\"algo-forecast-exponential\"

Overview

Exponential smoothing assigns exponentially decreasing weights to past observations. Three variants: Simple (SES, level only), Holt (level + trend), Holt-Winters (level + trend + seasonality). ETS framework (Error-Trend-Seasonality) provides a unified statistical model. Fast, interpretable, and competitive with complex models for short horizons.

When to Use

Trigger conditions:

• Quick forecasting with minimal configuration
• Short-horizon forecasts (1-2 seasonal cycles ahead)
• Data with clear level, trend, and/or seasonal components

When NOT to use:

• For long-range forecasts (uncertainty accumulates too fast)
• When external regressors are important (use regression or ML models)

Algorithm

IRON LAW: Smoothing Parameters Control the Bias-Variance Trade-Off
α (level), β (trend), γ (seasonality) range [0,1].
- α near 1: react quickly to changes, noisy forecasts (high variance)
- α near 0: smooth forecasts, slow to adapt (high bias)
Optimize via minimizing MSE on training data (or use information criteria).
Never hand-pick smoothing parameters without validation.

Phase 1: Input Validation

Identify components: level only (SES), level+trend (Holt), level+trend+seasonality (Holt-Winters). Determine: additive vs multiplicative trend/seasonality. Gate: Component structure identified, seasonal period known.

Phase 2: Core Algorithm

Holt-Winters (additive):

1. Initialize: level₀ = mean(first season), trend₀ = (mean(season 2) - mean(season 1))/s, seasonal₀ from first season deviations
2. Update equations at each t:
   • Level: ℓₜ = α(yₜ - sₜ₋ₛ) + (1-α)(ℓₜ₋₁ + bₜ₋₁)
   • Trend: bₜ = β(ℓₜ - ℓₜ₋₁) + (1-β)bₜ₋₁
   • Seasonal: sₜ = γ(yₜ - ℓₜ) + (1-γ)sₜ₋ₛ
3. Forecast: ŷₜ₊ₕ = ℓₜ + h×bₜ + sₜ₊ₕ₋ₛ

Phase 3: Verification

Check: in-sample RMSE, residual patterns. Compare against naive baselines (last value, seasonal naive). Gate: Beats naive baseline, residuals show no systematic pattern.

Phase 4: Output

Return forecasts with smoothed components.

Output Format

{
  "forecasts": [{"period": "2025-04", "forecast": 1150, "level": 1100, "trend": 20, "seasonal": 30}],
  "parameters": {"alpha": 0.3, "beta": 0.1, "gamma": 0.15},
  "metadata": {"method": "holt_winters_additive", "seasonal_period": 12, "rmse": 45}
}

Examples

Sample I/O

Input: 36 months of monthly sales, clear upward trend, December spike Expected: Holt-Winters additive. Forecast continues trend with repeated December seasonality.

Edge Cases

Input	Expected	Why
No trend, no seasonality	SES (α only)	Simplest variant suffices
Seasonal amplitude grows	Use multiplicative	Additive would underestimate peaks
Very short series (<2 seasons)	SES or Holt only	Can't estimate seasonality

Gotchas

• Additive vs multiplicative: If seasonal swings grow proportionally with level, use multiplicative. Wrong choice produces poor forecasts, especially at extremes.
• Initialization sensitivity: The first season's values set the baseline. Poor initialization from noisy early data propagates through the entire forecast.
• Damped trend: For long horizons, linear trend extrapolation is unrealistic. Use damped trend (φ parameter) to flatten the trend over time.
• Multiple seasonalities: Standard Holt-Winters handles one seasonal period. For daily data with weekly AND yearly patterns, use TBATS or STL+ETS.
• Outlier sensitivity: A single outlier can shift the level estimate significantly (especially with high α). Pre-detect and handle outliers.

References

• For ETS framework and model selection, see references/ets-framework.md
• For damped trend variants, see references/damped-trend.md