Time series and forecasting · insight-1213
Never add per-item quantile forecasts to get a quantile for the total
- Evidence type
- documented fact
- Difficulty
- medium
- Roles
- Data scientist, Data analyst
- Industries
- aviation, retail, logistics
Summary: situation, action, reason, boundary
Situation
A planner sums the 90th-percentile forecasts of 40 flights to size a network-wide buffer.
Action
Produce joint sample paths (or a simulation from the fitted models, with correlation), sum the samples, and take the quantile of the summed samples. Check the result against realised totals.
Reason
Quantiles are not additive: the sum of the quantiles equals the quantile of the sum only under perfect positive dependence.
Boundary
Aggregating over items, locations, or horizons for stock, capacity or budget buffers.
Symptom or risk
Item quantiles add up to far more than the quantile of the sum when items are not perfectly correlated, so the buffer is oversized by an amount that grows with the number of items.
Exceptions
Means and expectations add; point forecasts of the mean can be summed safely.
Trade-offs
Simulation needs a dependence assumption; wrong correlation misstates the total.
Pitfalls
Assuming independence when items share a demand driver, which understates the aggregate spread.
Example
Hypothetical: two independent items, each normal with mean 100 and sd 20. Summed 90th percentiles give 251.3; the 90th percentile of the total is 236.2.
Lessons and concepts
Claims and their sources
For reconciled distributional forecasts, bootstrapped sample paths are reconciled and percentiles of the paths give coherent prediction intervals; with normal base forecasts reconciled forecasts are also normal.