code: def _evaluate_grid_hyperparameters(
      forecaster,
      y: pd.Series,
      param_grid: dict,
      steps: int,
      metric: Union[str, Callable, list],
      initial_train_size: int,
      fixed_train_size: bool = True,
      gap: int = 0,
      allow_incomplete_fold: bool = True,
      exog: Optional[Union[pd.Series, pd.DataFrame]] = None,
      lags_grid: Optional[Union[list, dict]] = None,
      refit: Optional[Union[bool, int]] = False,
      return_best: bool = True,
      n_jobs: Optional[Union[int, str]] = "auto",
      verbose: bool = True,
      show_progress: bool = True,
  ) -> pd.DataFrame:
      """
      Evaluate parameter values for a Forecaster object using time series backtesting.
  
      Parameters
      ----------
      forecaster : ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect
          Forecaster model.
      y : pandas Series
          Training time series.
      param_grid : dict
          Dictionary with parameters names (`str`) as keys and lists of parameter
          settings to try as values.
      steps : int
          Number of steps to predict.
      metric : str, Callable, list
          Metric used to quantify the goodness of fit of the model.
  
          - If `string`: {'mean_squared_error', 'mean_absolute_error',
          'mean_absolute_percentage_error', 'mean_squared_log_error'}
          - If `Callable`: Function with arguments y_true, y_pred that returns
          a float.
          - If `list`: List containing multiple strings and/or Callables.
      initial_train_size : int
          Number of samples in the initial train split.
      fixed_train_size : bool, default `True`
          If True, train size doesn't increase but moves by `steps` in each iteration.
      gap : int, default `0`
          Number of samples to be excluded after the end of each training set and
          before the test set.
      allow_incomplete_fold : bool, default `True`
          Last fold is allowed to have a smaller number of samples than the
          `test_size`. If `False`, the last fold is excluded.
      exog : pandas Series, pandas DataFrame, default `None`
          Exogenous variable/s included as predictor/s. Must have the same
          number of observations as `y` and should be aligned so that y[i] is
          regressed on exog[i].
      lags_grid : list, dict, default `None`
          Lists of lags to try, containing int, lists, numpy ndarray, or range
          objects. If `dict`, the keys are used as labels in the `results`
          DataFrame, and the values are used as the lists of lags to try. Ignored
          if the forecaster is an instance of `ForecasterAutoregCustom` or
          `ForecasterAutoregMultiSeriesCustom`.
      refit : bool, int, default `False`
          Whether to re-fit the forecaster in each iteration. If `refit` is an integer,
          the Forecaster will be trained every that number of iterations.
      return_best : bool, default `True`
          Refit the `forecaster` using the best found parameters on the whole data.
      n_jobs : int, 'auto', default `'auto'`
          The number of jobs to run in parallel. If `-1`, then the number of jobs is
          set to the number of cores. If 'auto', `n_jobs` is set using the function
          skforecast.utils.select_n_jobs_backtesting.
          **New in version 0.9.0**
      verbose : bool, default `True`
          Print number of folds used for cv or backtesting.
      show_progress: bool, default `True`
          Whether to show a progress bar.
  
      Returns
      -------
      results : pandas DataFrame
          Results for each combination of parameters.
  
          - column lags: lags configuration for each iteration.
          - column params: parameters configuration for each iteration.
          - column metric: metric value estimated for each iteration.
          - additional n columns with param = value.
  
      """
  
      if return_best and exog is not None and (len(exog) != len(y)):
          raise ValueError(
              (
                  f"`exog` must have same number of samples as `y`. "
                  f"length `exog`: ({len(exog)}), length `y`: ({len(y)})"
              )
          )
  
      lags_grid, lags_label = initialize_lags_grid(
          forecaster, lags_grid
      )
  
      lags_list = []
      params_list = []
      if not isinstance(metric, list):
          metric = [metric]
      metric_dict = {
          (m if isinstance(m, str) else m.__name__): [] for m in metric
      }
  
      if len(metric_dict) != len(metric):
          raise ValueError(
              "When `metric` is a `list`, each metric name must be unique."
          )
  
      print(
          f"Number of models compared: {len(param_grid)*len(lags_grid)}."
      )
  
      if show_progress:
          lags_grid_tqdm = tqdm(
              lags_grid.items(), desc="lags grid", position=0
          )  # ncols=90
          param_grid = tqdm(
              param_grid, desc="params grid", position=1, leave=False
          )
      else:
          lags_grid_tqdm = lags_grid.items()
  
      for lags_k, lags_v in lags_grid_tqdm:
          if type(forecaster).__name__ != "ForecasterAutoregCustom":
              forecaster.set_lags(lags_v)
              lags_v = (
                  lags_k
                  if lags_label == "keys"
                  else forecaster.lags.copy()
              )
  
          for params in param_grid:
              forecaster.set_params(params)
              metrics_values = backtesting_forecaster(
                  forecaster=forecaster,
                  y=y,
                  steps=steps,
                  metric=metric,
                  initial_train_size=initial_train_size,
                  fixed_train_size=fixed_train_size,
                  gap=gap,
                  allow_incomplete_fold=allow_incomplete_fold,
                  exog=exog,
                  refit=refit,
                  interval=None,
                  n_jobs=n_jobs,
                  verbose=verbose,
                  show_progress=False,
              )[0]
              warnings.filterwarnings(
                  "ignore",
                  category=RuntimeWarning,
                  message="The forecaster will be fit.*",
              )
  
              lags_list.append(lags_v)
              params_list.append(params)
              for m, m_value in zip(metric, metrics_values):
                  m_name = m if isinstance(m, str) else m.__name__
                  metric_dict[m_name].append(m_value)
  
      results = pd.DataFrame(
          {"lags": lags_list, "params": params_list, **metric_dict}
      )
  
      results = results.sort_values(
          by=list(metric_dict.keys())[0], ascending=True
      )
      results = pd.concat(
          [results, results["params"].apply(pd.Series)], axis=1
      )
  
      if return_best:
          best_lags = results["lags"].iloc[0]
          best_params = results["params"].iloc[0]
          best_metric = results[list(metric_dict.keys())[0]].iloc[0]
  
          if lags_label == "keys":
              best_lags = lags_grid[best_lags]
  
          if type(forecaster).__name__ != "ForecasterAutoregCustom":
              forecaster.set_lags(best_lags)
              best_lags = forecaster.lags
          else:
              best_lags = "custom_predictors"
          forecaster.set_params(best_params)
  
          forecaster.fit(y=y, exog=exog, store_in_sample_residuals=True)
  
          print(
              f"`Forecaster` refitted using the best-found lags and parameters, "
              f"and the whole data set: \n"
              f"  Lags: {best_lags} \n"
              f"  Parameters: {best_params}\n"
              f"  Backtesting metric: {best_metric}\n"
          )
  
      return results
  
imports: from typing import Union, Tuple, Optional, Callable
  import numpy
  import pandas
  import warnings
  import logging
  from copy import deepcopy
  from joblib import Parallel, delayed, cpu_count
  from tqdm.auto import tqdm
  from sklearn.metrics import mean_squared_error, mean_absolute_error, mean_absolute_percentage_error, mean_squared_log_error
  from sklearn.model_selection import ParameterGrid
  from sklearn.model_selection import ParameterSampler
  import optuna
  from optuna.samplers import TPESampler, RandomSampler
  from ..exceptions import LongTrainingWarning
  from ..utils import check_backtesting_input
  from ..utils import initialize_lags_grid
  from ..utils import select_n_jobs_backtesting
label_desc: Use image node_id 7 for calling a global function with example usage: _evaluate_grid_hyperparameters(forecaster, y, param_grid, steps, metric, initial_train_size, fixed_train_size, gap, allow_incomplete_fold, exog, lags_grid, refit, return_best, n_jobs, verbose, show_progress) and returns: results