Reconciliation Methods

In hierarchical forecasting, we aim to create forecasts for many time series concurrently, whilst adhering to pre-specified hierarchical relationships that exist between the time series. We can enforce this coherence by performing a post-processing reconciliation step on the forecasts. The HierarchicalForecast package provides the most comprehensive collection of Python implementations of hierarchical forecasting algorithms that follow classic hierarchical reconciliation. All the methods have a reconcile function capable of reconciling base forecasts using numpy arrays.

Cross-sectional hierarchies

Traditionally, hierarchical forecasting methods reconcile cross-sectional aggregations. For example, we may have forecasts for individual product demand, but also for the overall product group, department and store, and we are interested in making sure these forecasts are coherent with each other. This can be formalized as:

\tilde{\textbf{Y}} = SP\hat{\textbf{Y}} \;,

where

\hat{\textbf{Y}} \in \mathbb{R}^{m \times p}

denotes the matrix of forecasts for all

m

time series for all

p

time steps in the hierarchy,

S \in \lbrace 0, 1 \rbrace^{m \times n}

is a matrix that defines the hierarchical relationship between the

n

bottom-level time series and the

m^* = m - n

aggregations,

P \in \mathbb{R}^{n \times m}

is a matrix that encapsulates the contribution of each forecast to the final estimate, and

\tilde{\textbf{Y}} \in \mathbb{R}^{m \times p}

is the matrix of reconciled forecasts. We can use the matrix

P

to define various forecast contribution scenarios. Cross-sectional reconciliation methods aim to find the optimal

P

matrix.

Temporal hierarchies

We can also perform temporal reconciliation. For example, we may have forecasts for daily demand, weekly, and monthly, and we are interested in making sure these forecasts are coherent with each other. We formalize the temporal hierarchical forecasting problem as:

\tilde{\textbf{Y}} = \left( S_{te} P_{te} \hat{\textbf{Y}}^{\intercal} \right)^{\intercal} \;,

where

S_{te} \in \lbrace 0, 1 \rbrace^{p \times k}

is a matrix that defines the hierarchical relationship between the

k

bottom-level time steps and the

p^* = p - k

aggregations and

P_{te} \in \mathbb{R}^{k \times p}

is a matrix that encapsulates the contribution of each forecast to the final estimate. We can use the matrix

P_{te}

to define various forecast contribution scenarios. Temporal reconciliation methods aim to find the optimal

P_{te}

matrix.

Cross-temporal reconciliation

We can combine cross-sectional and temporal hierarchical forecasting by performing cross-sectional reconciliation and temporal reconciliation in a two-step procedure.

References

-Hyndman, Rob. Notation for forecast reconciliation.

1. Bottom-Up

`BottomUp`

BottomUp()

Bases: HReconciler Bottom Up Reconciliation Class. The most basic hierarchical reconciliation is performed using an Bottom-Up strategy. It was proposed for the first time by Orcutt in 1968. The corresponding hierarchical “projection” matrix is defined as:

\mathbf{P}_{\\text{BU}} = [\mathbf{0}_{\mathrm{[b],[a]}}\;|\;\mathbf{I}_{\mathrm{[b][b]}}]

References:

Orcutt, G.H., Watts, H.W., & Edwards, J.B.(1968). “Data aggregation and information loss”. The American Economic Review, 58 , 773(787).

`BottomUp.fit`

fit(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

Bottom Up Fit Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`Optional[ndarray]`	In-sample values of size (`base`, `horizon`). Default is None.	`None`
`y_hat_insample`	`Optional[ndarray]`	In-sample forecast values of size (`base`, `horizon`). Default is None.	`None`
`sigmah`	`Optional[ndarray]`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`intervals_method`	`Optional[str]`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`Optional[int]`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`Optional[int]`	Seed for reproducibility. Default is None.	`None`
`tags`	`Optional[dict[str, ndarray]]`	Tags for hierarchical structure. Default is None.	`None`

Returns:

Name	Type	Description
`BottomUp`	`object`	fitted reconciler.

`BottomUp.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`BottomUp.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

BottomUp Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`Optional[ndarray]`	In-sample values of size (`base`, `insample_size`). Default is None.	`None`
`y_hat_insample`	`Optional[ndarray]`	In-sample forecast values of size (`base`, `insample_size`). Default is None.	`None`
`sigmah`	`Optional[ndarray]`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`
`intervals_method`	`Optional[str]`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`Optional[int]`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`Optional[int]`	Seed for reproducibility. Default is None.	`None`
`tags`	`Optional[dict[str, ndarray]]`	Tags for hierarchical structure. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated y_hat using the Bottom Up approach.

`BottomUp.sample`

sample(num_samples)

Sample probabilistic coherent distribution. Generates n samples from a probabilistic coherent distribution. The method uses fitted mean and probabilistic reconcilers, defined by the intervals_method selected during the reconciler’s instantiation. Currently available: normality, bootstrap, permbu. Parameters:

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

`BottomUpSparse`

BottomUpSparse()

Bases: BottomUp BottomUpSparse Reconciliation Class. This is the implementation of a Bottom Up reconciliation using the sparse matrix approach. It works much more efficient on datasets with many time series. [makoren: At least I hope so, I only checked up until ~20k time series, and there’s no real improvement, it would be great to check for smth like 1M time series, where the dense S matrix really stops fitting in memory] See the parent class for more details.

`BottomUpSparse.fit`

fit(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

Bottom Up Fit Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`Optional[ndarray]`	In-sample values of size (`base`, `horizon`). Default is None.	`None`
`y_hat_insample`	`Optional[ndarray]`	In-sample forecast values of size (`base`, `horizon`). Default is None.	`None`
`sigmah`	`Optional[ndarray]`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`intervals_method`	`Optional[str]`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`Optional[int]`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`Optional[int]`	Seed for reproducibility. Default is None.	`None`
`tags`	`Optional[dict[str, ndarray]]`	Tags for hierarchical structure. Default is None.	`None`

Returns:

Name	Type	Description
`BottomUp`	`object`	fitted reconciler.

`BottomUpSparse.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`BottomUpSparse.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

BottomUp Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`Optional[ndarray]`	In-sample values of size (`base`, `insample_size`). Default is None.	`None`
`y_hat_insample`	`Optional[ndarray]`	In-sample forecast values of size (`base`, `insample_size`). Default is None.	`None`
`sigmah`	`Optional[ndarray]`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`
`intervals_method`	`Optional[str]`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`Optional[int]`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`Optional[int]`	Seed for reproducibility. Default is None.	`None`
`tags`	`Optional[dict[str, ndarray]]`	Tags for hierarchical structure. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated y_hat using the Bottom Up approach.

`BottomUpSparse.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

2. Top-Down

`TopDown`

TopDown(method)

Bases: HReconciler Top Down Reconciliation Class. The Top Down hierarchical reconciliation method, distributes the total aggregate predictions and decomposes it down the hierarchy using proportions

\mathbf{p}_{\mathrm{[b]}}

that can be actual historical values or estimated.

\mathbf{P}=[\mathbf{p}_{\mathrm{[b]}}\;|\;\mathbf{0}_{\mathrm{[b][a,b\;-1]}}]

Parameters:

Name	Type	Description	Default
`method`	`str`	One of `forecast_proportions`, `average_proportions` and `proportion_averages`.	required

References:

`TopDown.fit`

fit(S, y_hat, y_insample, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

TopDown Fit Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray`	Insample values of size (`base`, `insample_size`). Optional for `forecast_proportions` method.	required
`y_hat_insample`	`ndarray`	Insample forecast values of size (`base`, `insample_size`). Optional for `forecast_proportions` method.	`None`
`sigmah`	`ndarray`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray]`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`TopDown`	`object`	fitted reconciler.

`TopDown.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`TopDown.fit_predict`

fit_predict(S, y_hat, tags, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None)

Top Down Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`tags`	`dict[str, ndarray]`	Each key is a level and each value its `S` indices.	required
`y_insample`	`ndarray`	Insample values of size (`base`, `insample_size`). Optional for `forecast_proportions` method. Default is None.	`None`
`y_hat_insample`	`ndarray`	Insample forecast values of size (`base`, `insample_size`). Optional for `forecast_proportions` method. Default is None.	`None`
`sigmah`	`ndarray`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`level`	`list[int]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`
`intervals_method`	`str`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`int`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`int`	Seed for reproducibility.	`None`

Returns:

Name	Type	Description
`y_tilde`	`ndarray`	Reconciliated y_hat using the Top Down approach.

`TopDown.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

`TopDownSparse`

Bases: TopDown TopDownSparse Reconciliation Class. This is an implementation of top-down reconciliation using the sparse matrix approach. It works much more efficiently on data sets with many time series. See the parent class for more details.

`TopDownSparse.fit`

fit(S, y_hat, y_insample, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

TopDown Fit Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray`	Insample values of size (`base`, `insample_size`). Optional for `forecast_proportions` method.	required
`y_hat_insample`	`ndarray`	Insample forecast values of size (`base`, `insample_size`). Optional for `forecast_proportions` method.	`None`
`sigmah`	`ndarray`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray]`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`TopDown`	`object`	fitted reconciler.

`TopDownSparse.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`TopDownSparse.fit_predict`

fit_predict(S, y_hat, tags, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None)

`TopDownSparse.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

cls_top_down(
                S=S, y_hat=S @ y_hat_bottom, y_insample=S @ y_bottom, tags=tags
            )["mean"]

cls_top_down = TopDownSparse(method="average_proportions")
test_fail(
    cls_top_down,
    contains="Top-down reconciliation requires strictly hierarchical structures.",
    args=(sparse.csr_matrix(S_non_hier), None, tags_non_hier),
)

3. Middle-Out

`MiddleOut`

MiddleOut(middle_level, top_down_method)

Bases: HReconciler Middle Out Reconciliation Class. This method is only available for strictly hierarchical structures. It anchors the base predictions in a middle level. The levels above the base predictions use the Bottom-Up approach, while the levels below use a Top-Down. Parameters:

Name	Type	Description	Default
`middle_level`	`str`	Middle level.	required
`top_down_method`	`str`	One of `forecast_proportions`, `average_proportions` and `proportion_averages`.	required

References:

Hyndman, R.J., & Athanasopoulos, G. (2021). “Forecasting: principles and practice, 3rd edition: Chapter 11: Forecasting hierarchical and grouped series”. OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on July 2022.

`MiddleOut.fit`

fit(**kwargs)

`MiddleOut.predict`

predict(**kwargs)

`MiddleOut.fit_predict`

fit_predict(S, y_hat, tags, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None)

Middle Out Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`tags`	`dict[str, ndarray]`	Each key is a level and each value its `S` indices.	required
`y_insample`	`ndarray`	Insample values of size (`base`, `insample_size`). Only used for `forecast_proportions`. Default is None.	`None`
`y_hat_insample`	`ndarray`	In-sample forecast values of size (`base`, `insample_size`). Only used for `forecast_proportions`. Default is None.	`None`
`sigmah`	`ndarray`	Estimated standard deviation of the conditional marginal distribution. Default is None.	`None`
`level`	`list[int]`	Confidence levels for prediction intervals. Default is None.	`None`
`intervals_method`	`str`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`. Default is None.	`None`
`num_samples`	`int`	Number of samples for probabilistic coherent distribution. Default is None.	`None`
`seed`	`int`	Seed for reproducibility. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`		Reconciliated y_hat using the Middle Out approach.

`MiddleOut.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

`MiddleOutSparse`

Bases: MiddleOut MiddleOutSparse Reconciliation Class. This is an implementation of middle-out reconciliation using the sparse matrix approach. It works much more efficiently on data sets with many time series. See the parent class for more details.

`MiddleOutSparse.fit`

fit(**kwargs)

`MiddleOutSparse.predict`

predict(**kwargs)

`MiddleOutSparse.fit_predict`

fit_predict(S, y_hat, tags, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None)

`MiddleOutSparse.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

4. Min-Trace

`MinTrace`

MinTrace(method, nonnegative=False, mint_shr_ridge=2e-08, num_threads=1)

Bases: HReconciler MinTrace Reconciliation Class. This reconciliation algorithm proposed by Wickramasuriya et al. depends on a generalized least squares estimator and an estimator of the covariance matrix of the coherency errors

\mathbf{W}_{h}

. The Min Trace algorithm minimizes the squared errors for the coherent forecasts under an unbiasedness assumption; the solution has a closed form.

\mathbf{P}_{\text{MinT}}=\left(\mathbf{S}^{\intercal}\mathbf{W}_{h}\mathbf{S}\right)^{-1}\mathbf{S}^{\intercal}\mathbf{W}^{-1}_{h}

Parameters:

Name	Type	Description	Default
`method`	`str`	One of `ols`, `wls_struct`, `wls_var`, `mint_shrink`, `mint_cov`, `emint`.	required
`nonnegative`	`bool`	Reconciled forecasts should be nonnegative?	`False`
`mint_shr_ridge`	`float`	Ridge numeric protection to MinTrace-shr covariance estimator.	`2e-08`
`num_threads`	`int`	Number of threads for the C++ covariance backend (OpenMP) and for solving the optimization problems (when nonnegative=True).	`1`

References:

`MinTrace.fit`

fit(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTrace Fit Method. Parameters:

Name	Type	Description	Default
`S`		Summing matrix of size (`base`, `bottom`).	required
`y_hat`		Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used with “wls_var”, “mint_cov”, “mint_shrink”.	`None`
`y_hat_insample`	`ndarray \| None`	Insample forecast values of size (`base`, `insample_size`). Only used with “wls_var”, “mint_cov”, “mint_shrink”	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`self`		object, fitted reconciler.

`MinTrace.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`MinTrace.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTrace Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`y_hat_insample`	`ndarray \| None`	Insample fitted values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`level`	`list[int] \| None`	float list 0-100, confidence levels for prediction intervals.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`y_tilde`		Reconciliated y_hat using the MinTrace approach.

`MinTrace.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

`MinTraceSparse`

MinTraceSparse(method, nonnegative=False, num_threads=1, qp=True)

Bases: MinTrace MinTraceSparse Reconciliation Class. This is the implementation of OLS and WLS estimators using sparse matrices. It is not guaranteed to give identical results to the non-sparse version, but works much more efficiently on data sets with many time series. See the parent class for more details. Parameters:

Name	Type	Description	Default
`method`	`str`	One of `ols`, `wls_struct`, or `wls_var`.	required
`nonnegative`	`bool`	Return non-negative reconciled forecasts.	`False`
`num_threads`	`int`	Number of threads for non-negative quadratic programming calls.	`1`
`qp`	`bool`	Implement non-negativity constraint with a quadratic programming approach. Setting	`True`

`MinTraceSparse.fit`

fit(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTraceSparse Fit Method. Parameters:

Name	Type	Description	Default
`S`	`csr_matrix`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used with “wls_var”.	`None`
`y_hat_insample`	`ndarray \| None`	Insample forecast values of size (`base`, `insample_size`). Only used with “wls_var”	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`self`	`MinTraceSparse`	object, fitted reconciler.

`MinTraceSparse.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`MinTraceSparse.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTrace Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`y_hat_insample`	`ndarray \| None`	Insample fitted values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`level`	`list[int] \| None`	float list 0-100, confidence levels for prediction intervals.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`y_tilde`		Reconciliated y_hat using the MinTrace approach.

`MinTraceSparse.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

5. Optimal Combination

`OptimalCombination`

OptimalCombination(method, nonnegative=False, num_threads=1)

Bases: MinTrace Optimal Combination Reconciliation Class. This reconciliation algorithm was proposed by Hyndman et al. 2011, the method uses generalized least squares estimator using the coherency errors covariance matrix. Consider the covariance of the base forecast

\\textrm{Var}(\epsilon_{h}) = \Sigma_{h}

, the

\mathbf{P}

matrix of this method is defined by:

\mathbf{P} = \\left(\mathbf{S}^{\intercal}\Sigma_{h}^{\dagger}\mathbf{S}\\right)^{-1}\mathbf{S}^{\intercal}\Sigma^{\dagger}_{h}

where

\Sigma_{h}^{\dagger}

denotes the variance pseudo-inverse. The method was later proven equivalent to MinTrace variants. Parameters:

Name	Type	Description	Default
`method`	`str`	str, allowed optimal combination methods: ‘ols’, ‘wls_struct’.	required
`nonnegative`	`bool`	bool, reconciled forecasts should be nonnegative?	`False`

`OptimalCombination.fit`

fit(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTrace Fit Method. Parameters:

Name	Type	Description	Default
`S`		Summing matrix of size (`base`, `bottom`).	required
`y_hat`		Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used with “wls_var”, “mint_cov”, “mint_shrink”.	`None`
`y_hat_insample`	`ndarray \| None`	Insample forecast values of size (`base`, `insample_size`). Only used with “wls_var”, “mint_cov”, “mint_shrink”	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`self`		object, fitted reconciler.

`OptimalCombination.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`OptimalCombination.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

MinTrace Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Insample values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`y_hat_insample`	`ndarray \| None`	Insample fitted values of size (`base`, `insample_size`). Only used by `wls_var`, `mint_cov`, `mint_shrink`	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`level`	`list[int] \| None`	float list 0-100, confidence levels for prediction intervals.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`y_tilde`		Reconciliated y_hat using the MinTrace approach.

`OptimalCombination.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

6. Emp. Risk Minimization

`ERM`

ERM(method, lambda_reg=0.01)

Bases: HReconciler Empirical Risk Minimization Reconciliation Class. The Empirical Risk Minimization reconciliation strategy relaxes the unbiasedness assumptions from previous reconciliation methods like MinT and optimizes square errors between the reconciled predictions and the validation data to obtain an optimal reconciliation matrix P. The exact solution for

\mathbf{P}

(method='closed') follows the expression:

\mathbf{P}^{*} = \\left(\mathbf{S}^{\intercal}\mathbf{S}\\right)^{-1}\mathbf{Y}^{\intercal}\hat{\mathbf{Y}}\\left(\hat{\mathbf{Y}}\hat{\mathbf{Y}}\\right)^{-1}

The alternative Lasso regularized

\mathbf{P}

solution (method='reg_bu') is useful when the observations of validation data is limited or the exact solution has low numerical stability.

\mathbf{P}^{*} = \\text{argmin}_{\mathbf{P}} ||\mathbf{Y}-\mathbf{S} \mathbf{P} \hat{Y} ||^{2}_{2} + \lambda ||\mathbf{P}-\mathbf{P}_{\\text{BU}}||_{1}

Parameters:

Name	Type	Description	Default
`method`	`str`	str, one of `closed`, `reg` and `reg_bu`.	required
`lambda_reg`	`float`	float, l1 regularizer for `reg` and `reg_bu`.	`0.01`

`ERM.fit`

fit(S, y_hat, y_insample, y_hat_insample, sigmah=None, intervals_method=None, num_samples=None, seed=None, tags=None)

ERM Fit Method. Parameters:

Name	Type	Description	Default
`S`		Summing matrix of size (`base`, `bottom`).	required
`y_hat`		Forecast values of size (`base`, `horizon`).	required
`y_insample`		Train values of size (`base`, `insample_size`).	required
`y_hat_insample`		Insample train predictions of size (`base`, `insample_size`).	required
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`self`		object, fitted reconciler.

`ERM.predict`

predict(S, y_hat, level=None)

Predict using reconciler. Predict using fitted mean and probabilistic reconcilers. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`level`	`Optional[list[int]]`	float list 0-100, confidence levels for prediction intervals. Default is None.	`None`

Returns:

Name	Type	Description
`y_tilde`	`dict`	Reconciliated predictions.

`ERM.fit_predict`

fit_predict(S, y_hat, y_insample=None, y_hat_insample=None, sigmah=None, level=None, intervals_method=None, num_samples=None, seed=None, tags=None)

ERM Reconciliation Method. Parameters:

Name	Type	Description	Default
`S`	`ndarray`	Summing matrix of size (`base`, `bottom`).	required
`y_hat`	`ndarray`	Forecast values of size (`base`, `horizon`).	required
`y_insample`	`ndarray \| None`	Train values of size (`base`, `insample_size`).	`None`
`y_hat_insample`	`ndarray \| None`	Insample train predictions of size (`base`, `insample_size`).	`None`
`sigmah`	`ndarray \| None`	Estimated standard deviation of the conditional marginal distribution.	`None`
`level`	`list[int] \| None`	float list 0-100, confidence levels for prediction intervals.	`None`
`intervals_method`	`str \| None`	Sampler for prediction intervals, one of `normality`, `bootstrap`, `permbu`, `conformal`.	`None`
`num_samples`	`int \| None`	Number of samples for probabilistic coherent distribution.	`None`
`seed`	`int \| None`	Seed for reproducibility.	`None`
`tags`	`dict[str, ndarray] \| None`	Each key is a level and each value its `S` indices.	`None`

Returns:

Name	Type	Description
`y_tilde`		Reconciliated y_hat using the ERM approach.

`ERM.sample`

sample(num_samples)

Name	Type	Description	Default
`num_samples`	`int`	number of samples generated from coherent distribution.	required

Returns:

Name	Type	Description
`samples`	`ndarray`	Coherent samples of size (`num_series`, `horizon`, `num_samples`).

​Cross-sectional hierarchies

​Temporal hierarchies

​Cross-temporal reconciliation

​References

​1. Bottom-Up

​BottomUp

​BottomUp.fit

​BottomUp.predict

​BottomUp.fit_predict

​BottomUp.sample

​BottomUpSparse

​BottomUpSparse.fit

​BottomUpSparse.predict

​BottomUpSparse.fit_predict

​BottomUpSparse.sample

​2. Top-Down

​TopDown

​TopDown.fit

​TopDown.predict

​TopDown.fit_predict

​TopDown.sample

​TopDownSparse

​TopDownSparse.fit

​TopDownSparse.predict

​TopDownSparse.fit_predict

​TopDownSparse.sample

​3. Middle-Out

​MiddleOut

​MiddleOut.fit

​MiddleOut.predict

​MiddleOut.fit_predict

​MiddleOut.sample

​MiddleOutSparse

​MiddleOutSparse.fit

​MiddleOutSparse.predict

​MiddleOutSparse.fit_predict

​MiddleOutSparse.sample

​4. Min-Trace

​MinTrace

​MinTrace.fit

​MinTrace.predict

​MinTrace.fit_predict

​MinTrace.sample

​MinTraceSparse

​MinTraceSparse.fit

​MinTraceSparse.predict

​MinTraceSparse.fit_predict

​MinTraceSparse.sample

​5. Optimal Combination

​OptimalCombination

​OptimalCombination.fit

​OptimalCombination.predict

​OptimalCombination.fit_predict

​OptimalCombination.sample

​6. Emp. Risk Minimization

​ERM

​ERM.fit

​ERM.predict

​ERM.fit_predict

​ERM.sample

​References

​General Reconciliation

​Hierarchical Probabilistic Coherent Predictions

Cross-sectional hierarchies

Temporal hierarchies

Cross-temporal reconciliation

References

1. Bottom-Up

`BottomUp`

`BottomUp.fit`

`BottomUp.predict`

`BottomUp.fit_predict`

`BottomUp.sample`

`BottomUpSparse`

`BottomUpSparse.fit`

`BottomUpSparse.predict`

`BottomUpSparse.fit_predict`

`BottomUpSparse.sample`

2. Top-Down

`TopDown`

`TopDown.fit`

`TopDown.predict`

`TopDown.fit_predict`

`TopDown.sample`

`TopDownSparse`

`TopDownSparse.fit`

`TopDownSparse.predict`

`TopDownSparse.fit_predict`

`TopDownSparse.sample`

3. Middle-Out

`MiddleOut`

`MiddleOut.fit`

`MiddleOut.predict`

`MiddleOut.fit_predict`

`MiddleOut.sample`

`MiddleOutSparse`

`MiddleOutSparse.fit`

`MiddleOutSparse.predict`

`MiddleOutSparse.fit_predict`

`MiddleOutSparse.sample`

4. Min-Trace

`MinTrace`

`MinTrace.fit`

`MinTrace.predict`

`MinTrace.fit_predict`

`MinTrace.sample`

`MinTraceSparse`

`MinTraceSparse.fit`

`MinTraceSparse.predict`

`MinTraceSparse.fit_predict`

`MinTraceSparse.sample`

5. Optimal Combination

`OptimalCombination`

`OptimalCombination.fit`

`OptimalCombination.predict`

`OptimalCombination.fit_predict`

`OptimalCombination.sample`

6. Emp. Risk Minimization

`ERM`

`ERM.fit`

`ERM.predict`

`ERM.fit_predict`

`ERM.sample`

References

General Reconciliation

Hierarchical Probabilistic Coherent Predictions