Forecasting in Conditions of Uncertainty

Problem

Use Croston’s method when a time series has many zero periods interspersed with sporadic non-zero demand. Standard exponential smoothing collapses because zeros dilute estimates and ARIMA struggles with the structural zeros.

Classic symptoms:

• ADI (Average Demand Interval) > 1.32
• CV² (squared coefficient of variation of non-zero demand) < 0.49

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Assumptions & Conditions

• Demand arrivals follow a Bernoulli process (independent across periods).
• Non-zero demand sizes are i.i.d. (no trend, no seasonality within bursts).
• Demand interval and demand size are independent of each other.

Breaks down when: demand has a trend, sizes cluster seasonally, or there’s a long-run structural shift.

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Approach

Croston splits the series into two separate ES processes:

$\hat{z}_t = \alpha \cdot z_t + (1 - \alpha) \cdot \hat{z}_{t-1}$

$\hat{p}_t = \alpha \cdot p_t + (1 - \alpha) \cdot \hat{p}_{t-1}$

where $z_t$ is the non-zero demand size at occurrence $t$, $p_t$ is the interval between occurrences, and $\alpha \in (0, 1)$ is the smoothing parameter.

Forecast per period:

$\hat{d} = \frac{\hat{z}}{\hat{p}}$

Bias correction (Syntetos-Boylan, SBA): Croston’s estimator is positively biased. SBA fixes this:

$\hat{d}_{SBA} = \left(1 - \frac{\alpha}{2}\right) \cdot \frac{\hat{z}}{\hat{p}}$

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Implementation

import numpy as np
from dataclasses import dataclass

@dataclass
class CrostonResult:
    forecast: float
    z_hat: float  # smoothed demand size
    p_hat: float  # smoothed interval


def croston(demand: np.ndarray, alpha: float = 0.1, variant: str = "sba") -> CrostonResult:
    """
    Croston's method for intermittent demand forecasting.

    Parameters
    ----------
    demand : 1-D array of demand values (zeros allowed)
    alpha  : smoothing parameter in (0, 1)
    variant: "classic" or "sba" (Syntetos-Boylan bias correction)

    Returns
    -------
    CrostonResult with per-period rate forecast
    """
    nonzero_idx = np.where(demand > 0)[0]
    if len(nonzero_idx) == 0:
        return CrostonResult(forecast=0.0, z_hat=0.0, p_hat=1.0)

    # Bootstrap from first occurrence
    z_hat = demand[nonzero_idx[0]]
    p_hat = float(nonzero_idx[0] + 1)  # periods until first demand

    for i in range(1, len(nonzero_idx)):
        interval = nonzero_idx[i] - nonzero_idx[i - 1]
        z_hat = alpha * demand[nonzero_idx[i]] + (1 - alpha) * z_hat
        p_hat = alpha * interval + (1 - alpha) * p_hat

    rate = z_hat / p_hat
    if variant == "sba":
        rate *= 1 - alpha / 2

    return CrostonResult(forecast=rate, z_hat=z_hat, p_hat=p_hat)

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Results & Tradeoffs

Property	Croston	SBA
Bias	Positive bias ~5–30%	Near-unbiased
Variance	Lower	Slightly higher
Best for	Never outperforms SBA	ADI > 1.32, CV² < 0.49

When to prefer alternatives:

• TSB (Teunter-Syntetos-Babai): Demand can become extinct (e.g., end-of-life parts). TSB updates probability of non-zero demand each period.
• ADIDA aggregation: Aggregate to non-zero-free bucket, forecast normally, then disaggregate. Simple and often competitive.
• ML on features: If you have rich features (promotions, seasonality), a gradient-boosted model on binned demand usually beats smoothing-based methods.

Typical error metrics: CSL (Cycle Service Level), Fill Rate, MAE on non-zero periods. Avoid MAPE — division by zero on empty periods.

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References

• Croston, J.D. (1972). Forecasting and Stock Control for Intermittent Demands. Operational Research Quarterly.
• Syntetos, A.A. & Boylan, J.E. (2005). The Accuracy of Intermittent Demand Estimates. International Journal of Forecasting.
• Teunter, R., Syntetos, A.A. & Babai, M.Z. (2011). Intermittent demand: Linking forecasting to inventory obsolescence. EJOR.