Activity31

Today, we’ll review the structure of ETS models, derive the forecasting equations, and then explore specific variants including ETS(M,N,A), ETS(M,A,A), ETS(A,Ad,A), and ETS(A,A,M).

Last class, we calculated the equations,

\[ \begin{align} y_t &= (\ell_{t-1}+s_{t-m})(1+\varepsilon_t),\\[1mm] \ell_t &= \ell_{t-1}+\alpha\,(\ell_{t-1}+s_{t-m})\,\varepsilon_t,\\[1mm] s_t &= s_{t-m}+\gamma\,(\ell_{t-1}+s_{t-m})\,\varepsilon_t, \end{align} \]

for the ETS(M,N,A) case and

\[ \begin{align} y_t &= (\ell_{t-1}+b_{t-1}+s_{t-m})(1+\varepsilon_t),\\[1mm] \ell_t &= \ell_{t-1}+b_{t-1}+\alpha\,(\ell_{t-1}+b_{t-1}+s_{t-m})\,\varepsilon_t,\\[1mm] b_t &= b_{t-1}+\beta\,(\ell_{t-1}+b_{t-1}+s_{t-m})\,\varepsilon_t,\\[1mm] s_t &= s_{t-m}+\gamma\,(\ell_{t-1}+b_{t-1}+s_{t-m})\,\varepsilon_t, \end{align} \]

for the ETS(M,A,A) case.

Derivation Outline

Observation Equation & Error Definition

The model starts with

\[ y_t = \mu_t\,(1+\varepsilon_t) \]

where the forecast is

\[\mu_t = \ell_{t-1}+s_{t-m}\quad \text{(or } \ell_{t-1}+b_{t-1}+s_{t-m}\quad\text{for ETS(M,A,A))},\]

so that

\[ \varepsilon_t = \frac{y_t}{\mu_t}-1. \]

State Updates

In a multiplicative-error model the updates use the forecast scale:
- Level update:
  
  \[ \ell_t = \ell_{t-1}+\alpha\,\mu_t\,\varepsilon_t. \]
- Seasonal update (no trend version):
  
  \[ s_t = s_{t-m}+\gamma\,\mu_t\,\varepsilon_t. \]
- For the ETS(M,A,A) model, the trend (or slope) is updated similarly:
  
  \[ b_t = b_{t-1}+\beta\,\mu_t\,\varepsilon_t. \]
Verification by Recursion

Suppose the state at time \(t-1\) is known. Then:
- The forecast is \(\mu_t = \ell_{t-1}+s_{t-m}\) (or \(\ell_{t-1}+b_{t-1}+s_{t-m}\)).
- The error is computed as
  
  \[ \varepsilon_t = \frac{y_t}{\mu_t}-1, \]
  
  so that by substitution
  
  \[ y_t-\mu_t=\mu_t\,\varepsilon_t. \]
- Therefore, the level update becomes
  
  \[ \ell_t = \ell_{t-1}+\alpha\,(y_t-\mu_t), \]
  
  and similarly for the seasonal and trend updates. This recursive structure ensures that any deviation of the observation from the forecast is proportionally fed back into the states—scaled by the forecast level.

1. Recursive Updating Equations Summary

Step 1: Compute the forecast \(\mu_t\) based on current state components.
Step 2: Calculate the error \(\varepsilon_t\) (multiplicative or additive).
Step 3: Update state components (level, trend, seasonal) using the error and smoothing parameters.
Step 4: Repeat for each new observation.

2. Deriving the ETS(M,N,A) Model

2.1 Model Structure

For ETS(M,N,A) (multiplicative error, no trend, additive seasonality), the equations are:

where:

\(y_t\) is the observed value at time \(t\).
\(\ell_{t-1}\) is the level component.
\(s_{t-m}\) is the seasonal component (with period \(m\)).
\(\varepsilon_t\) is the multiplicative error, defined as \[ \varepsilon_t = \frac{y_t}{\ell_{t-1}+s_{t-m}}-1. \]
\(\alpha\) and \(\gamma\) are the smoothing parameters.

2.2 Interpretation

Observation Equation: The forecast is given by the sum of the level and seasonal component, scaled by the multiplicative error.
Level Update: The level is adjusted by the error scaled by the current forecast.
Seasonal Update: The seasonal component is updated similarly, ensuring seasonal effects propagate correctly.

3. ETS(M,A,A) Model

3.1 Model Structure

For ETS(M,A,A) (multiplicative error, additive trend, additive seasonality), an additional trend component \(b_{t-1}\) is introduced:

3.2 Interpretation

Trend Component: The term \(b_{t-1}\) represents the additive trend.
Trend Update: The trend is adjusted with its own smoothing parameter \(\beta\).
The observation and seasonal equations are similar to ETS(M,N,A) but include the trend in the forecast.

4. Exploring Other Combinations

4.1 ETS(A,Ad,A) Model

The ETS(A,Ad,A) model uses an additive error, an additive damped trend, and additive seasonality. Its equations can be written as:

\[ \begin{align} y_t &= \ell_{t-1}+ \phi d_{t-1}+ s_{t-m}+\varepsilon_t,\\[1mm] \ell_t &= \ell_{t-1} + \phi d_{t-1} + \alpha \varepsilon_t, \\ d_t &= \phi d_{t-1} + \beta \varepsilon_t, \\ s_t &= s_{t-m} + \gamma \varepsilon_t, \end{align} \]

where:

\(\varepsilon_t\) is an additive error.
\(d_{t-1}\) represents the trend component.
\(\phi\) is the damping parameter, \(0 < \phi < 1\), that reduces the trend effect over time.

4.2 ETS(A,A,M) Model

For the ETS(A,A,M) model, the seasonal component enters multiplicatively while both the error and trend are additive:

\[ \begin{align} y_t &= (\ell_{t-1}+b_{t-1}+s_{t-m})(1+\varepsilon_t),\\[1mm] \ell_t &= \ell_{t-1}+b_{t-1}+\alpha\,\varepsilon_t,\\[1mm] b_t &= b_{t-1}+\beta\,\varepsilon_t,\\[1mm] s_t &= s_{t-m}\,(1+\gamma\,\varepsilon_t). \end{align} \]

4.3 Key Points in Model Variations

Error Type: Whether the error is additive or multiplicative will affect how deviations are incorporated into the state updates.
Trend Component: The presence (or absence) of a trend (additive or damped) changes the update equations. In a damped trend, the damping parameter \(\phi\) moderates the trend’s influence.
Seasonal Component: Additive seasonality adds or subtracts a fixed amount, while multiplicative seasonality scales the forecast.

5. Fitting ETS Models with Tidyverts

The Tidyverts framework in R (which includes packages such as fable and tsibble) provides a streamlined approach to fitting ETS models. Below is an example workflow.