algo-forecast-exponential
用指數平滑與 Holt-Winters 做簡單穩健的預測
中文翻譯
---
name: "algo-forecast-exponential"
description: "用指數平滑法（加權移動平均）做時間序列預測。當使用者需要簡單穩健的預測、為季節性資料實作 Holt-Winters，或想在不用複雜模型下建立輕量預測時使用，即使他們說的是『simple forecast』『moving average prediction』或『smoothing method』。"
metadata:
category: "WP-47 時間序列預測"
tags: ["forecasting", "exponential-smoothing", "holt-winters", "time-series"]
---
# 指數平滑
## 概述
指數平滑對過去的觀測值給予呈指數遞減的權重。三種變體：Simple（SES，只有水準）、Holt（水準 + 趨勢）、Holt-Winters（水準 + 趨勢 + 季節性）。ETS 框架（Error-Trend-Seasonality）提供一個統一的統計模型。快、可解釋，在短期預測上能和複雜模型一較高下。
## 何時使用
**觸發條件：**
- 要快速預測、設定越少越好
- 短期預測（往前 1-2 個季節週期）
- 資料有明確的水準、趨勢和（或）季節性成分
**不該使用的情況：**
- 長期預測（不確定性累積太快）
- 外部回歸變數很重要時（用迴歸或 ML 模型）
## 演算法
```
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.
```
### 階段 1：輸入驗證
辨識成分：只有水準（SES）、水準+趨勢（Holt）、水準+趨勢+季節性（Holt-Winters）。判斷：趨勢 / 季節性是加法還是乘法。
**關卡：** 已辨識成分結構，且知道季節週期。
### 階段 2：核心演算法
**Holt-Winters（加法）：**
1. 初始化：level₀ = mean(第一季)，trend₀ = (mean(第 2 季) - mean(第 1 季))/s，seasonal₀ 取自第一季的偏差
2. 每個 t 的更新公式：
- 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. 預測：ŷₜ₊ₕ = ℓₜ + h×bₜ + sₜ₊ₕ₋ₛ
### 階段 3：驗證
檢查：樣本內 RMSE、殘差模式。和 naive 基準（最後一個值、季節性 naive）比較。
**關卡：** 贏過 naive 基準，殘差沒有系統性模式。
### 階段 4：輸出
回傳預測值，附上平滑後的各成分。
## 輸出格式
```json
{
"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}
}
```
## 範例
### 輸入輸出範例
**輸入：** 36 個月的月銷售額，明顯上升趨勢，12 月有高峰
**預期：** Holt-Winters 加法。預測延續趨勢，並重複 12 月的季節性。
### 邊界情況
| 輸入 | 預期 | 原因 |
|-------|----------|-----|
| 沒有趨勢、沒有季節性 | SES（只有 α） | 最簡單的變體就夠了 |
| 季節振幅越來越大 | 用乘法 | 加法會低估高峰 |
| 序列非常短（<2 個季節） | 只能用 SES 或 Holt | 無法估計季節性 |
## 注意事項
- **加法 vs 乘法**：如果季節波動隨水準等比例放大，就用乘法。選錯會讓預測很差，尤其在極端值。
- **初始化的敏感度**：第一季的值設定了基準。早期資料雜訊大、初始化不好，會一路影響整個預測。
- **阻尼趨勢**：長期預測時，線性趨勢外推不切實際。用阻尼趨勢（φ 參數）讓趨勢隨時間變平。
- **多重季節性**：標準 Holt-Winters 只處理一個季節週期。如果日資料同時有週和年的模式，用 TBATS 或 STL+ETS。
- **對離群值敏感**：一個離群值就可能讓水準估計大幅位移（尤其 α 高的時候）。要先偵測並處理離群值。
## 參考資料
- ETS 框架與模型選擇，見 `references/ets-framework.md`
- 阻尼趨勢的變體，見 `references/damped-trend.md`英文原文
---
name: "algo-forecast-exponential"
description: "Apply exponential smoothing methods for time series forecasting with weighted moving averages. Use this skill when the user needs simple, robust forecasts, implement Holt-Winters for seasonal data, or build lightweight forecasting without complex models — even if they say 'simple forecast', 'moving average prediction', or 'smoothing method'."
metadata:
category: "WP-47 時間序列預測"
tags: ["forecasting", "exponential-smoothing", "holt-winters", "time-series"]
---
# 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.
## Output Format
```json
{
"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`