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

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