"algo-forecast-exponential"

Exponential Smoothing

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.