O(log num_timesteps)` sequential steps. The overall FLOP and memory cost may be larger than for the sequential implementations by a constant factor. Default value: `False`. |

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| Returns ||
|---|---|
| `joint_distribution` | joint distribution of model parameters and observed trajectories. If no `observed_time_series` was specified, this is an instance of `tfd.JointDistributionNamedAutoBatched` with a random variable for each model parameter (with names and order matching `self.parameters`), plus a final random variable `observed_time_series` representing a trajectory(ies) conditioned on the parameters. If `observed_time_series` was specified, the return value is given by `joint_distribution.experimental_pin( observed_time_series=observed_time_series)` where `joint_distribution` is as just described, so it defines an unnormalized posterior distribution over the parameters. |

<br />

#### Example:

The joint distribution can generate prior samples of parameters and
trajectories:

    from matplotlib import pylab as plt
    import tensorflow_probability as tfp

    # Sample and plot 100 trajectories from the prior.
    model = tfp.sts.LocalLinearTrend()
    prior_samples = model.joint_distribution(num_timesteps=200).sample([100])
    plt.plot(
      tf.linalg.matrix_transpose(prior_samples['observed_time_series'][..., 0]))

It also integrates with TFP inference APIs, providing a more flexible
alternative to the STS-specific fitting utilities.

    jd = model.joint_distribution(observed_time_series)

    # Variational inference.
    surrogate_posterior = (
      tfp.experimental.vi.build_factored_surrogate_posterior(
        event_shape=jd.event_shape,
        bijector=jd.experimental_default_event_space_bijector()))
    losses = tfp.vi.fit_surrogate_posterior(
      target_log_prob_fn=jd.unnormalized_log_prob,
      surrogate_posterior=surrogate_posterior,
      optimizer=tf_keras.optimizers.Adam(0.1),
      num_steps=200)
    parameter_samples = surrogate_posterior.sample(50)

    # No U-Turn Sampler.
    samples, kernel_results = tfp.experimental.mcmc.windowed_adaptive_nuts(
      n_draws=500, joint_dist=dist)

### `joint_log_prob`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L454-L520)

    joint_log_prob(
        observed_time_series
    )

Build the joint density `log p(params) + log p(y|params)` as a callable. (deprecated)

> [!WARNING]
> **Deprecated:** THIS FUNCTION IS DEPRECATED. It will be removed after 2022-03-01. Instructions for updating: Please use [`StructuralTimeSeries.joint_distribution(observed_time_series).log_prob`](https://www.tensorflow.org/probability/api_docs/python/tfp/sts/StructuralTimeSeries#joint_distribution)

<br />

<br />

<br />

| Args ||
|---|---|
| `observed_time_series` | Observed `Tensor` trajectories of shape `sample_shape + batch_shape + [num_timesteps, 1]` (the trailing `1` dimension is optional if `num_timesteps > 1`), where `batch_shape` should match `self.batch_shape` (the broadcast batch shape of all priors on parameters for this structural time series model). Any `NaN`s are interpreted as missing observations; missingness may be also be explicitly specified by passing a [`tfp.sts.MaskedTimeSeries`](https://www.tensorflow.org/probability/api_docs/python/tfp/sts/MaskedTimeSeries) instance. |

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| Returns ||
|---|---|
| `log_joint_fn` | A function taking a `Tensor` argument for each model parameter, in canonical order, and returning a `Tensor` log probability of shape `batch_shape`. Note that, *unlike* `tfp.Distributions` `log_prob` methods, the `log_joint` sums over the `sample_shape` from y, so that `sample_shape` does not appear in the output log_prob. This corresponds to viewing multiple samples in `y` as iid observations from a single model, which is typically the desired behavior for parameter inference. |

<br />

### `make_state_space_model`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L219-L248)

    make_state_space_model(
        num_timesteps,
        param_vals,
        initial_state_prior=None,
        initial_step=0,
        **linear_gaussian_ssm_kwargs
    )

Instantiate this model as a Distribution over specified `num_timesteps`.

<br />

<br />

<br />

| Args ||
|---|---|
| `num_timesteps` | Python `int` number of timesteps to model. |
| `param_vals` | a list of `Tensor` parameter values in order corresponding to `self.parameters`, or a dict mapping from parameter names to values. |
| `initial_state_prior` | an optional `Distribution` instance overriding the default prior on the model's initial state. This is used in forecasting ("today's prior is yesterday's posterior"). |
| `initial_step` | optional `int` specifying the initial timestep to model. This is relevant when the model contains time-varying components, e.g., holidays or seasonality. |
| `**linear_gaussian_ssm_kwargs` | Optional additional keyword arguments to to the base `tfd.LinearGaussianStateSpaceModel` constructor. |

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<br />

<br />

| Returns ||
|---|---|
| `dist` | a `LinearGaussianStateSpaceModel` Distribution object. |

<br />

### `prior_sample`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L408-L452)

    prior_sample(
        num_timesteps,
        initial_step=0,
        params_sample_shape=(),
        trajectories_sample_shape=(),
        seed=None
    )

Sample from the joint prior over model parameters and trajectories. (deprecated)

> [!WARNING]
> **Deprecated:** THIS FUNCTION IS DEPRECATED. It will be removed after 2022-03-01. Instructions for updating: Please use `StructuralTimeSeries.joint_distribution(...).sample()`

<br />

<br />

<br />

| Args ||
|---|---|
| `num_timesteps` | Scalar `int` `Tensor` number of timesteps to model. |
| `initial_step` | Optional scalar `int` `Tensor` specifying the starting timestep. Default value: 0. |
| `params_sample_shape` | Number of possible worlds to sample iid from the parameter prior, or more generally, `Tensor` `int` shape to fill with iid samples. Default value: `[]` (i.e., draw a single sample and don't expand the shape). |
| `trajectories_sample_shape` | For each sampled set of parameters, number of trajectories to sample, or more generally, `Tensor` `int` shape to fill with iid samples. Default value: `[]` (i.e., draw a single sample and don't expand the shape). |
| `seed` | PRNG seed; see [`tfp.random.sanitize_seed`](https://www.tensorflow.org/probability/api_docs/python/tfp/random/sanitize_seed) for details. Default value: `None`. |

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| Returns ||
|---|---|
| `trajectories` | `float` `Tensor` of shape `trajectories_sample_shape + params_sample_shape + [num_timesteps, 1]` containing all sampled trajectories. |
| `param_samples` | list of sampled parameter value `Tensor`s, in order corresponding to `self.parameters`, each of shape `params_sample_shape + prior.batch_shape + prior.event_shape`. |

<br />

### `__add__`

[View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/structural_time_series.py#L136-L177)

    __add__(
        other
    )

Models the sum of the series from the two components.