code: def test_naive_predict_var_backwards(
      strategy, sp, window_length, n_periods
  ):
      """Checks naive prediction variance computations.
  
      Test whether, given h=1 and large T, the forecast standard error
      is approximately equal to the residual standard errors.
      This property is noted in the Forecasting: Principles and
      Practice textbook (FPP3) [1]_.
  
      More specifically, predict_var computes the forecast standard errors
      (and hence variance) from the residuals standard errors times
      some constant. According to FPP3, this operation can be fully inverted.
      Hence, for this unit test, we redo our computations backwards and
      check that our results are approximately equal.
  
      References
      ----------
      .. [1] https://otexts.com/fpp3/prediction-intervals.html#benchmark-methods
      """
      mu, sigma = 0.0, 10.0
      fake_idx = pd.date_range("1980", periods=n_periods + 1, freq="H")
      np.random.seed(42)
      y = pd.Series(
          np.random.normal(mu, sigma, size=n_periods),
          index=fake_idx[:-1],
      )
  
      h = 1
      forecaster = NaiveForecaster(
          strategy, sp=sp, window_length=window_length
      )
      sigma2 = forecaster.fit(y).predict_var(fh=h)
      sigma = np.sqrt(sigma2).iloc[0, 0]
  
      T = len(y.dropna())
      if strategy == "last":
          # This is trivial because square root of (h) when h=1 is just 1
          sigma_res = sigma / np.sqrt(h)
      elif strategy == "mean":
          sigma_res = sigma / np.sqrt(1 + (1 / T))
      else:
          sigma_res = sigma / np.sqrt(1 + (1 / (T - 1)))
  
      upper_bound, lower_bound = (sigma + 0.001), (sigma - 0.001)
      assert lower_bound < sigma_res < upper_bound
  
imports: import numpy
  import pandas
  import pytest
  from sktime.datatypes._utilities import get_cutoff
  from sktime.forecasting.base import ForecastingHorizon
  from sktime.forecasting.naive import NaiveForecaster
  from sktime.forecasting.tests._config import TEST_OOS_FHS, TEST_SPS, TEST_WINDOW_LENGTHS_INT
  from sktime.utils._testing.forecasting import _assert_correct_pred_time_index
  from sktime.utils.validation.forecasting import check_fh
label_desc: Use image node_id 10 for calling a global function with example usage: test_naive_predict_var_backwards(strategy, sp, window_length, n_periods) without return types