## Probabilistic forecasting for univariate time series ------------------------------------------------------------------------ source ### prob_forecasts ``` python def prob_forecasts( model, # Any model with ``.target_col``, ``.fit(df)``, and ``.forecast(H, exog)`` attributes. H:int, # Forecast horizon. n_calibration:Union[int, None]=None, # Number of calibration windows for cross-validated residual estimation. If None, in sample residuals are used without cross-validation (Horizon-specific uncalibrated intervals may be too narrow in this case. This is recommended when data size is small as the model may not have enough data to fit well in each calibration fold). step_size:int=1, # Step size between consecutive calibration windows. random_state:int=42, # Seed for all internal random-number generators. n_iter:Union[int, None]=None, # Number of EM iterations during each calibration window. Only relevant for Markov-switching Autoregressive model ([`ms_arr`](https://mustafaslanCoto.github.io/peshbeen/modules/02_models/ms_arr.html#ms_arr)). A smaller value than the model's default speeds up calibration at the cost of convergence quality per fold — typically a value of 3–10 is sufficient for calibration windows where the model is already close to the solution. verbose:bool=False, # Print progress during calibration. ): ``` *Probabilistic forecasting wrapper for any point-forecasting model.*

	Type	Default	Details
model			Any model with .target_col, .fit(df), and .forecast(H, exog) attributes.
H	int		Forecast horizon.
n_calibration	Union[int, None]	None	Number of calibration windows for cross-validated residual estimation. If None, in sample residuals are used without cross-validation (Horizon-specific uncalibrated intervals may be too narrow in this case. This is recommended when data size is small as the model may not have enough data to fit well in each calibration fold).
step_size	int	1	Step size between consecutive calibration windows.
random_state	int	42	Seed for all internal random-number generators.
n_iter	Union[int, None]	None	Number of EM iterations during each calibration window. Only relevant for Markov-switching Autoregressive model (ms_arr). A smaller value than the model’s default speeds up calibration at the cost of convergence quality per fold — typically a value of 3–10 is sufficient for calibration windows where the model is already close to the solution.
verbose	bool	False	Print progress during calibration.

------------------------------------------------------------------------ source ### prob_forecasts.calibrate ``` python def calibrate( df:pd.DataFrame, # Calibration dataset. delta:Union[float, List[float]]=0.5, # Coverage level(s). A single float produces one symmetric interval; a list produces one interval per level. For example,``delta=0.9`` produces a 90 % prediction interval. )->'prob_forecasts': # The fitted object, with ``self.q_hat`` set to the calibrated ``` *Calibrate the conformal predictor.* Runs rolling-window cross-validation (if not already done) to collect non-conformity scores, then computes the per-horizon conformal quantile `q_hat` for each requested `delta` level.

	Type	Default	Details
df	pd.DataFrame		Calibration dataset.
delta	Union[float, List[float]]	0.5	Coverage level(s). A single float produces one symmetric interval; a list produces one interval per level. For example,delta=0.9 produces a 90 % prediction interval.
Returns	‘prob_forecasts’		The fitted object, with self.q_hat set to the calibrated

------------------------------------------------------------------------ source ### prob_forecasts.sample ``` python def sample( df:pd.DataFrame, # Training data. Residuals are computed via cross-validation if not yet available. n_samples:int=1000, # Number of sample paths to draw. method:str='empirical', # Sampling strategy (see above). future_exog:Union[pd.DataFrame, None]=None, # Future exogenous variables passed to ``forecast``. )->'prob_forecasts': ``` *Draw sample paths from the predictive distribution.* Three methods are available: - `"empirical"` — residuals are resampled with replacement independently at each horizon. - `"kde"` — a Gaussian KDE is fitted to each horizon’s residuals; samples are drawn from the smoothed distribution. - `"correlated"` — a multivariate normal is fitted to the full `H`-dimensional residual vectors, preserving cross-horizon correlation. Samples are drawn jointly. Results are stored on `self`: - `self.sample_paths` — `(n_samples, H)` array of sampled trajectories centred on the point forecast. - `self.point_forecast` — `(H,)` point forecast array. - `self.sample_paths_df` — the same data as a DataFrame with columns `h_1, …, h_H`.

	Type	Default	Details
df	pd.DataFrame		Training data. Residuals are computed via cross-validation if not yet available.
n_samples	int	1000	Number of sample paths to draw.
method	str	empirical	Sampling strategy (see above).
future_exog	Union[pd.DataFrame, None]	None	Future exogenous variables passed to forecast.
Returns	‘prob_forecasts’

------------------------------------------------------------------------ source ### prob_forecasts.sample_quantiles ``` python def sample_quantiles( quantiles:Union[float, List[float]], # Desired quantile levels (e.g. ``[0.1, 0.5, 0.9]``). )->pd.DataFrame: # Columns: ``point_forecast``, ``q_`` for each level. ``` *Compute quantiles from the sample paths generated by `sample`.* Works identically regardless of which `method` was passed to `sample`.

	Type	Details
quantiles	Union[float, List[float]]	Desired quantile levels (e.g. [0.1, 0.5, 0.9]).
Returns	pd.DataFrame	Columns: point_forecast, q_<level> for each level.

------------------------------------------------------------------------ source ### prob_forecasts.conformal_quantiles ``` python def conformal_quantiles( df:pd.DataFrame, # Training data for the final model fit. quantiles:Union[float, List[float]], # Desired quantile levels (e.g. ``[0.1, 0.5, 0.9]``). future_exog:Union[pd.DataFrame, None]=None, # Future exogenous variables. )->pd.DataFrame: # Columns: ``point_forecast``, ``q_`` for each level. ``` *Generate conformal prediction quantiles.* Requires `calibrate` to have been called first.

	Type	Default	Details
df	pd.DataFrame		Training data for the final model fit.
quantiles	Union[float, List[float]]		Desired quantile levels (e.g. [0.1, 0.5, 0.9]).
future_exog	Union[pd.DataFrame, None]	None	Future exogenous variables.
Returns	pd.DataFrame		Columns: point_forecast, q_<level> for each level.

## Probabilistic forecasting for multivariate time series ------------------------------------------------------------------------ source ### mv_prob_forecasts ``` python def mv_prob_forecasts( model, # Any model with ``.target_col``, ``.fit(df)``, and ``.forecast(H, exog)`` attributes. target_col:str, # Name of the target variable column in the input DataFrames. H:int, # Forecast horizon. n_calibration:Union[int, None]=None, # Number of calibration windows for cross-validated residual estimation. If None, in sample residuals are used without cross-validation (Horizon-specific uncalibrated intervals may be too narrow in this case. This is recommended when data size is small as the model may not have enough data to fit well in each calibration fold). step_size:int=1, # Step size between consecutive calibration windows. random_state:int=42, # Seed for all internal random-number generators. n_iter:Union[int, None]=None, # Number of EM iterations during each calibration window. Only relevant for Markov-switching Autoregressive model ([`ms_arr`](https://mustafaslanCoto.github.io/peshbeen/modules/02_models/ms_arr.html#ms_arr)). A smaller value than the model's default speeds up calibration at the cost of convergence quality per fold — typically a value of 3–10 is sufficient for calibration windows where the model is already close to the solution. verbose:bool=False, # Print progress during calibration. ): ``` *Probabilistic forecasting wrapper for any point-forecasting model.*

	Type	Default	Details
model			Any model with .target_col, .fit(df), and .forecast(H, exog) attributes.
target_col	str		Name of the target variable column in the input DataFrames.
H	int		Forecast horizon.
n_calibration	Union[int, None]	None	Number of calibration windows for cross-validated residual estimation. If None, in sample residuals are used without cross-validation (Horizon-specific uncalibrated intervals may be too narrow in this case. This is recommended when data size is small as the model may not have enough data to fit well in each calibration fold).
step_size	int	1	Step size between consecutive calibration windows.
random_state	int	42	Seed for all internal random-number generators.
n_iter	Union[int, None]	None	Number of EM iterations during each calibration window. Only relevant for Markov-switching Autoregressive model (ms_arr). A smaller value than the model’s default speeds up calibration at the cost of convergence quality per fold — typically a value of 3–10 is sufficient for calibration windows where the model is already close to the solution.
verbose	bool	False	Print progress during calibration.

------------------------------------------------------------------------ source ### mv_prob_forecasts.calibrate ``` python def calibrate( df:pd.DataFrame, # Calibration dataset. delta:Union[float, List[float]]=0.5, # Coverage level(s). A single float produces one symmetric interval; a list produces one interval per level. For example,``delta=0.9`` produces a 90 % prediction interval. )->'prob_forecasts': # The fitted object, with ``self.q_hat`` set to the calibrated ``` *Calibrate the conformal predictor.* Runs rolling-window cross-validation (if not already done) to collect non-conformity scores, then computes the per-horizon conformal quantile `q_hat` for each requested `delta` level.

	Type	Default	Details
df	pd.DataFrame		Calibration dataset.
delta	Union[float, List[float]]	0.5	Coverage level(s). A single float produces one symmetric interval; a list produces one interval per level. For example,delta=0.9 produces a 90 % prediction interval.
Returns	‘prob_forecasts’		The fitted object, with self.q_hat set to the calibrated

------------------------------------------------------------------------ source ### mv_prob_forecasts.sample ``` python def sample( df:pd.DataFrame, # Training data. Residuals are computed via cross-validation if not yet available. n_samples:int=1000, # Number of sample paths to draw. method:str='empirical', # Sampling strategy (see above). future_exog:Union[pd.DataFrame, None]=None, # Future exogenous variables passed to ``forecast``. )->'prob_forecasts': ``` *Draw sample paths from the predictive distribution.* Three methods are available: - `"empirical"` — residuals are resampled with replacement independently at each horizon. - `"kde"` — a Gaussian KDE is fitted to each horizon’s residuals; samples are drawn from the smoothed distribution. - `"correlated"` — a multivariate normal is fitted to the full `H`-dimensional residual vectors, preserving cross-horizon correlation. Samples are drawn jointly. Results are stored on `self`: - `self.sample_paths` — `(n_samples, H)` array of sampled trajectories centred on the point forecast. - `self.point_forecast` — `(H,)` point forecast array. - `self.sample_paths_df` — the same data as a DataFrame with columns `h_1, …, h_H`.

	Type	Default	Details
df	pd.DataFrame		Training data. Residuals are computed via cross-validation if not yet available.
n_samples	int	1000	Number of sample paths to draw.
method	str	empirical	Sampling strategy (see above).
future_exog	Union[pd.DataFrame, None]	None	Future exogenous variables passed to forecast.
Returns	‘prob_forecasts’

------------------------------------------------------------------------ source ### mv_prob_forecasts.sample_quantiles ``` python def sample_quantiles( quantiles:Union[float, List[float]], # Desired quantile levels (e.g. ``[0.1, 0.5, 0.9]``). )->pd.DataFrame: # Columns: ``point_forecast``, ``q_`` for each level. ``` *Compute quantiles from the sample paths generated by `sample`.* Works identically regardless of which `method` was passed to `sample`.

	Type	Details
quantiles	Union[float, List[float]]	Desired quantile levels (e.g. [0.1, 0.5, 0.9]).
Returns	pd.DataFrame	Columns: point_forecast, q_<level> for each level.

------------------------------------------------------------------------ source ### mv_prob_forecasts.conformal_quantiles ``` python def conformal_quantiles( df:pd.DataFrame, # Training data for the final model fit. quantiles:Union[float, List[float]], # Desired quantile levels (e.g. ``[0.1, 0.5, 0.9]``). future_exog:Union[pd.DataFrame, None]=None, # Future exogenous variables. )->pd.DataFrame: # Columns: ``point_forecast``, ``q_`` for each level. ``` *Generate conformal prediction quantiles.* Requires `calibrate` to have been called first.

	Type	Default	Details
df	pd.DataFrame		Training data for the final model fit.
quantiles	Union[float, List[float]]		Desired quantile levels (e.g. [0.1, 0.5, 0.9]).
future_exog	Union[pd.DataFrame, None]	None	Future exogenous variables.
Returns	pd.DataFrame		Columns: point_forecast, q_<level> for each level.