216. Summary
217. ETS taxonomy (state space form)
■ level ■ trend ■ seasonality
Each model has an observation equation (top, separated by a line) and state equations for each component plus a point-forecast formula. is the one-step-ahead innovation, assumed mean zero. is the seasonal period. picks the appropriate seasonal index for an -step-ahead forecast. Damped models use . Smoothing parameters .
Every one of the 30 cells gives you the same two-step pattern:
- Compute the innovation from the observation equation, solving for :
- Additive errors:
- Multiplicative errors:
where is whatever the observation equation says equals when (i.e., the conditional mean).
- Update the states by plugging into the state equations.
That’s it. The same loop works for all 30 models — only the specific formulas for and the state updates change.
-
Error (E):
- Additive (A)
- Multiplicative (M)
-
Trend (T):
- None (N)
- Additive (A)
- Additive damped (Ad)
- Multiplicative (M)
- Multiplicative damped (Md)
-
Seasonal (S):
- None (N)
- Additive (A)
- Multiplicative (M)
217.1. Trend = N (no trend)
217.2. Trend = A (additive trend)
217.3. Trend = Ad (damped additive trend)
217.4. Trend = M (multiplicative trend)
217.5. Trend = Md (damped multiplicative trend)
| Model | Name |
| ETS(A,N,N) | Simple exponential smoothing |
| ETS(A,N,A) | SES with additive seasonality |
| ETS(A,N,M) | SES with multiplicative seasonality |
| ETS(A,A,N) | Holt’s linear trend |
| ETS(A,A,A) | Additive Holt-Winters |
| ETS(A,A,M) | Holt-Winters with multiplicative seasonality |
| ETS(A,Ad,N) | Damped linear trend |
| ETS(A,Ad,A) | Damped additive Holt-Winters |
| ETS(A,Ad,M) | Damped Holt-Winters with multiplicative seasonality |
| ETS(A,M,N) | Multiplicative trend |
| ETS(A,M,A) | Multiplicative trend with additive seasonality |
| ETS(A,M,M) | Multiplicative trend and seasonality |
| ETS(A,Md,N) | Damped multiplicative trend |
| ETS(A,Md,A) | Damped multiplicative trend with additive seasonality |
| ETS(A,Md,M) | Damped multiplicative trend and seasonality |
| ETS(M,N,N) | SES with multiplicative errors |
| ETS(M,N,A) | SES with additive seasonality, multiplicative errors |
| ETS(M,N,M) | SES with multiplicative seasonality and errors |
| ETS(M,A,N) | Holt’s linear trend with multiplicative errors |
| ETS(M,A,A) | Additive Holt-Winters with multiplicative errors |
| ETS(M,A,M) | Multiplicative Holt-Winters |
| ETS(M,Ad,N) | Damped linear trend with multiplicative errors |
| ETS(M,Ad,A) | Damped additive Holt-Winters with multiplicative errors |
| ETS(M,Ad,M) | Damped multiplicative Holt-Winters |
| ETS(M,M,N) | Multiplicative trend with multiplicative errors |
| ETS(M,M,A) | Multiplicative trend, additive seasonality, multiplicative errors |
| ETS(M,M,M) | Fully multiplicative |
| ETS(M,Md,N) | Damped multiplicative trend with multiplicative errors |
| ETS(M,Md,A) | Damped multiplicative trend, additive seasonality, mult. errors |
| ETS(M,Md,M) | Fully multiplicative with damped trend |
217.6. ETS (Exponential Smoothing)
ETS stands for Error, Trend, Seasonality — a family of forecasting models that decomposes a time series into up to three components and updates each one recursively from new observations.
217.6.1. The weighting-scheme axis
Before exponential smoothing, ask: how much should past observations count? Two extremes and two interpolators:
Naïve () and Cumulative () are the two extremes of a single “how much history counts” axis. MA sits between them with a finite equal-weight window; SES sits between them with infinite geometrically-decaying weights. The rest of this section builds on SES.