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

{
  "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