pastas.stats.tests.durbin_watson
================================

.. py:function:: pastas.stats.tests.durbin_watson(series: pandas.Series) -> float

   
   Durbin-Watson test for autocorrelation.

   :param series: residuals series.
   :type series: pandas.Series, optional

   :returns: **dw_stat** -- The method returns the Durbin-Watson test statistic.
   :rtype: float

   .. rubric:: Notes

   The Durban Watson statistic ([R0fd492ec46e4-durbin_1951]_, [R0fd492ec46e4-Fahidy_2004]_) tests the
   null-hypothesis that the correlation between the noise values at lag one equals
   zero. The formula to calculate the Durbin-Watson statistic (DW) is:

   .. math::

       DW = \frac{\sum_{t=2}^{n}(\upsilon_t-\upsilon_{t-1}^2)}
       {\sum_{t=1}^{n}\upsilon_t^2}

   where $n$ is the number of values in the noise series. The test-statistic has a
   range :math:`0 \geq DW \leq 4`, where values of $DW < 2$ indicate a positive
   correlation and values of $DW > 2$ indicates negative autocorrelation. The
   Durbin-Watson test requires a constant time interval of the noise series and
   tests for autocorrelation at a lag of 1 time step.

   **Considerations for this test:**

   - The time series should have equidistant time steps.
   - The Durbin-Watson test tests for autocorrelation at lag 1 but not for
     larger time lags.
   - The test statistic for this test is difficult to compute and is usually
     obtained from pre-calculated tables.

   .. rubric:: References

   .. [R0fd492ec46e4-durbin_1951] Durbin, J., & Watson, G. S. (1951). Testing for serial
      correlation in least squares regression. II. Biometrika, 38(1/2), 159-177.

   .. [R0fd492ec46e4-Fahidy_2004] Fahidy, T. Z. (2004). On the Application of Durbin-Watson
     Statistics to Time-Series-Based Regression Models. CHEMICAL ENGINEERING
     EDUCATION, 38(1), 22-25.

   .. rubric:: Examples

   >>> data = pd.Series(index=pd.date_range(start=0, periods=1000, freq="D"),
   >>>                data=np.random.rand(1000))
   >>> result = ps.stats.durbin_watson(data)















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