pastas.stats.tests.runs_test
============================

.. py:function:: pastas.stats.tests.runs_test(series: pandas.Series, cutoff: str = 'median') -> tuple[float, float]

   
   Runs test for autocorrelation.

   :param series: Time series to test for autocorrelation.
   :type series: pandas.Series
   :param cutoff: String set to "mean", "median", or a float value to use as the cutoff.
   :type cutoff: str or float, optional

   :returns: * **z_stat** (*float*) -- Runs test statistic.
             * **pval** (*float*) -- p-value for the test statistic, based on a normal distribution.

   .. rubric:: Notes

   Wald and Wolfowitz developed [R04c1b8fcc520-wald_1943]_ developed a distribution free test (
   i.e., no normal distribution is assumed) to test for autocorrelation. This test
   is also appropriate for non-equidistant time steps in the residuals time series.
   The Null-hypothesis is that the residual time series is a random sequence of
   positive and negative values. The alternative hypothesis is that they are
   non-random. The test statistic is computed as follows:

   .. math::

       Z = \frac{R-\bar{R}}{\sigma_R}

   where $R$ is the number of runs, :math:`\bar{R}` the expected number of runs and
   :math:`\sigma_R` the standard deviation of the number of runs. A run is defined
   as the number of sequences of exclusively positive and negative values in the
   time series.

   **Considerations for this test:**

   - Test is also applicable to time series with non-equidistant time steps.

   .. rubric:: References

   .. [R04c1b8fcc520-wald_1943] Wald, A., & Wolfowitz, J. (1943). An exact test for randomness in
      the non-parametric case based on serial correlation. The Annals of
      Mathematical Statistics, 14(4), 378-388.

   .. rubric:: Examples

   >>> res = pd.Series(index=pd.date_range(start=0, periods=1000, freq="D"),
   >>>                 data=np.random.rand(1000))
   >>> stat, pval = ps.stats.runs_test(res)
   >>> if p > alpha:
   >>>     print("Failed to reject the Null-hypothesis, no significant"
   >>>           "autocorrelation. p =", p.round(2))
   >>> else:
   >>>     print("Reject the Null-hypothesis")















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