algo-forecast-arima
建立 ARIMA 模型做時間序列預測，含平穩性檢定
中文翻譯
---
name: "algo-forecast-arima"
description: "建立 ARIMA 模型做時間序列預測，包含趨勢與季節性分解。當使用者需要從歷史序列資料預測未來數值、檢定平穩性或挑選 ARIMA 參數時使用，即使他們說的是『time series forecast』『predict next month sales』或『ARIMA model』。"
metadata:
category: "WP-47 時間序列預測"
tags: ["forecasting", "arima", "time-series", "statistics"]
---
# ARIMA 時間序列模型
## 概述
ARIMA(p,d,q) 結合自迴歸（AR）、差分（I）和移動平均（MA）來做時間序列預測。季節性變體：SARIMA(p,d,q)(P,D,Q,s)。資料必須平穩（透過差分達成）。最適合有明確趨勢 / 季節性模式的單變量序列。
## 何時使用
**觸發條件：**
- 預測單變量時間序列（銷售、需求、流量）
- 資料有明確的趨勢和（或）季節性模式
- 需要具備統計性質、可解釋的模型
**不該使用的情況：**
- 有很多外部特徵的多變量預測（用 ML 模型）
- 非常長期的預測（ARIMA 的信賴區間會快速變寬）
- 不規則 / 事件驅動的資料（用因果模型）
## 演算法
```
IRON LAW: ARIMA Requires STATIONARY Data
Non-stationary data (trend, changing variance) violates ARIMA assumptions.
Test stationarity with ADF test (p < 0.05 = stationary).
If non-stationary: difference the series (d=1 usually suffices).
If still non-stationary after d=2, ARIMA may not be appropriate.
```
### 階段 1：輸入驗證
檢查：時間間隔固定、沒有缺值（需要的話做補值）、至少 50 筆觀測值（理想上有 2 個以上完整的季節週期）。用 ADF 檢定測試平穩性。
**關卡：** 資料間隔規律、長度足夠、已評估平穩性。
### 階段 2：核心演算法
1. **平穩性**：ADF 檢定。如果 p > 0.05，就做差分（d=1），再檢定一次。
2. **參數選擇**：檢視 ACF/PACF 圖。或用 auto_arima（以 AIC 為基礎的網格搜尋）。
- p（AR 項）：PACF 截尾的 lag
- q（MA 項）：ACF 截尾的 lag
- d：需要的差分次數
3. **擬合模型**：最大概似估計
4. **預測**：產生預測值與信賴區間
### 階段 3：驗證
檢查殘差：應該是白噪音（沒有自相關）。Ljung-Box 檢定（p > 0.05 = 沒有自相關）。殘差呈常態分布。
**關卡：** 殘差通過 Ljung-Box 檢定，沒有殘留的模式。
### 階段 4：輸出
回傳預測值與信賴區間。
## 輸出格式
```json
{
"forecasts": [{"period": "2025-04", "forecast": 1250, "lower_95": 1100, "upper_95": 1400}],
"model": {"order": [1,1,1], "seasonal_order": [1,1,1,12], "aic": 520.3},
"metadata": {"training_periods": 60, "forecast_horizon": 12}
}
```
## 範例
### 輸入輸出範例
**輸入：** 12 筆呈上升趨勢的月資料：[10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]
**步驟 1：** 一階差分 = [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]（常數 → 平穩，d=1 就夠）
**步驟 2：** 最簡單能擬合的模型是 ARIMA(0,1,0)，也就是漂移 μ=2 的隨機漫步。
**預期預測（ARIMA(0,1,0)，drift=2）：**
- 第 13 期：32 + 2 = **34**
- 第 14 期：32 + 4 = **36**
- 第 15 期：32 + 6 = **38**
驗證：差分後的序列是常數（2）→ 不需要 AR/MA 項。殘差剛好是 0 → 完美擬合（玩具範例）。真實資料的殘差應該要通過 Ljung-Box（p > 0.05）。
### 邊界情況
| 輸入 | 預期 | 原因 |
|-------|----------|-----|
| 沒有趨勢、沒有季節性 | ARIMA(p,0,q) | 不需要差分 |
| 只有強烈趨勢 | ARIMA(p,1,q) | 一次差分就能去除線性趨勢 |
| 多重季節性 | ARIMA 可能吃力 | 改考慮 Prophet 或 TBATS |
## 注意事項
- **過度差分**：d=1 就夠卻用 d=2，會引入不必要的雜訊。再次差分之前，先檢查一階差分是不是已經平穩。
- **Auto-ARIMA 不是魔法**：以 AIC 為基礎的選擇可能挑出過擬合的模型。不管是不是自動選的，都一定要檢查殘差診斷。
- **信賴區間變寬很快**：多步預測會累積不確定性。超過 2-3 個季節週期的點預測就不要相信。
- **日曆效應**：工作日、假日、閏年會影響月 / 週資料。ARIMA 原生不處理這些，要加回歸變數或改用 Prophet。
- **結構性斷點**：ARIMA 假設資料產生過程是穩定的。COVID、市場衝擊或政策改變都會打破這個假設。
## 參考資料
- ACF/PACF 的判讀指南，見 `references/acf-pacf.md`
- SARIMA 季節性參數的選擇，見 `references/seasonal-arima.md`英文原文
---
name: "algo-forecast-arima"
description: "Build ARIMA models for time series forecasting with trend and seasonality decomposition. Use this skill when the user needs to forecast future values from historical sequential data, test for stationarity, or select ARIMA parameters — even if they say 'time series forecast', 'predict next month sales', or 'ARIMA model'."
metadata:
category: "WP-47 時間序列預測"
tags: ["forecasting", "arima", "time-series", "statistics"]
---
# ARIMA Time Series Model
## Overview
ARIMA(p,d,q) combines autoregression (AR), differencing (I), and moving average (MA) for time series forecasting. Seasonal variant: SARIMA(p,d,q)(P,D,Q,s). Requires stationary data (achieved through differencing). Best for univariate series with clear trend/seasonality patterns.
## When to Use
**Trigger conditions:**
- Forecasting univariate time series (sales, demand, traffic)
- Data has clear trend and/or seasonal patterns
- Need interpretable model with statistical properties
**When NOT to use:**
- For multivariate forecasting with many external features (use ML models)
- For very long-range forecasts (ARIMA confidence intervals widen rapidly)
- For irregular/event-driven data (use causal models)
## Algorithm
```
IRON LAW: ARIMA Requires STATIONARY Data
Non-stationary data (trend, changing variance) violates ARIMA assumptions.
Test stationarity with ADF test (p < 0.05 = stationary).
If non-stationary: difference the series (d=1 usually suffices).
If still non-stationary after d=2, ARIMA may not be appropriate.
```
### Phase 1: Input Validation
Check: regular time intervals, no missing values (impute if needed), minimum 50 observations (ideally 2+ full seasonal cycles). Test stationarity with ADF test.
**Gate:** Data is regular, sufficient length, stationarity assessed.
### Phase 2: Core Algorithm
1. **Stationarity**: ADF test. If p > 0.05, difference (d=1). Retest.
2. **Parameter selection**: Examine ACF/PACF plots. Or use auto_arima (AIC-based grid search).
- p (AR terms): PACF cutoff lag
- q (MA terms): ACF cutoff lag
- d: number of differences needed
3. **Fit model**: Maximum likelihood estimation
4. **Forecast**: Generate predictions with confidence intervals
### Phase 3: Verification
Check residuals: should be white noise (no autocorrelation). Ljung-Box test (p > 0.05 = no autocorrelation). Residuals normally distributed.
**Gate:** Residuals pass Ljung-Box test, no remaining patterns.
### Phase 4: Output
Return forecasts with confidence intervals.
## Output Format
```json
{
"forecasts": [{"period": "2025-04", "forecast": 1250, "lower_95": 1100, "upper_95": 1400}],
"model": {"order": [1,1,1], "seasonal_order": [1,1,1,12], "aic": 520.3},
"metadata": {"training_periods": 60, "forecast_horizon": 12}
}
```
## Examples
### Sample I/O
**Input:** 12 monthly observations with upward trend: [10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]
**Step 1:** First difference = [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] (constant → stationary, d=1 sufficient)
**Step 2:** ARIMA(0,1,0) random walk with drift μ=2 is the simplest fitting model.
**Expected forecast (ARIMA(0,1,0) with drift=2):**
- Period 13: 32 + 2 = **34**
- Period 14: 32 + 4 = **36**
- Period 15: 32 + 6 = **38**
Verify: differenced series is constant (2) → no AR/MA terms needed. Residuals are exactly 0 → perfect fit (toy example). On real data, residuals should pass Ljung-Box (p > 0.05).
### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| No trend, no seasonality | ARIMA(p,0,q) | No differencing needed |
| Strong trend only | ARIMA(p,1,q) | Single difference removes linear trend |
| Multiple seasonalities | ARIMA may struggle | Consider Prophet or TBATS instead |
## Gotchas
- **Over-differencing**: d=2 when d=1 suffices introduces unnecessary noise. Check if first difference is stationary before differencing again.
- **Auto-ARIMA isn't magic**: AIC-based selection can pick overfit models. Always check residual diagnostics regardless of auto selection.
- **Confidence intervals widen fast**: Multi-step forecasts accumulate uncertainty. Don't trust point forecasts beyond 2-3 seasonal cycles.
- **Calendar effects**: Business days, holidays, and leap years affect monthly/weekly data. ARIMA doesn't handle these natively — add regressors or use Prophet.
- **Structural breaks**: ARIMA assumes the data-generating process is stable. COVID, market shocks, or policy changes break this assumption.
## References
- For ACF/PACF interpretation guide, see `references/acf-pacf.md`
- For SARIMA seasonal parameter selection, see `references/seasonal-arima.md`