model_selection

backtesting_forecaster(forecaster, y, steps, metric, initial_train_size, fixed_train_size=True, exog=None, refit=False, interval=None, n_boot=500, random_state=123, in_sample_residuals=True, verbose=False)

Backtesting of forecaster model.

If refit is False, the model is trained only once using the initial_train_size first observations. If refit is True, the model is trained in each iteration increasing the training set. A copy of the original forecaster is created so it is not modified during the process.

Parameters:

Name	Type	Description	Default
forecaster	ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,		required
ForecasterAutoregMultiOutput	None	Forecaster model.	required
y	Series	Training time series values.	required
steps	int	Number of steps to predict.	required
metric	Union[str, <built-in function 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 several strings and/or callable.	required
initial_train_size	Optional[int]	Number of samples in the initial train split. If None and forecaster is already trained, no initial train is done and all data is used to evaluate the model. However, the first len(forecaster.last_window) observations are needed to create the initial predictors, so no predictions are calculated for them. None is only allowed when refit is False.	required
fixed_train_size	bool	If True, train size doesn't increases but moves by steps in each iteration.	True
exog	Union[pandas.core.series.Series, pandas.core.frame.DataFrame]	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].	None
refit	bool	Whether to re-fit the forecaster in each iteration.	False
interval	Optional[list]	Confidence of the prediction interval estimated. Sequence of percentiles to compute, which must be between 0 and 100 inclusive. For example, interval of 95% should be as interval = [2.5, 97.5]. If None, no intervals are estimated. Only available for forecaster of type ForecasterAutoreg and ForecasterAutoregCustom.	None
n_boot	int	Number of bootstrapping iterations used to estimate prediction intervals.	500
random_state	int	Sets a seed to the random generator, so that boot intervals are always deterministic.	123
in_sample_residuals	bool	If True, residuals from the training data are used as proxy of prediction error to create prediction intervals. If False, out_sample_residuals are used if they are already stored inside the forecaster.	True
verbose	bool	Print number of folds and index of training and validation sets used for backtesting.	False

Returns:

Type	Description
Tuple[Union[float, list], pandas.core.frame.DataFrame]	Value(s) of the metric(s).

Source code in skforecast/model_selection/model_selection.py

def backtesting_forecaster(
    forecaster,
    y: pd.Series,
    steps: int,
    metric: Union[str, callable, list],
    initial_train_size: Optional[int],
    fixed_train_size: bool=True,
    exog: Optional[Union[pd.Series, pd.DataFrame]]=None,
    refit: bool=False,
    interval: Optional[list]=None,
    n_boot: int=500,
    random_state: int=123,
    in_sample_residuals: bool=True,
    verbose: bool=False
) -> Tuple[Union[float, list], pd.DataFrame]:
    """
    Backtesting of forecaster model.

    If `refit` is False, the model is trained only once using the `initial_train_size`
    first observations. If `refit` is True, the model is trained in each iteration
    increasing the training set. A copy of the original forecaster is created so 
    it is not modified during the process.

    Parameters
    ----------
    forecaster : ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,
    ForecasterAutoregMultiOutput
        Forecaster model.

    y : pandas Series
        Training time series 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 several strings and/or callable.

    initial_train_size : int, default `None`
        Number of samples in the initial train split. If `None` and `forecaster` is already 
        trained, no initial train is done and all data is used to evaluate the model. However, 
        the first `len(forecaster.last_window)` observations are needed to create the 
        initial predictors, so no predictions are calculated for them.

        `None` is only allowed when `refit` is `False`.

    fixed_train_size : bool, default `True`
        If True, train size doesn't increases but moves by `steps` in each iteration.

    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].

    refit : bool, default `False`
        Whether to re-fit the forecaster in each iteration.

    interval : list, default `None`
        Confidence of the prediction interval estimated. Sequence of percentiles
        to compute, which must be between 0 and 100 inclusive. For example, 
        interval of 95% should be as `interval = [2.5, 97.5]`. If `None`, no
        intervals are estimated. Only available for forecaster of type 
        ForecasterAutoreg and ForecasterAutoregCustom.

    n_boot : int, default `500`
        Number of bootstrapping iterations used to estimate prediction
        intervals.

    random_state : int, default `123`
        Sets a seed to the random generator, so that boot intervals are always 
        deterministic.

    in_sample_residuals : bool, default `True`
        If `True`, residuals from the training data are used as proxy of
        prediction error to create prediction intervals.  If `False`, out_sample_residuals
        are used if they are already stored inside the forecaster.

    verbose : bool, default `False`
        Print number of folds and index of training and validation sets used for backtesting.

    Returns 
    -------
    metrics_value : float, list
        Value(s) of the metric(s).

    backtest_predictions : pandas DataFrame
        Value of predictions and their estimated interval if `interval` is not `None`.
            column pred = predictions.
            column lower_bound = lower bound of the interval.
            column upper_bound = upper bound interval of the interval.

    """

    if initial_train_size is not None and initial_train_size > len(y):
        raise Exception(
            'If used, `initial_train_size` must be smaller than length of `y`.'
        )

    if initial_train_size is not None and initial_train_size < forecaster.window_size:
        raise Exception(
            f"`initial_train_size` must be greater than "
            f"forecaster's window_size ({forecaster.window_size})."
        )

    if initial_train_size is None and not forecaster.fitted:
        raise Exception(
            '`forecaster` must be already trained if no `initial_train_size` is provided.'
        )

    if not isinstance(refit, bool):
        raise Exception(
            f'`refit` must be boolean: True, False.'
        )

    if initial_train_size is None and refit:
        raise Exception(
            f'`refit` is only allowed when there is a initial_train_size.'
        )

    if interval is not None and isinstance(forecaster, (ForecasterAutoregDirect,
    ForecasterAutoregMultiOutput)):
        raise Exception(
            ('Interval prediction is only available when forecaster is of type '
            'ForecasterAutoreg or ForecasterAutoregCustom.')
        )

    if isinstance(forecaster, ForecasterAutoregMultiSeries):
        raise Exception(
            ('For `forecaster` of type `ForecasterAutoregMultiSeries`, use the '
             'functions available in the model_selection_multiseries module.')
        )

    if refit:
        metrics_values, backtest_predictions = _backtesting_forecaster_refit(
            forecaster          = forecaster,
            y                   = y,
            steps               = steps,
            metric              = metric,
            initial_train_size  = initial_train_size,
            fixed_train_size    = fixed_train_size,
            exog                = exog,
            interval            = interval,
            n_boot              = n_boot,
            random_state        = random_state,
            in_sample_residuals = in_sample_residuals,
            verbose             = verbose
        )
    else:
        metrics_values, backtest_predictions = _backtesting_forecaster_no_refit(
            forecaster          = forecaster,
            y                   = y,
            steps               = steps,
            metric              = metric,
            initial_train_size  = initial_train_size,
            exog                = exog,
            interval            = interval,
            n_boot              = n_boot,
            random_state        = random_state,
            in_sample_residuals = in_sample_residuals,
            verbose             = verbose
        )

    return metrics_values, backtest_predictions

grid_search_forecaster(forecaster, y, param_grid, steps, metric, initial_train_size, fixed_train_size=True, exog=None, lags_grid=None, refit=False, return_best=True, verbose=True)

Exhaustive search over specified parameter values for a Forecaster object.

Validation is done using time series backtesting.

Parameters:

Name	Type	Description	Default
forecaster	ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,		required
ForecasterAutoregMultiOutput	None	Forcaster model.	required
y	Series	Training time series values.	required
param_grid	dict	Dictionary with parameters names (str) as keys and lists of parameter settings to try as values.	required
steps	int	Number of steps to predict.	required
metric	Union[str, <built-in function 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 several strings and/or callable.	required
initial_train_size	int	Number of samples in the initial train split.	required
fixed_train_size	bool	If True, train size doesn't increases but moves by steps in each iteration.	True
exog	Union[pandas.core.series.Series, pandas.core.frame.DataFrame]	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].	None
lags_grid	Optional[list]	Lists of lags to try. Only used if forecaster is an instance of ForecasterAutoreg, ForecasterAutoregDirect or ForecasterAutoregMultiOutput.	None
refit	bool	Whether to re-fit the forecaster in each iteration of backtesting.	False
return_best	bool	Refit the forecaster using the best found parameters on the whole data.	True
verbose	bool	Print number of folds used for cv or backtesting.	True

Returns:

Type	Description
DataFrame	Results for each combination of parameters. column lags = predictions. column params = lower bound of the interval. column metric = metric value estimated for the combination of parameters. additional n columns with param = value.

Source code in skforecast/model_selection/model_selection.py

def grid_search_forecaster(
    forecaster,
    y: pd.Series,
    param_grid: dict,
    steps: int,
    metric: Union[str, callable, list],
    initial_train_size: int,
    fixed_train_size: bool=True,
    exog: Optional[Union[pd.Series, pd.DataFrame]]=None,
    lags_grid: Optional[list]=None,
    refit: bool=False,
    return_best: bool=True,
    verbose: bool=True
) -> pd.DataFrame:
    """
    Exhaustive search over specified parameter values for a Forecaster object.
    Validation is done using time series backtesting.

    Parameters
    ----------
    forecaster : ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,
    ForecasterAutoregMultiOutput
        Forcaster model.

    y : pandas Series
        Training time series values. 

    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 several strings and/or callable.

    initial_train_size : int 
        Number of samples in the initial train split.

    fixed_train_size : bool, default `True`
        If True, train size doesn't increases but moves by `steps` in each iteration.

    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 of int, lists, numpy ndarray or range, default `None`
        Lists of `lags` to try. Only used if forecaster is an instance of 
        `ForecasterAutoreg`, `ForecasterAutoregDirect` or `ForecasterAutoregMultiOutput`.

    refit : bool, default `False`
        Whether to re-fit the forecaster in each iteration of backtesting.

    return_best : bool, default `True`
        Refit the `forecaster` using the best found parameters on the whole data.

    verbose : bool, default `True`
        Print number of folds used for cv or backtesting.

    Returns 
    -------
    results : pandas DataFrame
        Results for each combination of parameters.
            column lags = predictions.
            column params = lower bound of the interval.
            column metric = metric value estimated for the combination of parameters.
            additional n columns with param = value.

    """

    param_grid = list(ParameterGrid(param_grid))

    results = _evaluate_grid_hyperparameters(
        forecaster          = forecaster,
        y                   = y,
        param_grid          = param_grid,
        steps               = steps,
        metric              = metric,
        initial_train_size  = initial_train_size,
        fixed_train_size    = fixed_train_size,
        exog                = exog,
        lags_grid           = lags_grid,
        refit               = refit,
        return_best         = return_best,
        verbose             = verbose
    )

    return results

random_search_forecaster(forecaster, y, param_distributions, steps, metric, initial_train_size, fixed_train_size=True, exog=None, lags_grid=None, refit=False, n_iter=10, random_state=123, return_best=True, verbose=True)

Random search over specified parameter values or distributions for a Forecaster object.

Validation is done using time series backtesting.

Parameters:

Name	Type	Description	Default
forecaster	ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,		required
ForecasterAutoregMultiOutput	None	Forcaster model.	required
y	Series	Training time series values.	required
param_distributions	dict	Dictionary with parameters names (str) as keys and distributions or lists of parameters to try.	required
steps	int	Number of steps to predict.	required
metric	Union[str, <built-in function 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 several strings and/or callable.	required
initial_train_size	int	Number of samples in the initial train split.	required
fixed_train_size	bool	If True, train size doesn't increases but moves by steps in each iteration.	True
exog	Union[pandas.core.series.Series, pandas.core.frame.DataFrame]	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].	None
lags_grid	Optional[list]	Lists of lags to try. Only used if forecaster is an instance of ForecasterAutoreg, ForecasterAutoregDirect or ForecasterAutoregMultiOutput.	None
refit	bool	Whether to re-fit the forecaster in each iteration of backtesting.	False
n_iter	int	Number of parameter settings that are sampled. n_iter trades off runtime vs quality of the solution.	10
random_state	int	Sets a seed to the random sampling for reproducible output.	123
return_best	bool	Refit the forecaster using the best found parameters on the whole data.	True
verbose	bool	Print number of folds used for cv or backtesting.	True

Returns:

Type	Description
DataFrame	Results for each combination of parameters. column lags = predictions. column params = lower bound of the interval. column metric = metric value estimated for the combination of parameters. additional n columns with param = value.

Source code in skforecast/model_selection/model_selection.py

def random_search_forecaster(
    forecaster,
    y: pd.Series,
    param_distributions: dict,
    steps: int,
    metric: Union[str, callable, list],
    initial_train_size: int,
    fixed_train_size: bool=True,
    exog: Optional[Union[pd.Series, pd.DataFrame]]=None,
    lags_grid: Optional[list]=None,
    refit: bool=False,
    n_iter: int=10,
    random_state: int=123,
    return_best: bool=True,
    verbose: bool=True
) -> pd.DataFrame:
    """
    Random search over specified parameter values or distributions for a Forecaster object.
    Validation is done using time series backtesting.

    Parameters
    ----------
    forecaster : ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,
    ForecasterAutoregMultiOutput
        Forcaster model.

    y : pandas Series
        Training time series values. 

    param_distributions : dict
        Dictionary with parameters names (`str`) as keys and 
        distributions or lists of parameters to try.

    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 several strings and/or callable.

    initial_train_size : int 
        Number of samples in the initial train split.

    fixed_train_size : bool, default `True`
        If True, train size doesn't increases but moves by `steps` in each iteration.

    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 of int, lists, numpy ndarray or range, default `None`
        Lists of `lags` to try. Only used if forecaster is an instance of 
        `ForecasterAutoreg`, `ForecasterAutoregDirect` or `ForecasterAutoregMultiOutput`.

    refit : bool, default `False`
        Whether to re-fit the forecaster in each iteration of backtesting.

    n_iter : int, default `10`
        Number of parameter settings that are sampled. 
        n_iter trades off runtime vs quality of the solution.

    random_state : int, default `123`
        Sets a seed to the random sampling for reproducible output.

    return_best : bool, default `True`
        Refit the `forecaster` using the best found parameters on the whole data.

    verbose : bool, default `True`
        Print number of folds used for cv or backtesting.

    Returns 
    -------
    results : pandas DataFrame
        Results for each combination of parameters.
            column lags = predictions.
            column params = lower bound of the interval.
            column metric = metric value estimated for the combination of parameters.
            additional n columns with param = value.

    """

    param_grid = list(ParameterSampler(param_distributions, n_iter=n_iter, random_state=random_state))

    results = _evaluate_grid_hyperparameters(
        forecaster          = forecaster,
        y                   = y,
        param_grid          = param_grid,
        steps               = steps,
        metric              = metric,
        initial_train_size  = initial_train_size,
        fixed_train_size    = fixed_train_size,
        exog                = exog,
        lags_grid           = lags_grid,
        refit               = refit,
        return_best         = return_best,
        verbose             = verbose
    )

    return results

bayesian_search_forecaster(forecaster, y, search_space, steps, metric, initial_train_size, fixed_train_size=True, exog=None, lags_grid=None, refit=False, n_trials=10, random_state=123, return_best=True, verbose=True, engine='skopt', kwargs_create_study={}, kwargs_study_optimize={}, kwargs_gp_minimize={})

Bayesian optimization for a Forecaster object using time series backtesting and

optuna or skopt library.

Parameters:

Name	Type	Description	Default
forecaster	ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,		required
ForecasterAutoregMultiOutput	None	Forcaster model.	required
y	Series	Training time series values.	required
search_space	Union[<built-in function callable>, dict]	If optuna engine: callable Function with argument trial which returns a dictionary with parameters names (str) as keys and Trial object from optuna (trial.suggest_float, trial.suggest_int, trial.suggest_categorical) as values. If skopt engine: dict Dictionary with parameters names (str) as keys and Space object from skopt (Real, Integer, Categorical) as values.	required
steps	int	Number of steps to predict.	required
metric	Union[str, <built-in function 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 several strings and/or callable.	required
initial_train_size	int	Number of samples in the initial train split.	required
fixed_train_size	bool	If True, train size doesn't increases but moves by steps in each iteration.	True
exog	Union[pandas.core.series.Series, pandas.core.frame.DataFrame]	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].	None
lags_grid	Optional[list]	Lists of lags to try. Only used if forecaster is an instance of ForecasterAutoreg, ForecasterAutoregDirect or ForecasterAutoregMultiOutput.	None
refit	bool	Whether to re-fit the forecaster in each iteration of backtesting.	False
n_trials	int	Number of parameter settings that are sampled in each lag configuration. When using engine "skopt", the minimum value is 10.	10
random_state	int	Sets a seed to the sampling for reproducible output.	123
return_best	bool	Refit the forecaster using the best found parameters on the whole data.	True
verbose	bool	Print number of folds used for cv or backtesting.	True
engine	str	If 'optuna': Bayesian optimization runs through the optuna library If 'skopt': Bayesian optimization runs through the skopt library	'skopt'
kwargs_create_study	dict	Only applies to engine='optuna'. Keyword arguments (key, value mappings) to pass to optuna.create_study.	{}
kwargs_study_optimize	dict	Only applies to engine='optuna'. Other keyword arguments (key, value mappings) to pass to study.optimize().	{}
kwargs_gp_minimize	dict	Only applies to engine='skopt'. Other keyword arguments (key, value mappings) to pass to skopt.gp_minimize().	{}

Returns:

Type	Description
Tuple[pandas.core.frame.DataFrame, object]	Results for each combination of parameters. column lags = predictions. column params = lower bound of the interval. column metric = metric value estimated for the combination of parameters. additional n columns with param = value.

Source code in skforecast/model_selection/model_selection.py

def bayesian_search_forecaster(
    forecaster,
    y: pd.Series,
    search_space: Union[callable, dict],
    steps: int,
    metric: Union[str, callable, list],
    initial_train_size: int,
    fixed_train_size: bool=True,
    exog: Optional[Union[pd.Series, pd.DataFrame]]=None,
    lags_grid: Optional[list]=None,
    refit: bool=False,
    n_trials: int=10,
    random_state: int=123,
    return_best: bool=True,
    verbose: bool=True,
    engine: str='skopt',
    kwargs_create_study: dict={},
    kwargs_study_optimize: dict={},
    kwargs_gp_minimize: dict={},
) -> Tuple[pd.DataFrame, object]:
    """
    Bayesian optimization for a Forecaster object using time series backtesting and 
    optuna or skopt library.

    Parameters
    ----------
    forecaster : ForecasterAutoreg, ForecasterAutoregCustom, ForecasterAutoregDirect,
    ForecasterAutoregMultiOutput
        Forcaster model.

    y : pandas Series
        Training time series values. 

    search_space : callable (optuna), dict (skopt)
        If optuna engine: callable
            Function with argument `trial` which returns a dictionary with parameters names 
            (`str`) as keys and Trial object from optuna (trial.suggest_float, 
            trial.suggest_int, trial.suggest_categorical) as values.

        If skopt engine: dict
            Dictionary with parameters names (`str`) as keys and Space object from skopt 
            (Real, Integer, Categorical) 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 several strings and/or callable.


    initial_train_size : int 
        Number of samples in the initial train split.

    fixed_train_size : bool, default `True`
        If True, train size doesn't increases but moves by `steps` in each iteration.

    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 of int, lists, numpy ndarray or range, default `None`
        Lists of `lags` to try. Only used if forecaster is an instance of 
        `ForecasterAutoreg`, `ForecasterAutoregDirect` or `ForecasterAutoregMultiOutput`.

    refit : bool, default `False`
        Whether to re-fit the forecaster in each iteration of backtesting.

    n_trials : int, default `10`
        Number of parameter settings that are sampled in each lag configuration.
        When using engine "skopt", the minimum value is 10.

    random_state : int, default `123`
        Sets a seed to the sampling for reproducible output.

    return_best : bool, default `True`
        Refit the `forecaster` using the best found parameters on the whole data.

    verbose : bool, default `True`
        Print number of folds used for cv or backtesting.

    engine : str, default `'skopt'`
        If 'optuna':
            Bayesian optimization runs through the optuna library 

        If 'skopt':
            Bayesian optimization runs through the skopt library

    kwargs_create_study : dict, default `{'direction':'minimize', 'sampler':TPESampler(seed=123)}`
        Only applies to engine='optuna'.
            Keyword arguments (key, value mappings) to pass to optuna.create_study.

    kwargs_study_optimize : dict, default `{}`
        Only applies to engine='optuna'.
            Other keyword arguments (key, value mappings) to pass to study.optimize().

    kwargs_gp_minimize : dict, default `{}`
        Only applies to engine='skopt'.
            Other keyword arguments (key, value mappings) to pass to skopt.gp_minimize().

    Returns 
    -------
    results : pandas DataFrame
        Results for each combination of parameters.
            column lags = predictions.
            column params = lower bound of the interval.
            column metric = metric value estimated for the combination of parameters.
            additional n columns with param = value.

    results_opt_best : optuna object (optuna), scipy object (skopt)   
        If optuna engine:
            The best optimization result returned as a FrozenTrial optuna object.

        If skopt engine:
            The best optimization result returned as a OptimizeResult object.

    """

    if engine not in ['optuna', 'skopt']:
        raise ValueError(
                f"""`engine` only allows 'optuna' or 'skopt', got {engine}."""
              )

    if engine == 'optuna':
        results, results_opt_best = _bayesian_search_optuna(
                                        forecaster            = forecaster,
                                        y                     = y,
                                        exog                  = exog,
                                        lags_grid             = lags_grid,
                                        search_space          = search_space,
                                        steps                 = steps,
                                        metric                = metric,
                                        refit                 = refit,
                                        initial_train_size    = initial_train_size,
                                        fixed_train_size      = fixed_train_size,
                                        n_trials              = n_trials,
                                        random_state          = random_state,
                                        return_best           = return_best,
                                        verbose               = verbose,
                                        kwargs_create_study   = kwargs_create_study,
                                        kwargs_study_optimize = kwargs_study_optimize
                                    )
    else:
        results, results_opt_best = _bayesian_search_skopt(
                                        forecaster         = forecaster,
                                        y                  = y,
                                        exog               = exog,
                                        lags_grid          = lags_grid,
                                        search_space       = search_space,
                                        steps              = steps,
                                        metric             = metric,
                                        refit              = refit,
                                        initial_train_size = initial_train_size,
                                        fixed_train_size   = fixed_train_size,
                                        n_trials           = n_trials,
                                        random_state       = random_state,
                                        return_best        = return_best,
                                        verbose            = verbose,
                                        kwargs_gp_minimize = kwargs_gp_minimize
                                    )

    return results, results_opt_best