"algo-forecast-arima"

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.