Linear Reconciliation#

This page documents the available functions for reconciliation. Canonical modules are under bayesreconpy.linear; legacy module paths remain available for backward compatibility.

bayesreconpy.linear.conditioning#

Conditioning and sampling-based linear reconciliation methods.

bayesreconpy.linear.conditioning.reconc_buis(A, base_forecasts, in_type, distr, num_samples=20000, suppress_warnings=False, seed=None)#

BUIS for Probabilistic Reconciliation of Forecasts via Conditioning

Uses the Bottom-Up Importance Sampling (BUIS) algorithm to draw samples from the reconciled forecast distribution obtained via conditioning.

Parameters:

  • A (ndarray) – Aggregation matrix with shape (n_upper, n_bottom). Each column represents a bottom-level forecast, and each row represents an upper-level forecast. A value of 1 indicates a contribution of a bottom-level forecast to an upper-level forecast.

  • base_forecasts (List[Union[Dict[str, float], ndarray]]) –

    A list containing n_upper + n_bottom elements. The first n_upper elements represent the upper-level base forecasts (in the order of rows in A), and the remaining elements represent the bottom-level base forecasts (in the order of columns in A).

    • If in_type[i] == “samples”, base_forecasts[i] is a NumPy array containing samples from the base forecast distribution.

    • If in_type[i] == “params”, base_forecasts[i] is a dictionary containing parameters for the specified distribution in distr[i]:

      • ’gaussian’: {“mean”: float, “sd”: float}

      • ’poisson’: {“lambda”: float}

      • ’nbinom’: {“size”: float, “prob”: float} or {“size”: float, “mu”: float}

  • in_type (Union[str, List[str]]) –

    Specifies the input type for each base forecast. If a string, the same input type is applied to all forecasts. If a list, in_type[i] specifies the type for the i-th forecast:

    • ’samples’: The forecast is provided as samples.

    • ’params’: The forecast is provided as parameters.

  • distr (Union[str, List[str]]) –

    Specifies the distribution type for each base forecast. If a string, the same distribution is applied to all forecasts. If a list, distr[i] specifies the distribution for the i-th forecast:

    • ’continuous’ or ‘discrete’ if in_type[i] == “samples”.

    • ’gaussian’, ‘poisson’, or ‘nbinom’ if in_type[i] == “params”.

  • num_samples (int) – Number of samples to draw from the reconciled distribution. Ignored if in_type == “samples”, in which case the number of reconciled samples matches the number of samples in the base forecasts. Default is 20,000.

  • suppress_warnings (bool) – Whether to suppress warnings during the importance sampling step. Default is False.

  • seed (Optional[int]) – Random seed for reproducibility. Default is None.

Returns:

A dictionary containing the reconciled forecasts:

  • ’bottom_reconciled_samples’: numpy.ndarray

    A matrix of shape (n_bottom, num_samples) containing the reconciled samples for the bottom-level forecasts.

  • ’upper_reconciled_samples’: numpy.ndarray

    A matrix of shape (n_upper, num_samples) containing the reconciled samples for the upper-level forecasts.

  • ’reconciled_samples’: numpy.ndarray

    A matrix of shape (n, num_samples) containing the reconciled samples for all forecasts.

Return type:

Dict[str, ndarray]

Notes

  • Warnings are triggered during the importance sampling step if:

    • All weights are zero (the corresponding upper forecast is ignored).

    • Effective sample size is less than 200.

    • Effective sample size is less than 1% of the total number of samples.

  • Such warnings indicate potential issues with the base forecasts. Check the inputs in case of warnings.

Examples

Example 1: Gaussian base forecasts (in params form)
>>> A = np.array([
...     [1, 0, 0],
...     [0, 1, 1]
... ])
>>> base_forecasts = [
...     {"mean": 9.0, "sd": 3.0},  # Upper forecast
...     {"mean": 2.0, "sd": 2.0},  # Bottom forecast 1
...     {"mean": 4.0, "sd": 2.0}   # Bottom forecast 2
... ]
>>> result = reconc_buis(A, base_forecasts, in_type="params", distr="gaussian", num_samples=10000, seed=42)
>>> print(result['reconciled_samples'].shape)
(3, 10000)
Example 2: Poisson base forecasts (in params form)
>>> A = np.array([
...     [1, 0, 0],
...     [0, 1, 1]
... ])
>>> base_forecasts = [
...     {"lambda": 9.0},  # Upper forecast
...     {"lambda": 2.0},  # Bottom forecast 1
...     {"lambda": 4.0}   # Bottom forecast 2
... ]
>>> result = reconc_buis(A, base_forecasts, in_type="params", distr="poisson", num_samples=10000, seed=42)
>>> print(result['reconciled_samples'].shape)
(3, 10000)
bayesreconpy.linear.conditioning.reconc_mcmc(A, base_forecasts, distr, num_samples=10000, tuning_int=100, init_scale=1, burn_in=1000, seed=None)#

MCMC for Probabilistic Reconciliation of forecasts via conditioning.

Uses a Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the reconciled forecast distribution, obtained via conditioning. This implementation uses the Metropolis-Hastings algorithm.

Note: This implementation only supports Poisson or Negative Binomial base forecasts. Gaussian distributions are not supported for MCMC.

Parameters:

  • A (ndarray) – Aggregation matrix of shape (n_upper, n_bottom). Each column represents a bottom-level forecast, and each row represents an upper-level forecast. A value of 1 in A[i, j] indicates that bottom-level forecast j contributes to upper-level forecast i.

  • base_forecasts (list) –

    A list containing the parameters of the base forecast distributions. The first n_upper elements correspond to the upper base forecasts (in the order of rows in A), and the remaining n_bottom elements correspond to the bottom base forecasts (in the order of columns in A). Each element is a dictionary:

    • For ‘poisson’: {“lambda”: float}

    • For ‘nbinom’: {“size”: float, “prob”: float} or {“size”: float, “mu”: float}

  • distr (str) –

    Type of predictive distribution. Supported values:

    • ’poisson’

    • ’nbinom’

  • num_samples (int) – Number of samples to draw using MCMC. Default is 10,000.

  • tuning_int (int) – Number of iterations between scale updates of the proposal. Default is 100.

  • init_scale (float) – Initial scale of the proposal distribution. Default is 1.0.

  • burn_in (int) – Number of initial samples to discard. Default is 1,000.

  • seed (Optional[int]) – Random seed for reproducibility. Default is None.

Returns:

A dictionary containing the reconciled forecasts:

  • ’bottom_reconciled_samples’: numpy.ndarray

    A matrix of shape (n_bottom, num_samples) containing reconciled samples for the bottom-level time series.

  • ’upper_reconciled_samples’: numpy.ndarray

    A matrix of shape (n_upper, num_samples) containing reconciled samples for the upper-level time series.

  • ’reconciled_samples’: numpy.ndarray

    A matrix of shape (n, num_samples) containing the reconciled samples for all time series, where n = n_upper + n_bottom.

Return type:

Dict[str, ndarray]

Notes

  • This is a bare-bones implementation of the Metropolis-Hastings algorithm.

  • We recommend using additional tools to assess the convergence of the MCMC chains.

  • The reconc_buis function is generally faster for most hierarchies.

Examples

Example: Simple hierarchy with Poisson base forecasts
>>> import numpy as np
>>> from bayesreconpy.reconc_buis import reconc_buis
>>>
>>> # Create a minimal hierarchy with 1 upper and 2 bottom variables
>>> A = np.array([[1, 1]])  # Aggregation matrix
>>>
>>> # Set the parameters of the Poisson base forecast distributions
>>> lambda_vals = [9, 2, 4]
>>> base_forecasts = [{"lambda": lam} for lam in lambda_vals]
>>>
>>> # Perform MCMC reconciliation
>>> mcmc_result = reconc_mcmc(A, base_forecasts, distr="poisson", num_samples=30000, seed=42)
>>>
>>> # Access reconciled samples
>>> samples_mcmc = mcmc_result['reconciled_samples']
>>>
>>> # Compare reconciled means with those from BUIS
>>> buis_result = reconc_buis(A, base_forecasts, in_type="params", distr="poisson", num_samples=100000, seed=42)
>>> samples_buis = buis_result['reconciled_samples']
>>>
>>> print("MCMC Reconciled Means:", np.mean(samples_mcmc, axis=1))
>>> print("BUIS Reconciled Means:", np.mean(samples_buis, axis=1))

References

  • Corani, G., Azzimonti, D., Rubattu, N. (2024). Probabilistic reconciliation of count time series. International Journal of Forecasting, 40(2), 457-469.

See also

reconc_buis

Faster reconciliation method for most hierarchies.

bayesreconpy.linear.conditioning.reconc_mix_cond(A, fc_bottom, fc_upper, bottom_in_type='pmf', distr=None, num_samples=20000, return_type='pmf', suppress_warnings=False, seed=None)#

Probabilistic forecast reconciliation of mixed hierarchies via conditioning.

Uses importance sampling to draw samples from the reconciled forecast distribution, obtained via conditioning, in the case of a mixed hierarchy.

Parameters:

  • A (ndarray) – Aggregation matrix with shape (n_upper, n_bottom). Each column represents a bottom-level forecast, and each row represents an upper-level forecast. A value of 1 in A[i, j] indicates that bottom-level forecast j contributes to upper-level forecast i.

  • fc_bottom (Union[array, dict]) –

    Base bottom forecasts. The format depends on bottom_in_type:

    • If bottom_in_type == “pmf”: A dictionary where each value is a PMF object (probability mass function).

    • If bottom_in_type == “samples”: A dictionary or array where each value contains samples.

    • If bottom_in_type == “params”: A dictionary where each value contains parameters:

      • ’poisson’: {“lambda”: float}

      • ’nbinom’: {“size”: float, “prob”: float} or {“size”: float, “mu”: float}

  • fc_upper (dict) –

    Base upper forecasts, represented as parameters of a multivariate Gaussian distribution:

    • ’mu’: A vector of length n_upper containing the means.

    • ’Sigma’: A covariance matrix of shape (n_upper, n_upper).

  • bottom_in_type (str) –

    Specifies the type of the base bottom forecasts. Possible values are:

    • ’pmf’: Bottom base forecasts are provided as PMF objects.

    • ’samples’: Bottom base forecasts are provided as samples.

    • ’params’: Bottom base forecasts are provided as estimated parameters.

    Default is ‘pmf’.

  • distr (Optional[str]) –

    Specifies the distribution type for the bottom base forecasts. Only used if bottom_in_type == “params”. Possible values:

    • ’poisson’

    • ’nbinom’

  • num_samples (int) – Number of samples to draw from the reconciled distribution. Ignored if bottom_in_type == “samples”. In this case, the number of reconciled samples equals the number of samples in the base forecasts. Default is 20,000.

  • return_type (str) –

    Specifies the return type of the reconciled distributions. Possible values:

    • ’pmf’: Returns reconciled marginal PMF objects.

    • ’samples’: Returns reconciled multivariate samples.

    • ’all’: Returns both PMF objects and samples.

    Default is ‘pmf’.

  • suppress_warnings (bool) – Whether to suppress warnings about importance sampling weights. Default is False.

  • seed (Optional[int]) – Random seed for reproducibility. Default is None.

Returns:

A dictionary containing the reconciled forecasts:

  • ’bottom_reconciled’: Contains the reconciled forecasts for the bottom-level variables.

    • If return_type == “pmf”: A list of PMF objects.

    • If return_type == “samples”: A matrix of shape (n_bottom, num_samples).

    • If return_type == “all”: Contains both PMF objects and samples.

  • ’upper_reconciled’: Contains the reconciled forecasts for the upper-level variables.

    • If return_type == “pmf”: A list of PMF objects.

    • If return_type == “samples”: A matrix of shape (n_upper, num_samples).

    • If return_type == “all”: Contains both PMF objects and samples.

  • ’ESS’: Effective sample size after importance sampling.

Return type:

Dict[str, Union[Dict[str, Union[List, ndarray]], float]]

Notes

  • A PMF object is a numerical vector where each element corresponds to the probability of integers from 0 to the last value in the support.

  • Warnings are triggered during the importance sampling step if:

    • All weights are zero, causing the upper forecast to be ignored.

    • Effective sample size (ESS) is less than 200.

    • ESS is less than 1% of the total sample size.

Examples

Simple hierarchy with PMF inputs
>>> import numpy as np
>>> from scipy.stats import poisson
>>>
>>> # Simple hierarchy with one upper and two bottom nodes
>>> A = np.array([[1, 1]])  # Aggregation matrix
>>>
>>> # Bottom forecasts as Poisson distributions
>>> lambda_val = 15
>>> n_tot = 60
>>> fc_bottom = {
...     0: np.array([poisson.pmf(x, lambda_val) for x in range(n_tot + 1)]),
...     1: np.array([poisson.pmf(x, lambda_val) for x in range(n_tot + 1)])
... }
>>>
>>> # Upper forecast as Gaussian parameters
>>> fc_upper = {
...     "mu": np.array([40]),  # Mean
...     "Sigma": np.array([[25]])  # Variance (standard deviation squared)
... }
>>>
>>> # Perform reconciliation
>>> result = reconc_mix_cond(A, fc_bottom, fc_upper, bottom_in_type="pmf", return_type="all")
>>>
>>> # Check the results
>>> print(result['bottom_reconciled']['pmf'][0])
>>> print(result['bottom_reconciled']['pmf'][1])
>>> print(result['upper_reconciled']['pmf'][0])

References

  • Zambon, L., Azzimonti, D., Rubattu, N., Corani, G. (2024). Probabilistic reconciliation of mixed-type hierarchical time series. 40th Conference on Uncertainty in Artificial Intelligence.

See also

reconc_td_cond

Probabilistic reconciliation via top-down conditioning.

reconc_buis

Reconciliation using Bottom-Up Importance Sampling (BUIS).

bayesreconpy.linear.conditioning.reconc_td_cond(A, fc_bottom, fc_upper, bottom_in_type='pmf', distr=None, num_samples=20000, return_type='pmf', suppress_warnings=False, seed=None, return_num_samples_ok=False)#

Probabilistic forecast reconciliation of mixed hierarchies via top-down conditioning.

Uses the top-down conditioning algorithm to draw samples from the reconciled forecast distribution. Reconciliation is performed in two steps: 1. Upper base forecasts are reconciled via conditioning using the hierarchical constraints between upper variables. 2. Bottom distributions are updated via a probabilistic top-down procedure.

Parameters:

  • A (ndarray) – Aggregation matrix with shape (n_upper, n_bottom). Each column represents a bottom-level forecast, and each row represents an upper-level forecast. A value of 1 in A[i, j] indicates that bottom-level forecast j contributes to upper-level forecast i.

  • fc_bottom (Union[array, dict]) –

    Base bottom forecasts. The format depends on bottom_in_type:

    • If bottom_in_type == “pmf”: A dictionary where each value is a PMF object (probability mass function).

    • If bottom_in_type == “samples”: A dictionary or array where each value contains samples.

    • If bottom_in_type == “params”: A dictionary where each value contains parameters:

      • ’poisson’: {“lambda”: float}

      • ’nbinom’: {“size”: float, “prob”: float} or {“size”: float, “mu”: float}

  • fc_upper (dict) –

    Base upper forecasts, represented as parameters of a multivariate Gaussian distribution:

    • ’mu’: A vector of length n_upper containing the means.

    • ’Sigma’: A covariance matrix of shape (n_upper, n_upper).

  • bottom_in_type (str) –

    Specifies the type of the base bottom forecasts. Possible values are:

    • ’pmf’: Bottom base forecasts are provided as PMF objects.

    • ’samples’: Bottom base forecasts are provided as samples.

    • ’params’: Bottom base forecasts are provided as estimated parameters.

    Default is ‘pmf’.

  • distr (Optional[str]) –

    Specifies the distribution type for the bottom base forecasts. Only used if bottom_in_type == “params”. Possible values:

    • ’poisson’

    • ’nbinom’

  • num_samples (int) – Number of samples to draw from the reconciled distribution. Ignored if bottom_in_type == “samples”. In this case, the number of reconciled samples equals the number of samples in the base forecasts. Default is 20,000.

  • return_type (str) –

    Specifies the return type of the reconciled distributions. Possible values:

    • ’pmf’: Returns reconciled marginal PMF objects.

    • ’samples’: Returns reconciled multivariate samples.

    • ’all’: Returns both PMF objects and samples.

    Default is ‘pmf’.

  • suppress_warnings (bool) – Whether to suppress warnings about samples lying outside the support of the bottom-up distribution. Default is False.

  • seed (Optional[int]) – Random seed for reproducibility. Default is None.

  • return_num_samples_ok (bool) – Whether to return the number of samples that were valid after reconciliation. Default is False.

Returns:

A dictionary containing the reconciled forecasts:

  • ’bottom_reconciled’: Contains the reconciled forecasts for the bottom-level variables.

    • If return_type == “pmf”: A list of PMF objects.

    • If return_type == “samples”: A matrix of shape (n_bottom, num_samples).

    • If return_type == “all”: Contains both PMF objects and samples.

  • ’upper_reconciled’: Contains the reconciled forecasts for the upper-level variables.

    • If return_type == “pmf”: A list of PMF objects.

    • If return_type == “samples”: A matrix of shape (n_upper, num_samples).

    • If return_type == “all”: Contains both PMF objects and samples.

  • If return_num_samples_ok is True, a tuple is returned.

  • The first element is the above dictionary.

  • The second element is the number of valid samples after reconciliation.

Return type:

Union[Dict[str, Dict[str, Union[List, ndarray]]], Tuple[Dict, int]]

Notes

  • A PMF object is a numerical vector where each element corresponds to the probability of integers from 0 to the last value in the support.

  • Samples lying outside the support of the bottom-up distribution are discarded, and a warning is issued if suppress_warnings is False.

Examples

Simple hierarchy with PMF inputs
>>> import numpy as np
>>> from scipy.stats import poisson
>>>
>>> # Simple hierarchy with one upper and two bottom nodes
>>> A = np.array([[1, 1]])  # Aggregation matrix
>>>
>>> # Bottom forecasts as Poisson distributions
>>> lambda_val = 15
>>> n_tot = 60
>>> fc_bottom = {
...     0: np.array([poisson.pmf(x, lambda_val) for x in range(n_tot + 1)]),
...     1: np.array([poisson.pmf(x, lambda_val) for x in range(n_tot + 1)])
... }
>>>
>>> # Upper forecast as Gaussian parameters
>>> fc_upper = {
...     "mu": np.array([40]),  # Mean
...     "Sigma": np.array([[25]])  # Variance (standard deviation squared)
... }
>>>
>>> # Perform reconciliation
>>> result = reconc_td_cond(A, fc_bottom, fc_upper, bottom_in_type="pmf", return_type="all")
>>>
>>> # Check the results
>>> print(result['bottom_reconciled']['pmf'][0])
>>> print(result['bottom_reconciled']['pmf'][1])
>>> print(result['upper_reconciled']['pmf'][0])

References

  • Zambon, L., Azzimonti, D., Rubattu, N., Corani, G. (2024). Probabilistic reconciliation of mixed-type hierarchical time series. 40th Conference on Uncertainty in Artificial Intelligence.

See also

reconc_mix_cond

Reconciliation for mixed-type hierarchies.

reconc_buis

Reconciliation using Bottom-Up Importance Sampling (BUIS).

bayesreconpy.linear.gaussian#

Gaussian linear reconciliation.

bayesreconpy.linear.gaussian.reconc_gaussian(A, base_forecasts_mu, base_forecasts_Sigma)#

Analytical reconciliation of Gaussian base forecasts. Reconciles forecasts using a closed-form computation for Gaussian base forecasts.

Parameters:

  • A (ndarray) – Aggregation matrix with shape (n_upper, n_bottom). Each column represents a bottom-level forecast, and each row represents an upper-level forecast. A value of 1 in A[i, j] indicates that the bottom-level forecast j contributes to the upper-level forecast i.

  • base_forecasts_mu (List[float]) –

    A 1D list containing the means of the base forecasts. The order is:

    • First, the upper-level means (in the order of rows in A).

    • Then, the bottom-level means (in the order of columns in A).

  • base_forecasts_Sigma (ndarray) – A 2D covariance matrix representing the uncertainties in the base forecasts. The order of rows and columns must match the order of base_forecasts_mu. Shape: (n, n), where n = n_upper + n_bottom.

Returns:

A dictionary containing:

  • ’bottom_reconciled_mean’numpy.ndarray

    A 1D array of the reconciled means for the bottom-level forecasts. Shape: (n_bottom,).

  • ’bottom_reconciled_covariance’numpy.ndarray

    A 2D covariance matrix of the reconciled bottom-level forecasts. Shape: (n_bottom, n_bottom).

Return type:

Dict[str, ndarray]

Notes

  • The function assumes that base forecasts follow a multivariate Gaussian distribution.

  • The covariance matrix base_forecasts_Sigma should be symmetric and positive semi-definite.

  • The order of elements in base_forecasts_mu and rows/columns in base_forecasts_Sigma is critical: first the upper-level forecasts (in the order of rows in A), followed by the bottom-level forecasts (in the order of columns in A).

  • The function returns only the reconciled parameters for the bottom-level forecasts. Reconciled parameters for upper-level forecasts and the entire hierarchy can be derived using the reconciliation matrix A.

Examples

Example 1: Minimal hierarchy with Gaussian base forecasts
>>> A = np.array([
...     [1, 0, 0],
...     [0, 1, 1]
... ])
>>> base_forecasts_mu = [9.0, 2.0, 4.0]
>>> base_forecasts_Sigma = np.diag([9.0, 4.0, 4.0])
>>> result = reconc_gaussian(A, base_forecasts_mu, base_forecasts_Sigma)
>>> print(result['bottom_reconciled_mean'])
[2.5, 4.0]
>>> print(result['bottom_reconciled_covariance'])
[[2.25, 1.5 ],
 [1.5 , 2.0 ]]

References

  • Corani, G., Azzimonti, D., Augusto, J.P.S.C., Zaffalon, M. (2021). Probabilistic Reconciliation of Hierarchical Forecast via Bayes’ Rule. ECML PKDD 2020. Lecture Notes in Computer Science, vol 12459. https://doi.org/10.1007/978-3-030-67664-3_13

  • Zambon, L., Agosto, A., Giudici, P., Corani, G. (2024). Properties of the reconciled distributions for Gaussian and count forecasts. International Journal of Forecasting (in press). https://doi.org/10.1016/j.ijforecast.2023.12.004

bayesreconpy.linear.projection#

Projection-based linear reconciliation methods.

bayesreconpy.linear.projection.estimate_cov_matrix(res, upper=False, n_u=None)#

Estimate the covariance matrix using the Schafer-Strimmer method.

Parameters: - res: array-like, residuals or data for covariance estimation. - upper: bool, whether to use only the upper portion of the data. - n_u: int, number of upper rows/columns to include (required if upper=True).

Returns: - ss: Covariance matrix (shrinkage-based or regular).

bayesreconpy.linear.projection.get_S_from_A(A)#
bayesreconpy.linear.projection.reconc_mint(A, base_forecasts, res=None, W=None, samples=False)#

Reconciles base forecasts using the MinT (Minimum Trace) approach with a shrinkage-based covariance estimator.

The MinT method adjusts base forecasts to ensure coherence with a given aggregation structure, minimizing the total variance of the reconciliation errors using the residual covariance matrix.

Parameters:#

Anp.ndarray

The aggregation constraint matrix (typically of shape [n_agg_levels, n_total_series]), defining how the bottom-level time series aggregate into upper levels.

base_forecastsnp.ndarray

The base forecasts to be reconciled: - of shape [n_total_series, n_time, n_samples] if samples=True, or - of shape [n_total_series, n_time] if samples=False.

resnp.ndarray

Residuals (forecast errors) from a previous model, used to estimate the covariance matrix. Should be of shape [n_total_series, n_time].

Wnp.ndarray, optional

Pre-computed covariance/weight matrix (W_h). If provided, ‘res’ is ignored.

samplesbool, optional (default=False)

Indicates whether the base forecasts include multiple samples. If True, reconciliation is applied sample-by-sample along the third axis.

Returns:#

:

y_tilde_meannp.ndarray

The reconciled forecast means, with the same shape as base_forecasts.

y_tilde_varnp.ndarray or list of np.ndarray

The reconciled forecast variances: - A single matrix of shape [n_total_series, n_total_series] if samples=False. - A list of such matrices, one per sample, if samples=True.

bayesreconpy.linear.projection.reconc_ols(A, base_forecasts, samples=False)#

Reconciles base forecasts using Ordinary Least Squares (OLS) based on a summing matrix.

This function applies forecast reconciliation using the OLS method, ensuring that the reconciled forecasts are coherent with the aggregation structure defined by the matrix A.

Parameters:#

Anp.ndarray

The aggregation constraint matrix (usually of shape [n_agg_levels, n_total_series]). It defines how bottom-level time series are aggregated into upper levels.

base_forecastsnp.ndarray

The base forecasts before reconciliation. Should be: - of shape [n_total_series, n_time, n_samples] if samples=True, or - of shape [n_total_series, n_time] if samples=False.

samplesbool, optional (default=False)

Indicates whether the base forecasts contain multiple samples (i.e., are probabilistic). If True, reconciliation is applied to each sample individually along the third axis.

Returns:#

:

y_tilde_meannp.ndarray

The reconciled forecasts: - of shape [n_total_series, n_time, n_samples] if samples=True, or - of shape [n_total_series, n_time] if samples=False.