Hierarchical Time Series

A defining property where lower-level series must mathematically sum to higher-level totals. In supply chains, daily SKU-level demand at a specific warehouse must equal the weekly regional demand for that product category.

• Summing matrix: A linear operator that maps bottom-level forecasts to all higher aggregation nodes
• Coherency violation: Occurs when independently generated forecasts at different levels produce conflicting totals
• Cross-sectional hierarchy: Product, geographic, and temporal dimensions create intersecting aggregation paths
• Example: National sales forecast ≠ sum of regional forecasts, creating confusion for inventory planners

Methods for adjusting base forecasts to enforce mathematical coherence across the hierarchy without sacrificing accuracy at any level.

• Bottom-up: Generate forecasts at the most granular level and aggregate upward—preserves detail but amplifies noise
• Top-down: Forecast at the highest level and disaggregate using historical proportions—stable but loses local patterns
• Middle-out: Start at an intermediate level, forecast up and down—balances granularity and stability
• Optimal reconciliation: Uses generalized least squares to produce coherent forecasts that minimize total error variance across all levels

The statistical dependencies that exist between series at different hierarchical levels, which naive independent forecasting ignores.

• Shared shocks: A supply disruption at a regional distribution center simultaneously impacts all downstream store-level series
• Information pooling: Higher-level aggregates often exhibit more stable patterns, providing useful regularization for noisy bottom-level forecasts
• Covariance estimation: Optimal reconciliation requires estimating the full covariance matrix of base forecast errors across all nodes
• Example: A national promotion lifts all regional demand simultaneously, creating correlated forecast errors that reconciliation must account for

Aggregation constraints that operate across time dimensions, where daily forecasts must sum to weekly, monthly, and quarterly totals.

• Non-overlapping aggregation: Daily → Weekly → Monthly → Quarterly → Annual
• Seasonal alignment: Weekly patterns must be preserved when aggregating to monthly buckets
• Forecast horizon consistency: Short-term daily forecasts and long-term annual projections must not contradict each other
• Example: A demand sensing model predicting daily sales must reconcile with the S&OP process's monthly volume forecast

Extending reconciliation beyond point forecasts to ensure that entire predictive distributions are consistent across hierarchy levels.

• Distributional reconciliation: Adjusts quantile forecasts so that the sum of lower-level distributions matches the upper-level distribution
• Copula-based methods: Model the dependence structure between hierarchical nodes to preserve joint distributional properties
• Gaussian aggregation: Under normality assumptions, the mean and variance of aggregated forecasts can be derived analytically
• Example: The 95th percentile of national demand must equal the convolution of regional demand distributions, not their simple sum

A diagnostic framework for measuring whether hierarchical reconciliation improves or degrades accuracy at each node compared to base forecasts.

• Node-level metrics: Evaluate reconciliation impact using CRPS or MASE at every aggregation point
• Reconciliation penalty: Some nodes may experience accuracy loss to achieve global coherence—quantify this trade-off
• Disaggregation gain: Bottom-level forecasts often improve when informed by stable top-level patterns
• Example: A retail chain finds that reconciliation reduces regional forecast error by 12% but increases store-level error by 3%, requiring a business decision on acceptable trade-offs

The mathematical process of adjusting forecasts generated independently at different hierarchy levels to ensure aggregate consistency. Without reconciliation, bottom-level SKU forecasts will not sum to the category total, creating planning chaos.

• Top-down: National forecast is disaggregated to lower levels using historical proportions
• Bottom-up: Detailed forecasts are summed upward to create higher-level views
• Middle-out: A middle level is forecast, then both aggregated up and disaggregated down
• Optimal reconciliation uses the covariance structure of forecast errors to produce the most accurate coherent forecasts across all levels

An attention-based deep learning architecture purpose-built for interpretable multi-horizon forecasting with native support for hierarchical structures. Unlike black-box models, TFT provides variable selection networks and attention weights that explain which features drive predictions.

• Handles static metadata (product category, location), known future inputs (holidays, promotions), and observed historical inputs (past sales) simultaneously
• Quantile outputs produce full prediction intervals, not just point estimates
• Naturally extends to hierarchical settings by sharing learning across related time series

Two fundamental strategies for generating coherent hierarchical forecasts, each with distinct trade-offs in accuracy and computational complexity.

• Bottom-up: Forecast at the most granular level (SKU-store-day), then aggregate upward. Preserves local patterns but accumulates noise and may miss macro trends
• Top-down: Forecast at the aggregate level, then distribute downward using historical proportions. Stable and fast but obliterates granular signals
• Middle-out hybrid approaches often outperform both extremes in practice, especially when combined with optimal reconciliation

Amazon's autoregressive recurrent neural network that produces probabilistic forecasts by learning a parametric distribution (e.g., negative binomial for count data) from hundreds or thousands of related time series. DeepAR excels in hierarchical settings because it shares global parameters while maintaining local context.

• Outputs full probability distributions, enabling quantile-based safety stock calculations
• Handles cold-start items by leveraging learned similarities across the hierarchy
• Particularly effective for intermittent and lumpy demand patterns common in spare parts supply chains

The direct bridge between probabilistic hierarchical forecasts and operational inventory decisions. A prediction interval specifies the range within which future demand will fall with a given probability (e.g., 95%).

• A 95% prediction interval means there is only a 5% chance actual demand exceeds the upper bound
• Service level targets (e.g., 98% fill rate) map directly to specific quantiles of the forecast distribution
• Hierarchical reconciliation must preserve the coherence of uncertainty — the variance of the aggregate forecast should equal the sum of lower-level variances plus covariance terms

The strictly proper scoring rule used to evaluate probabilistic hierarchical forecasts. CRPS measures the integrated squared difference between the predicted cumulative distribution function and the observed outcome, rewarding both calibration and sharpness.

• Unlike point-error metrics (MAE, RMSE), CRPS penalizes overconfident narrow intervals that miss the truth
• A CRPS of zero indicates a perfect probabilistic forecast
• Essential for model selection when comparing DeepAR, TFT, and other probabilistic methods across hierarchy levels