Lesson 1 - From ARIMA to SARIMA

Welcome to From ARIMA to SARIMA#

Module 5 ended on a precise, measured failure: a non-seasonal ARIMA couldn’t beat a one-line seasonal baseline on Cyclepath, because it had no term for seasonality. The diagnosis pointed straight at the cure — give the model an explicit seasonal mechanism. That’s SARIMA: Seasonal ARIMA. It doesn’t replace anything you learned; it duplicates the ARIMA machinery at the seasonal lag, so the model can say “this month depends on its recent neighbors and on the same month a year ago.”

By the end of this lesson, you will be able to:

• Read the full SARIMA(p,d,q)(P,D,Q)[s] notation
• Explain what the seasonal period s is and how to choose it
• Map each seasonal order (P, D, Q) to its non-seasonal twin (p, d, q)
• Explain why the seasonal terms are the mechanism ARIMA lacked

Let’s read the notation.

The SARIMA Notation#

A SARIMA model is written with two sets of orders:

SARIMA(p, d, q)(P, D, Q)[s]
       \_______/\_________/
      non-seasonal  seasonal

• (p, d, q) — the familiar non-seasonal orders from Module 5: p AR terms, d differences, q MA terms, all operating at short lags (1, 2, 3, …).
• (P, D, Q) — the seasonal orders (capital letters), applying the same three ideas but at the seasonal lag: P seasonal AR terms, D seasonal differences, Q seasonal MA terms.
• [s] — the seasonal period: how many time steps make up one full cycle. For Cyclepath’s monthly data with a yearly pattern, s = 12.

So SARIMA(1, 1, 0)(1, 1, 0)[12] — the model this module builds toward — means: one non-seasonal AR term, one first difference, no non-seasonal MA; plus one seasonal AR term, one seasonal difference, no seasonal MA; with a 12-month season. Six numbers, but every one of them is an idea you already know, applied at one of two scales.

Each Seasonal Term Mirrors Its Non-Seasonal Twin#

The elegance of SARIMA is that the seasonal orders aren’t new concepts — they’re the Module 3 and Module 5 concepts applied a full cycle apart:

You’ve already met two of these directly. Seasonal differencing (D) is exactly y.diff(12) from Module 3 — the transformation that stationarized Cyclepath. And seasonal AR (P) is what Module 4’s preview model used: SARIMA(0,0,0)(1,1,0)[12], the seasonal-AR(1) that scored far better than any non-seasonal candidate on AIC. SARIMA just gives all six terms a single, unified home.

Why This Is the Missing Piece#

Module 5’s non-seasonal ARIMA failed on Cyclepath for a structural reason: its terms only reached back a few short lags, so it could model month-to-month persistence but had no way to connect July 2023 to July 2022 — twelve steps apart, far beyond any sensible non-seasonal p or q. You saw the desperate workaround: ARIMA(2,1,2) tried to synthesize a 12-month cycle by pushing its roots onto the unit circle, a fragile hack that triggered convergence warnings.

SARIMA’s seasonal terms make that connection directly and stably. A seasonal AR(1) term (P=1) explicitly relates each month to the same month one cycle earlier — no contortion required. That’s the entire reason SARIMA can beat the seasonal-naive baseline where ARIMA couldn’t: it has a term built for exactly the “same month last year” relationship the baseline exploits, and it can refine that relationship with the trend-and-persistence machinery of the non-seasonal part layered on top.

Practice Exercises#

Exercise 1: Read a specification#

What does SARIMA(0, 1, 1)(0, 1, 1)[12] mean, term by term?

Exercise 2: Pick the seasonal period#

You have hourly electricity-demand data with a strong daily pattern (demand peaks every afternoon, dips every night). What seasonal period s should a SARIMA use?

Exercise 3: Why not just increase p?#

If July depends on last July (12 months back), why not just fit a non-seasonal ARIMA with p = 12 to reach that lag, instead of using a seasonal term?

Summary#

A SARIMA model is written SARIMA(p, d, q)(P, D, Q)[s] — the familiar non-seasonal orders plus a seasonal set (P, D, Q) that applies the identical AR, differencing, and MA ideas at the seasonal lag s (12 for monthly-yearly data). Each seasonal term mirrors a non-seasonal twin: seasonal differencing D is Module 3’s y.diff(12), seasonal AR P is the “this July depends on last July” relationship Module 4’s preview used, and seasonal MA Q is the same for shocks. These seasonal terms are precisely the mechanism a non-seasonal ARIMA lacked — they connect each period to the same period one cycle earlier with a single coefficient, directly and stably, rather than the fragile unit-root workaround Module 5 exposed.

Key Concepts#

• SARIMA(p,d,q)(P,D,Q)[s] — non-seasonal orders plus seasonal orders at lag s.
• Seasonal period s — set by the calendar (12 monthly, 4 quarterly, 7 daily-weekly, 24 hourly-daily), not estimated.
• Seasonal terms mirror non-seasonal ones — D is seasonal differencing, P/Q are AR/MA at the seasonal lag.
• Parsimony — a seasonal term reaches the seasonal lag with one coefficient, where a large non-seasonal p would waste eleven.

Why This Matters#

SARIMA is where the entire course converges: the stationarity work (d and D), the ACF/PACF reading (p, q, P, Q), the seasonal period (s), and the ARIMA fitting machinery all combine into one model that can finally handle a series like Cyclepath properly. Understanding that the seasonal orders are just the non-seasonal ideas applied a cycle apart means SARIMA’s six-number notation isn’t six new things to learn — it’s two things you know, at two scales. Next, you’ll build the seasonal AR and MA terms concretely and see exactly what a seasonal coefficient says about the relationship between one season and the next.

Next Steps#

Continue Building Your Skills#

You now know the full SARIMA notation and that its seasonal orders are the non-seasonal ideas applied one cycle apart — the exact mechanism ARIMA lacked. Next you’ll build the seasonal AR and MA terms concretely, seeing how a seasonal-AR coefficient captures the relationship between this season and the last, and how seasonal differencing sets the stage for them.