code: def test_cross_validation(self, ts_short, parallel_method, backend):
      m = Prophet(stan_backend=backend)
      m.fit(ts_short)
      # Calculate the number of cutoff points(k)
      horizon = pd.Timedelta("4 days")
      period = pd.Timedelta("10 days")
      initial = pd.Timedelta("115 days")
      df_cv = diagnostics.cross_validation(
          m,
          horizon="4 days",
          period="10 days",
          initial="115 days",
          parallel=parallel_method,
      )
      assert len(np.unique(df_cv["cutoff"])) == 3
      assert max(df_cv["ds"] - df_cv["cutoff"]) == horizon
      assert min(df_cv["cutoff"]) >= min(ts_short["ds"]) + initial
      dc = df_cv["cutoff"].diff()
      dc = dc[dc > pd.Timedelta(0)].min()
      assert dc >= period
      assert (df_cv["cutoff"] < df_cv["ds"]).all()
      # Each y in df_cv and ts_short with same ds should be equal
      df_merged = pd.merge(df_cv, ts_short, "left", on="ds")
      assert np.sum(
          (df_merged["y_x"] - df_merged["y_y"]) ** 2
      ) == pytest.approx(0.0)
      df_cv = diagnostics.cross_validation(
          m, horizon="4 days", period="10 days", initial="135 days"
      )
      assert len(np.unique(df_cv["cutoff"])) == 1
      with pytest.raises(ValueError):
          diagnostics.cross_validation(
              m, horizon="10 days", period="10 days", initial="140 days"
          )
  
imports: import datetime
  import itertools
  import numpy
  import pandas
  import pytest
  from prophet import Prophet, diagnostics
label_desc: Use image node_id 1 for calling the TestCrossValidation obj's underlying member method code with example usage: obj.test_cross_validation(ts_short, parallel_method, backend) without return types