Forecast Error Distribution

The systematic tendency of a forecast to consistently over-predict or under-predict actual demand. Mean Error (ME) quantifies this directional bias.

• Positive Bias: Forecasts are persistently higher than actuals, leading to excess inventory buildup.
• Negative Bias: Forecasts are persistently lower than actuals, causing frequent stockouts.
• A well-calibrated model should have a mean error approaching zero, indicating no systematic skew.
• Bias is distinct from accuracy; a forecast can be unbiased but highly imprecise.

The measure of the average magnitude of forecast errors, regardless of direction. Root Mean Squared Error (RMSE) and Mean Absolute Deviation (MAD) are the primary metrics.

• RMSE penalizes large errors more heavily due to squaring, making it sensitive to outliers.
• MAD provides a linear, more interpretable measure of average error size.
• In a normal distribution, safety stock is directly proportional to the standard deviation of forecast error.
• High dispersion during demand volatility clustering requires adaptive buffer sizing.

A measure of the asymmetry of the error distribution around its mean. Skewness indicates whether extreme errors tend to be positive or negative.

• Positive Skew: A long right tail indicates occasional massive under-forecasts, where actual demand far exceeds predictions. This is the most dangerous scenario for stockouts.
• Negative Skew: A long left tail indicates occasional massive over-forecasts, leading to dead stock risk.
• Assuming normality when skewness is present leads to under-protection against the tail risk, making quantile forecasting essential.

A measure of the propensity for extreme, unexpected errors. Excess Kurtosis quantifies how fat the tails of the distribution are compared to a normal distribution.

• Leptokurtic (Fat-Tailed): Indicates a higher probability of extreme forecast misses than a normal distribution would predict. Common in intermittent demand patterns.
• Platykurtic (Thin-Tailed): Indicates fewer extreme events, suggesting a more stable, predictable demand pattern.
• High kurtosis invalidates standard safety stock formulas that rely on z-scores from a normal distribution, requiring Monte Carlo buffer simulation.

The correlation of a forecast error at one point in time with errors at previous time steps. Durbin-Watson statistic tests for this pattern.

• Positive Autocorrelation: Errors cluster together; an over-forecast today is likely followed by an over-forecast tomorrow. Indicates the model is failing to capture a trend or seasonality.
• Negative Autocorrelation: Errors oscillate; an over-forecast is systematically followed by an under-forecast. Suggests over-correction.
• Persistent autocorrelation signals concept drift and triggers automated model retraining to restore statistical independence of residuals.

A condition where the variance of forecast errors is not constant but changes systematically with the level of demand or over time.

• Demand-Level Dependent: Error variance increases with the magnitude of demand. High-volume periods exhibit proportionally larger absolute errors.
• Time-Dependent: Error variance spikes during specific seasons, promotions, or supply disruptions.
• Standard safety stock calculations assume homoscedasticity (constant variance). Heteroscedasticity requires time-phased safety stock or variance-stabilizing transformations.

A buffer calculation method that models demand and lead time as probability distributions rather than fixed values. Unlike deterministic approaches, stochastic safety stock directly uses the forecast error distribution to set buffer quantities that achieve a specific service level target under real-world variability.

A probabilistic prediction method that estimates specific percentiles of the future demand distribution rather than a single-point average. By generating the full forecast error distribution, quantile forecasting enables precise buffer sizing:

• 50th percentile: Most likely demand
• 95th percentile: Buffer for 95% service level
• 99th percentile: Buffer for critical items

A phenomenon where large demand fluctuations tend to be followed by more large fluctuations, violating the assumption of constant variance in simple safety stock models. When volatility clustering is present, the forecast error distribution exhibits heteroskedasticity, requiring adaptive safety stock algorithms that increase buffers during turbulent periods and reduce them during calm periods.

A computational technique that runs thousands of randomized demand-supply scenarios to empirically determine the safety stock required for a target service level. By sampling directly from the forecast error distribution, Monte Carlo simulation captures complex interactions that closed-form formulas miss:

• Non-normal error distributions
• Correlated demand and lead time
• Multi-echelon ripple effects

The degradation of a safety stock model's accuracy over time as the underlying statistical properties of demand or supply change. When concept drift occurs, the forecast error distribution shifts—mean error increases, variance expands, or the shape transforms—triggering automated retraining requirements. Monitoring error distribution stability is essential for maintaining buffer integrity.

The statistical process of matching historical supplier delivery data to a theoretical probability distribution to accurately model replenishment uncertainty. Combined with the forecast error distribution, lead time distribution fitting enables the calculation of combined demand-supply uncertainty for precise safety stock:

• Normal distribution: Stable suppliers
• Gamma distribution: Skewed lead times
• Weibull distribution: Reliability-focused modeling