Croston's Method

Croston's Method separates the forecasting problem into two independent components: demand size (the magnitude of non-zero demand occurrences) and demand interval (the time between demand occurrences). Each component is updated independently using simple exponential smoothing, but only when demand actually occurs. This decomposition prevents the forecast from being artificially depressed during long sequences of zero-demand periods, which is the fundamental flaw of standard exponential smoothing when applied to intermittent demand patterns.

A defining characteristic of Croston's Method is that parameter updates occur exclusively in periods with positive demand. When a zero-demand period is observed, neither the demand size estimate nor the interval estimate is revised. This contrasts sharply with standard exponential smoothing, which would update the level component at every time step and systematically drive the forecast toward zero during extended demand droughts. The update-only-on-demand rule preserves the integrity of the non-zero demand signal.

The original Croston's Method produces forecasts that are positively biased—they systematically overestimate expected demand. The Syntetos-Boylan Approximation (SBA) corrects this by applying a multiplicative bias reduction factor of (1 − α/2), where α is the smoothing parameter for the interval estimate. This adjustment is critical for inventory optimization, as uncorrected Croston forecasts lead to excess safety stock and inflated holding costs. The SBA variant is now considered standard practice in production forecasting systems.

Croston's Method operates under the assumption that demand occurrences are independent and identically distributed following a Bernoulli process, and that demand sizes are independent of the intervals between them. This assumption simplifies the mathematics but can break down in practice when demand exhibits autocorrelation, trend, or seasonality. In such cases, extensions like the Croston-TSB model (which incorporates probability of demand occurrence) or negative binomial Croston variants may be required to capture more complex intermittent patterns.

Croston's Method is tightly coupled with inventory replenishment logic, particularly in spare parts management and aftermarket supply chains. The forecast output—a rate of demand per period—directly feeds into reorder point calculations and safety stock determinations. When combined with a compound Poisson demand model, Croston-based forecasts enable accurate estimation of fill rates and stockout probabilities for slow-moving SKUs where traditional normal distribution assumptions fail. This integration makes it a cornerstone of service-parts logistics systems.

Evaluating Croston's Method requires metrics suited to intermittent demand. Standard measures like Mean Absolute Percentage Error (MAPE) become undefined or infinite when actual demand is zero. Instead, practitioners use Mean Absolute Scaled Error (MASE), Periods in Stock simulations, or inventory-oriented metrics that measure the financial impact of forecast errors on holding costs and service levels. The ultimate test of a Croston forecast is not point accuracy but whether it minimizes total inventory cost while meeting target service levels.

The specific demand pattern that Croston's Method is designed to solve. It is characterized by sporadic demand occurrences separated by frequent periods of zero demand.

• Key Metric: High zero-inflation (often >30% of periods have no demand)
• Common in: Spare parts, aftermarket services, and capital goods
• Failure of standard models: Simple Exponential Smoothing produces biased forecasts by updating estimates during zero-demand periods

One of the two independent components estimated by Croston's decomposition. It represents the average time between non-zero demand occurrences.

• Updated only when demand occurs, preventing bias from zero periods
• Typically modeled using a simple exponential smoothing process
• Directly determines the frequency component of the final forecast
• A long interval signals sporadic, lumpy demand patterns

The second component in Croston's decomposition, representing the magnitude of demand when it occurs. It is estimated independently from the interval.

• Also updated only after a non-zero demand observation
• Uses simple exponential smoothing on the positive demand values
• Separating size from interval prevents the forecast from being dragged down by zero periods
• Critical for sizing safety stock for lumpy items

The Syntetos-Boylan Approximation, a bias correction to Croston's original method. Croston's estimator is mathematically biased because the expectation of a ratio does not equal the ratio of expectations.

• Applies a correction factor of (1 - α/2) to the final forecast
• Reduces the positive bias inherent in the original Croston formula
• Published in 2005, it has become the standard variant used in practice
• Maintains the same decomposition structure while improving accuracy

The Teunter-Syntetos-Babai method, which extends Croston's framework by separately modeling the probability of demand occurrence using a different smoothing constant.

• Introduces an explicit demand probability component updated every period
• Allows the probability to decay during long zero-demand sequences
• Addresses obsolescence risk: items that haven't sold in many periods get a declining probability
• Superior for items approaching end-of-life or with non-stationary demand intervals

A systematic tendency to over-predict or under-predict demand. Croston's original method exhibits positive bias that the SBA and TSB variants correct.

• Croston's bias: Arises from the mathematical inequality E(X/Y) ≠ E(X)/E(Y)
• Detection: Track cumulative forecast error over a rolling horizon
• Impact: Biased forecasts lead to excess inventory holding costs
• Remediation: Apply the SBA correction factor or switch to an unbiased probabilistic model