Forecast Reconciliation

Ensures that forecasts generated at different aggregation levels—such as SKU-level, product category, and national totals—are mathematically coherent. The core principle is that lower-level forecasts must sum exactly to higher-level totals.

• Bottom-Up: Aggregates detailed SKU forecasts upward
• Top-Down: Disaggregates strategic forecasts downward using historical proportions
• Middle-Out: Combines both approaches from an intermediate hierarchy level

This prevents the classic disconnect where regional sales forecasts contradict corporate financial projections.

Uses statistical weighting to produce the most accurate reconciled forecasts by leveraging information from all hierarchy levels simultaneously. MinT (Minimum Trace) reconciliation minimizes the total forecast error variance.

• OLS (Ordinary Least Squares): Simple but ignores scale differences between levels
• WLS (Weighted Least Squares): Accounts for varying forecast accuracy across series
• MinT (Sample): Uses historical forecast error covariance for optimal weighting
• MinT (Shrinkage): Applies regularization when error covariance is poorly estimated

These methods consistently outperform naive bottom-up or top-down approaches in empirical studies.

Reconciliation bridges the gap between operational planning and financial planning by ensuring a single source of truth. Without reconciliation, procurement buys to one forecast while finance budgets to another.

• S&OP Alignment: Unifies Sales & Operations Planning with a coherent demand signal
• Inventory Optimization: Prevents overstocking from inflated top-down forecasts
• Supplier Collaboration: Provides consistent demand signals to upstream partners
• Executive Reporting: Eliminates confusion from conflicting departmental projections

This coherence is critical for autonomous supply chain systems making real-time decisions.

Extends traditional point forecast reconciliation to full probability distributions, preserving uncertainty quantification across the hierarchy. This ensures that prediction intervals at the aggregate level are consistent with lower-level uncertainty.

• Copula-Based Methods: Model dependence between hierarchical nodes
• Bayesian Reconciliation: Incorporate prior beliefs about hierarchical relationships
• Gaussian Aggregation: Sum lower-level distributions to form coherent aggregate distributions
• Quantile Coherence: Ensure quantile forecasts satisfy hierarchical constraints

This is essential for safety stock calculations that depend on accurate demand variability estimates at every echelon.

Ensures coherence across time granularities, so that daily forecasts sum to weekly totals, which in turn sum to monthly and quarterly projections. This is critical for planning horizons that span operational to strategic timeframes.

• Temporal Hierarchies: Define aggregation constraints across time buckets
• Rolling Horizon Consistency: Maintain coherence as forecasts update with new data
• Lead Time Alignment: Match forecast granularity to supplier lead times
• Seasonal Decomposition: Preserve seasonal patterns across aggregation levels

Temporal reconciliation prevents the common problem where short-term demand sensing contradicts medium-term trend forecasts.

Reconciled forecasts must be evaluated using metrics that account for hierarchical structure. Standard accuracy measures applied independently at each level can be misleading.

• Hierarchical RMSE: Weighted error across all hierarchy levels
• Reconciliation Bias: Measures systematic over- or under-forecasting after reconciliation
• Information Loss: Quantifies how much signal is lost during aggregation
• Coherence Ratio: Proportion of time that reconciled forecasts remain consistent

Proper evaluation ensures that reconciliation improves decision-making rather than just enforcing mathematical constraints.

A structured collection of time series organized by aggregation constraints—product SKUs roll up to categories, regional demand sums to national totals. Reconciliation ensures that forecasts generated independently at each level satisfy these linear summation constraints, eliminating inconsistencies between bottom-up operational plans and top-down financial targets.

The Minimum Trace (MinT) estimator produces the most accurate reconciled forecasts by weighting base forecasts according to their covariance structure. Unlike naive top-down or bottom-up approaches, MinT:

• Accounts for forecast error correlations between series
• Minimizes the variance of the reconciled forecasts
• Adapts to the relative accuracy of each base forecasting model

Two fundamental reconciliation strategies:

Bottom-Up: Generate forecasts at the most granular level and sum upward. Preserves local patterns but amplifies noise at higher levels.

Top-Down: Forecast at the aggregate level and disaggregate downward using historical proportions. Stable at the top but loses granular signal.

Middle-Out: A hybrid approach that forecasts from an intermediate level outward, balancing both perspectives.

The summing matrix mathematically encodes the hierarchical structure, mapping bottom-level series to all aggregate nodes. For a hierarchy with n bottom-level series and m total nodes, this m × n matrix defines the linear constraints that reconciled forecasts must satisfy: ỹ = S * b̃, where ỹ are reconciled forecasts and b̃ are bottom-level predictions.

Extends point forecast reconciliation to full probability distributions, ensuring that the joint predictive distribution respects hierarchical constraints. Techniques include:

• Copula-based reconciliation that preserves dependency structures
• Bayesian hierarchical models with coherent priors
• Score-optimal reconciliation that minimizes the Continuous Ranked Probability Score (CRPS) across all levels simultaneously

While often conflated, these are distinct operations:

Combination: Weighted averaging of multiple forecasts for the same series to reduce model risk.

Reconciliation: Adjusting forecasts across different series to satisfy aggregation constraints.

In practice, both techniques are applied sequentially—first combine models per series, then reconcile across the hierarchy.