Can I add monthly mean forecasts?

Yes, when they refer to compatible quantities and the same information set.

Can I add monthly medians?

The sum need not be the median of the total.

Does aggregation remove uncertainty?

No. It changes the target and the relevant uncertainty calculation.

Should I sum month-end inventory balances?

Only if that sum is genuinely the quantity needed; it is not the same as total demand or final inventory.

AI Glossary / Time series and signal processing / Intermediate

Cumulative forecasting

Definition

Cumulative forecasting predicts the total amount of a variable over several future periods. Instead of focusing only on separate monthly or weekly values, it addresses a combined quantity, such as total sales during the next quarter. The aggregation window, starting point and meaning of the underlying variable must be specified.

What it means

A business decision may depend on a total over a lead time rather than on a single period. A supplier might need enough stock for the entire interval before the next delivery. A project manager may need the expected spending across the remaining stages of a project.

Cumulative forecasting describes the target being forecast, not one particular algorithm. A total can be modeled directly or constructed from forecasts for its component periods. These approaches need not produce identical fitted forecasts. The appropriate target also depends on measurement: flows such as monthly sales can be summed, whereas adding successive month-end account balances usually answers a different and less useful question.

How it works

Define the future periods included in the total and ensure their quantities are additive and use compatible units. One approach produces a joint forecast for those periods and sums each predicted path. Another fits a model specifically for the aggregate target using an appropriate historical construction.

If the period forecasts are conditional means based on the same information, their sum is the conditional mean of the total. This follows from linearity of expectation and does not require independent periods. Medians and other quantiles generally do not add in the same way.

Uncertainty requires more care. The variance of a sum includes each component variance and twice every distinct pairwise covariance. Simulating coherent future paths can preserve these dependencies. Evaluate performance for the cumulative horizon that the decision actually uses, including overlap between successive evaluation windows.

Three practical examples

1. Inventory planning: A distributor forecasts demand across a delivery lead time and examines uncertainty in the total. This supports a replenishment decision while keeping service targets separate from the mean forecast.

2. Project budgeting: A manager forecasts spending over the next three reporting periods. The total helps estimate funding requirements, while the period breakdown remains useful for payment timing.

3. Capacity planning: A service organization forecasts the combined workload during a holiday interval. It also retains daily forecasts because identical totals can create very different peak staffing needs.

Think of it this way

Think of planning water for a journey. The amount needed for each day matters, but the total needed before the next refill answers a separate planning question. Summing expected daily use is straightforward; uncertainty in the total depends partly on whether unusually high-use days tend to occur together.

Worked example

In an original example, forecasts of the next three monthly sales amounts are 100, 120 and 130 units. Treat them as conditional means using the same forecast-origin information. The forecast of the three-month total is 100 + 120 + 130 = 350 units.

Suppose the specified joint forecast-error model gives variance 100 square units for each month. Let the pairwise error covariances be 20 for months one and two, 10 for months one and three, and 20 for months two and three. The total-error variance is 100 + 100 + 100 + 2 × (20 + 10 + 20) = 400, so its standard deviation is 20 units.

Ignoring those covariances would instead give a standard deviation of the square root of 300, approximately 17.32 units. The numbers are assumed teaching inputs, not estimates from a real series. A standard deviation alone does not determine a valid prediction interval without further distributional assumptions or a suitable empirical construction.

Illustrative case study

In a fictional case, Clearbank Supplies forecasts each week separately and adds the point forecasts for its four-week replenishment window. Its uncertainty calculation initially assumes independent weekly errors, although promotions tend to affect several consecutive weeks.

The analyst replaces that assumption with a jointly simulated forecast and checks cumulative forecast errors at historical order dates. Managers continue to examine weekly peaks because the warehouse has limited daily handling capacity. The fictional case illustrates why an aggregate target needs an aggregate uncertainty assessment. It does not claim that one method universally improves stock availability or reduces costs.

Common misconceptions and mistakes

Cumulative forecasting is not simply forecasting the last value in a window. Nor does every variable have a meaningful cumulative interpretation: stock levels, rates and percentages require careful treatment rather than automatic summation.

Adding individual lower and upper interval limits does not generally produce an interval with the desired coverage for the total. Summing medians does not generally yield the median total. Dependence may increase or reduce aggregate uncertainty, depending on its sign and structure. A small cumulative error can also conceal large offsetting errors across periods, which may still matter operationally.

Business use

Use cumulative forecasts when commitments depend on an amount over a defined interval. Align the window with procurement lead times, budget periods or contractual obligations, and document whether the forecast concerns a future total or a running total that includes known actuals.

Keep both aggregate and period-level views when timing matters. Backtest at realistic decision dates and compare with simple benchmarks using the same target. Report uncertainty alongside the point forecast, with assumptions about dependence made explicit. A forecast of expected demand is not automatically the right order quantity: costs, capacity, service objectives and the consequences of shortage still influence the decision.

Check your understanding

What is the example's cumulative point forecast?

Answer: It is 350 units.

Why is the total-error standard deviation 20 rather than 17.32?

Answer: The specified positive cross-period covariances contribute to the variance.

Can a correct total guarantee correct timing?

Answer: No. Period errors can offset each other.

Related terms

Forecast horizon
Conditional expectation
Covariance
Prediction interval
lead-time demand
temporal aggregation
forecast reconciliation
simulation

Learn next

Learn next how joint forecast paths represent dependence, how cumulative forecast errors are evaluated, and how temporal aggregation differs from aggregation across products. Then connect uncertainty in lead-time demand with the separate decision about stock or capacity.

Sources

Makridakis, Wheelwright and Hyndman, Glossary of Forecasting Terms, Cumulative forecasting: https://robjhyndman.com/mwh3/FG4.pdf . The primary author-hosted glossary was consulted for the multi-period-total definition. The covariance calculation and planning examples are original, independently checked teaching material; they do not reproduce an empirical forecasting study.

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