tfp.sts.one_step_predictive

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Compute one-step-ahead predictive distributions for all timesteps.

Used in the notebooks

Used in the tutorials

Given samples from the posterior over parameters, return the predictive distribution over observations at each time T, given observations up through time T-1.

model An instance of StructuralTimeSeries representing a time-series model. This represents a joint distribution over time-series and their parameters with batch shape [b1, ..., bN].
observed_time_series float Tensor of shape concat([sample_shape, model.batch_shape, [num_timesteps, 1]]) wheresample_shapecorresponds to i.i.d. observations, and the trailing[1]dimension may (optionally) be omitted ifnum_timesteps > 1. May optionally be an instance of <a href="../../tfp/sts/MaskedTimeSeries"><code>tfp.sts.MaskedTimeSeries</code></a> including a maskTensorto encode the locations of missing observations. </td> </tr><tr> <td>parameter_samples</td> <td> PythonlistofTensorsrepresenting posterior samples of model parameters, with shapes[concat([[num_posterior_draws], param.prior.batch_shape, param.prior.event_shape]) for param in model.parameters]. This may optionally also be a map (Pythondict) of parameter names toTensor` values.

forecast_dist a tfd.MixtureSameFamily instance with event shape [num_timesteps] and batch shape concat([sample_shape, model.batch_shape]), with num_posterior_draws mixture components. The tth step represents the forecast distribution p(observed_time_series[t] | observed_time_series[0:t-1], parameter_samples).

Examples

Suppose we've built a model and fit it to data using HMC:

  day_of_week = tfp.sts.Seasonal(
      num_seasons=7,
      observed_time_series=observed_time_series,
      name='day_of_week')
  local_linear_trend = tfp.sts.LocalLinearTrend(
      observed_time_series=observed_time_series,
      name='local_linear_trend')
  model = tfp.sts.Sum(components=[day_of_week, local_linear_trend],
                      observed_time_series=observed_time_series)

  samples, kernel_results = tfp.sts.fit_with_hmc(model, observed_time_series)

Passing the posterior samples into one_step_predictive, we construct a one-step-ahead predictive distribution:

  one_step_predictive_dist = tfp.sts.one_step_predictive(
    model, observed_time_series, parameter_samples=samples)

  predictive_means = one_step_predictive_dist.mean()
  predictive_scales = one_step_predictive_dist.stddev()

If using variational inference instead of HMC, we'd construct a forecast using samples from the variational posterior:

  surrogate_posterior = tfp.sts.build_factored_surrogate_posterior(
    model=model)
  loss_curve = tfp.vi.fit_surrogate_posterior(
    target_log_prob_fn=model.joint_log_prob(observed_time_series),
    surrogate_posterior=surrogate_posterior,
    optimizer=tf.optimizers.Adam(learning_rate=0.1),
    num_steps=200)
  samples = surrogate_posterior.sample(30)

  one_step_predictive_dist = tfp.sts.one_step_predictive(
    model, observed_time_series, parameter_samples=samples)

We can visualize the forecast by plotting:

  from matplotlib import pylab as plt
  def plot_one_step_predictive(observed_time_series,
                               forecast_mean,
                               forecast_scale):
    plt.figure(figsize=(12, 6))
    num_timesteps = forecast_mean.shape[-1]
    c1, c2 = (0.12, 0.47, 0.71), (1.0, 0.5, 0.05)
    plt.plot(observed_time_series, label="observed time series", color=c1)
    plt.plot(forecast_mean, label="one-step prediction", color=c2)
    plt.fill_between(np.arange(num_timesteps),
                     forecast_mean - 2 * forecast_scale,
                     forecast_mean + 2 * forecast_scale,
                     alpha=0.1, color=c2)
    plt.legend()

  plot_one_step_predictive(observed_time_series,
                           forecast_mean=predictive_means,
                           forecast_scale=predictive_scales)

To detect anomalous timesteps, we check whether the observed value at each step is within a 95% predictive interval, i.e., two standard deviations from the mean:

  z_scores = ((observed_time_series[..., 1:] - predictive_means[..., :-1])
               / predictive_scales[..., :-1])
  anomalous_timesteps = tf.boolean_mask(
      tf.range(1, num_timesteps),
      tf.abs(z_scores) > 2.0)