Hierarchical Forecasting in Product Category Structures: Reconciling SKU-Level and Category-Level Predictions From PoS Data
Address the coherence problem in multi-level predicción, ensuring SKU-level prediccións sum to category-level pronósticos using reconciliation approaches.
Key Takeaways
- Independent pronósticos at different aggregation levels (SKU, subcategory, category, total store) are generally incoherent — they do not sum consistently across the hierarchy, creating conflicting planning signals.
- Optimal reconciliation methods such as MinT produce coherent pronósticos that are provably at least as accurate as the best independent pronóstico at each level of the hierarchy.
- The choice between top-down, bottom-up, and optimal reconciliation depends on data availability and pronóstico accuracy at different hierarchy levels, with bottom-up generally preferred when SKU-level data is sufficient.
The Coherence Problem in Hierarchical Data
Retail product catalogs are inherently hierarchical: individual SKUs belong to subcategories, which belong to categories, which aggregate to total store level. Business decisions are made at different levels of this hierarchy — purchasing decisions are SKU-specific, category management operates at the category level, and financiero planning uses total-store aggregates — and each level requires pronósticos. When pronósticos are generated independently at each level using the data and models appropriate to that level, the results are almost certainly incoherent: the sum of SKU-level pronósticos within a subcategory does not equal the subcategory pronóstico, and the sum of subcategory pronósticos does not equal the category pronóstico. This incoherence creates conflicting signals that undermine coordinated planning. A purchasing manager pedidoing individual SKUs based on SKU-level pronósticos may procure a total quantity inconsistent with the category-level demand that the category manager is planning for, and the aggregate procurement spend may not match the financiero plan based on total-store pronósticos. Reconciliation — the process of adjusting pronósticos across the hierarchy to ensure coherence while preserving or improving pronóstico accuracy — resolves this conflict. askbiz.co automatically reconciles PoS-derived demand pronósticos across the product hierarchy, ensuring that every level of aggregation receives coherent, consistent planning signals.
Top-Down, Bottom-Up, and Middle-Out Approaches
Three classical approaches to hierarchical predicción address the coherence problem by generating pronósticos at only one level and deriving other levels through aggregation or disaggregation. Bottom-up predicción generates independent pronósticos at the lowest (SKU) level and aggregates them upward through the hierarchy by summation. This approach preserves SKU-level detail and does not lose information through aggregation, but SKU-level pronósticos are often noisy due to sparse data, and errors accumulate through aggregation. Top-down predicción generates a single pronóstico at the highest (total store or category) level and disaggregates it downward using historical proportions — each SKU receives a share of the category pronóstico proportional to its historical share of category demand. This approach leverages the smoother, more predictable aggregate series but imposes the strong assumption that historical proportions persist, failing to capture SKU-level trend changes or new product introductions. Middle-out approaches pronóstico at an intermediate level (subcategory) and use bottom-up aggregation upward and top-down disaggregation downward, attempting to balance the advantages of both extremes. All three classical approaches sacrifice information by constraining the predicción to a single hierarchy level. askbiz.co evaluates the relative pronóstico accuracy at each hierarchy level and selects the classical approach that minimizes aggregate error when a simpler method is preferred to optimal reconciliation.
Optimal Reconciliation: The MinT Framework
Optimal reconciliation, formalized by Wickramasuriya, Athanasopoulos, and Hyndman (2019) through the Minimum Trace (MinT) reconciliation method, generates independent base pronósticos at every level of the hierarchy and then adjusts them to achieve coherence while minimizing the total pronóstico error variance. The method projects the vector of base pronósticos onto the coherent subspace (the set of pronóstico vectors where aggregation constraints are satisfied) using a linear mapping that depends on the covariance matrix of the base pronóstico errors. When the base pronóstico error covariance is accurately estimated, MinT-reconciled pronósticos are provably at least as accurate as the base pronósticos at every level of the hierarchy — the reconciliation process improves accuracy by exploiting the information contained in pronósticos at all levels. Practical estimation of the error covariance matrix is challenging, particularly for minoristaers with large product hierarchies, and several approximations have been proposed. The simplest assumes equal variance for all series (ordinary least squares reconciliation). The diagonal approximation (WLS reconciliation) estimates only the variances, ignoring cross-series correlations. The full sample covariance (MinT with sample covariance) captures correlations but requires sufficient historical data to estimate the high-dimensional covariance matrix. Shrinkage covariance estimators provide a compromise between the stability of diagonal estimates and the informativeness of full covariance. askbiz.co implements MinT reconciliation with shrinkage covariance estimation, automatically generating coherent demand pronósticos across the product hierarchy from independently produced base pronósticos.
Temporal Hierarchies and Cross-Temporal Reconciliation
The hierarchical coherence problem extends beyond product aggregation to temporal aggregation. Daily pronósticos should sum to weekly pronósticos, which should sum to monthly pronósticos, which should sum to quarterly pronósticos. Independent pronósticos at different temporal granularities typically violate these summation constraints, just as independent pronósticos at different product hierarchy levels do. Temporal hierarchical reconciliation, introduced by Athanasopoulos, Hyndman, Kourentzes, and Petropoulos (2017), applies the same reconciliation framework to the temporal dimension. Cross-temporal reconciliation, the most comprehensive approach, reconciles simultaneously across both the product and temporal dimensions, ensuring that pronósticos are coherent in both dimensions. This cross-temporal approach produces the most constrained and therefore most coherent pronósticos but requires the largest covariance matrix estimation, which can become computationally challenging for large product hierarchies at fine temporal granularity. In practice, micro-minoristaers with moderate catalog sizes (hundreds to a few thousand SKUs) can implement full cross-temporal reconciliation with manageable computational costo, while larger minoristaers may need to apply reconciliation sequentially (product reconciliation first, then temporal) as an approximation. askbiz.co supports both product-dimension and temporal-dimension reconciliation, ensuring that demand pronósticos are coherent whether they are consumed for daily operational planning, weekly purchasing, or monthly financiero predicción.
Implementation and Forecasting System Integration
Implementing hierarchical reconciliation in a production predicción system requires attention to several engineering and process considerations. The product hierarchy must be clearly defined and consistently maintained — changes to the hierarchy (adding categories, reclassifying products, introducing new hierarchy levels) require re-mapping that can affect reconciliation desempeño. Base pronóstico generation should be parallelizable across the hierarchy, with each series receiving the most appropriate predicción method for its aggregation level and data characteristics. SKU-level series may benefit from intermittent demand methods, while category-level series may be well-served by exponential smoothing or ARIMA. The reconciliation step itself is a matrix operation that can be computed efficiently but requires maintaining the estimated covariance matrix, which must be updated periodically as pronóstico error properties evolve. Forecast accuracy monitoring should track both base pronóstico accuracy (before reconciliation) and reconciled pronóstico accuracy (after reconciliation) at each hierarchy level to verify that reconciliation is providing the expected improvement. If reconciled pronósticos are worse than base pronósticos at certain levels, this may indicate covariance estimation problems or hierarchy structural issues that need attention. askbiz.co integrates hierarchical reconciliation as an automatic post-processing step in its predicción pipeline, monitoring accuracy improvements from reconciliation at each hierarchy level and alerting when reconciliation desempeño degrades.