SARIMA (Seasonal ARIMA)

SARIMA extends ARIMA to handle seasonality by adding seasonal AR, differencing, and MA terms: SARIMA(\(p, d, q\))(\(P, D, Q, s\)).

The Seasonal Parameters

Parameter	Meaning
\(P\)	Seasonal autoregressive order
\(D\)	Seasonal differencing order (usually 0 or 1)
\(Q\)	Seasonal moving average order
\(s\)	Length of the seasonal cycle (e.g., 12 for monthly data with yearly seasonality, 7 for daily data with weekly seasonality)

When to Use SARIMA vs ARIMA

• Use ARIMA when the series has a trend but no repeating seasonal pattern.
• Use SARIMA when the series has a clear seasonal cycle (e.g., monthly sales peaking every December).

Implementation

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.statespace.sarimax import SARIMAX

# Generate synthetic monthly data with yearly seasonality
rng = np.random.default_rng(42)
dates = pd.date_range("2018-01-01", periods=72, freq="MS")
trend = np.linspace(100, 160, 72)
seasonal = 15 * np.sin(2 * np.pi * np.arange(72) / 12)
noise = rng.normal(0, 3, 72)

ts = pd.Series(trend + seasonal + noise, index=dates, name="value")

# Fit SARIMA(1,1,1)(1,1,0,12)
model = SARIMAX(ts, order=(1, 1, 1), seasonal_order=(1, 1, 0, 12))
fitted = model.fit(disp=False)
print(fitted.summary())

# Forecast 12 months
forecast = fitted.forecast(steps=12)

plt.figure(figsize=(10, 4))
ts.plot(label="Observed")
forecast.plot(label="Forecast", color="red", linestyle="--")
plt.legend()
plt.title("SARIMA(1,1,1)(1,1,0,12) Forecast")
plt.tight_layout()
plt.show()

Model Selection

Use AIC (Akaike Information Criterion) to compare different orders:

# Compare several model orders
best_aic = float("inf")
best_order = None

for p in range(3):
    for q in range(3):
        try:
            m = SARIMAX(ts, order=(p, 1, q), seasonal_order=(1, 1, 0, 12))
            r = m.fit(disp=False)
            if r.aic < best_aic:
                best_aic, best_order = r.aic, (p, 1, q)
        except Exception:
            continue

print(f"Best order: {best_order}, AIC: {best_aic:.2f}")

Workplace Tip

Use pmdarima.auto_arima(seasonal=True, m=12) for automated SARIMA order selection. This handles the grid search for you and is far more robust than manual iteration.

KSB Mapping

KSB	Description	How This Addresses It
K4.1	Statistical models and methods	ARIMA, SARIMA, and exponential smoothing foundations
K4.2	Predictive analytics and ML techniques	Time series forecasting and model comparison
K5.3	Common patterns in real-world data	Identifying trends, seasonality, and stationarity
S1	Scientific methods and hypothesis testing	Stationarity testing, model diagnostics, forecast validation
S4	Analysis and models to inform outcomes	Building forecasts to support business planning
B5	Impartial, hypothesis-driven approach	Honest evaluation of forecast accuracy and limitations