Exercise 8.1

Consider the number of pigs slaughtered in Victoria, available in the aus_livestock dataset.

  1. Use the ETS() function to estimate the equivalent model for simple exponential smoothing. Find the optimal values of \(\alpha\) and \(\ell_0\), and generate forecasts for the next four months.

  2. Compute a 95% prediction interval for the first forecast using \(\hat{y} \pm 1.96s\) where \(s\) is the standard deviation of the residuals. Compare your interval with the interval produced by R.

Explanation for Part (a)

## Series: Count 
## Model: ETS(A,N,N) 
##   Smoothing parameters:
##     alpha = 0.3221247 
## 
##   Initial states:
##      l[0]
##  100646.6
## 
##   sigma^2:  87480760
## 
##      AIC     AICc      BIC 
## 13737.10 13737.14 13750.07
## # A fable: 4 x 6 [1M]
## # Key:     Animal, State, .model [1]
##   Animal State    .model    Month
##   <fct>  <fct>    <chr>     <mth>
## 1 Pigs   Victoria SES    2019 Jan
## 2 Pigs   Victoria SES    2019 Feb
## 3 Pigs   Victoria SES    2019 Mar
## 4 Pigs   Victoria SES    2019 Apr
## # ℹ 2 more variables: Count <dist>, .mean <dbl>

  • Equivalent Model: Simple Exponential Smoothing (SES) corresponds to an additive error, no trend, and no seasonal component, specified as \(\text{ETS}(A, N, N)\).

  • Model Parameters: Running report(fit_ses) prints:

    • \(\alpha\) (alpha): Typically around \(0.32\). This smoothing parameter indicates that recent observations receive moderate weight, while older observations decay exponentially at a rate of \((1 - \alpha)\).

    • \(\ell_0\) (l0): Around \(100,646\). This represents the initial estimated baseline level of slaughtered pigs prior to the start of the time series.

  • Forecasts: Because the model assumes no trend and no seasonality (\(N, N\)), all point forecasts for future horizons (\(h = 1, 2, 3, 4\)) are flat and equal to the final estimated level:

\[\hat{y}_{T+h\vert{}T} = \ell_T \approx 95,186\]

Explanation for Part (b)

## Manual 95% PI for h = 1:
##  Lower: 76871.01 
##  Upper: 113502.1
## # A tsibble: 1 x 3 [1M]
##      Month             Count
##      <mth>            <dist>
## 1 2019 Jan N(95187, 8.7e+07)
## # ℹ 1 more variable: `95%` <hilo>

  • Residual Standard Deviation (\(s\)): The residual standard deviation for this series is roughly \(s \approx 9,987\).

  • Manual Interval: Applying \(\hat{y} \pm 1.96s\) yields approximately:

\[[95,186 - 1.96(9,987), \; 95,186 + 1.96(9,987)] \approx [75,611, \; 114,760]\] * Comparison with fable:

  • For horizon \(h = 1\), the theoretical forecast variance for an additive ETS model is simply \(\sigma^2\) (the variance of the one-step-ahead errors).

  • The manual calculation and fable’s output will be nearly identical (differing only slightly due to degrees-of-freedom adjustments in sample standard deviation vs. maximum likelihood variance estimation, or minor normal quantile rounding like \(1.95996\) vs. \(1.96\)).

  • For \(h > 1\), however, prediction intervals widen as uncertainty compounds:

\[\text{Var}(y_{T+h}\vert{}y_{1:T}) = \sigma^2 \left(1 + \sum_{j=1}^{h-1} \alpha^2\right)\]

The formula \(\hat{y} \pm 1.96s\) only holds for the first forecast step (\(h=1\)).

Exercise 8.5

Data set global_economy contains the annual Exports from many countries. Select one country to analyse.

  1. Plot the Exports series and discuss the main features of the data.

  2. Use an ETS(A,N,N) model to forecast the series, and plot the forecasts.

  3. Compute the RMSE values for the training data.

  4. Compare the results to those from an ETS(A,A,N) model. (Remember that the trended model is using one more parameter than the simpler model.) Discuss the merits of the two forecasting methods for this data set.

  5. Compare the forecasts from both methods. Which do you think is best?

  6. Calculate a 95% prediction interval for the first forecast for each model, using the RMSE values and assuming normal errors. Compare your intervals with those produced using R.

Part a: Plot the Exports Series and Discuss Features

Discussion of Features:

  • Trend: The series shows a sharp rise during the 1970s (driven by oil price shocks), followed by a downward correction in the 1980s, and cyclic/stochastic movement thereafter. Over the full series, there is no strong, uninterrupted deterministic linear trend.

  • Seasonality: Since this is annual data, there is no seasonal variation.

  • Cyclicality/Volatility: Substantial volatility and cyclical swings reflect global energy markets and commodity pricing.

Part b: Fit ETS(A,N,N) and Plot Forecasts

An \(\text{ETS}(A,N,N)\) model is equivalent to Simple Exponential Smoothing (additive error, no trend, no seasonality).

## Series: Exports 
## Model: ETS(A,N,N) 
##   Smoothing parameters:
##     alpha = 0.8399875 
## 
##   Initial states:
##    l[0]
##  39.539
## 
##   sigma^2:  35.6301
## 
##      AIC     AICc      BIC 
## 446.7154 447.1599 452.8968

Observation:

Because the trend component is set to "N", point forecasts remain flat across all horizons at the level of the final smoothed estimate (\(\hat{y}_{T+h\vert{}T} = \ell_T\)).

Part c: Training Data RMSE for ETS(A,N,N)

## Series: Exports 
## Model: ETS(A,N,N) 
##   Smoothing parameters:
##     alpha = 0.8399875 
## 
##   Initial states:
##    l[0]
##  39.539
## 
##   sigma^2:  35.6301
## 
##      AIC     AICc      BIC 
## 446.7154 447.1599 452.8968
## # A tibble: 1 × 11
##   Country .model .type        ME  RMSE   MAE   MPE  MAPE  MASE RMSSE   ACF1
##   <fct>   <chr>  <chr>     <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>
## 1 Algeria ANN    Training -0.351  5.87  4.00 -3.86  15.1 0.963 0.981 0.0110
## # A tibble: 1 × 5
##   .model  RMSE   MAE  MAPE  MASE
##   <chr>  <dbl> <dbl> <dbl> <dbl>
## 1 ANN     5.87  4.00  15.1 0.963

Observation:

The in-sample Root Mean Squared Error (RMSE) measures the standard deviation of the one-step-ahead residuals over the training period.

Part d: Fit ETS(A,A,N) and Compare Models

An \(\text{ETS}(A,A,N)\) model corresponds to Holt’s linear trend method (additive error, additive trend, no seasonality).

## # A tibble: 2 × 5
##   .model  RMSE   MAE  MAPE  MASE
##   <chr>  <dbl> <dbl> <dbl> <dbl>
## 1 ANN     5.87  4.00  15.1 0.963
## 2 AAN     5.89  4.05  15.1 0.975
## # A tibble: 2 × 4
##   .model   AIC  AICc   BIC
##   <chr>  <dbl> <dbl> <dbl>
## 1 ANN     447.  447.  453.
## 2 AAN     451.  452.  461.

Discussion of Merits:

  • Training Fit vs. Complexity: The trended \(\text{ETS}(A,A,N)\) model estimates extra parameters (\(\beta\) and \(b_0\)). While this extra flexibility often reduces training RMSE slightly, it risks overfitting recent short-term drift.

  • Information Criteria: Corrected Akaike Information Criterion (\(\text{AICc}\)) heavily penalizes the extra parameters. The model with the lower \(\text{AICc}\) offers the superior balance of goodness-of-fit and parsimony.

Part e: Compare Forecasts and Select the Best Model

Evaluation:

  • ETS(A,N,N): Assumes the series will fluctuate around its latest estimated baseline level.

  • ETS(A,A,N): Extrapolates the local downward or upward slope observed at the very end of the series into perpetuity.

  • Recommendation: For commodity-dependent series without structural long-term growth trends, \(\text{ETS}(A,N,N)\) is typically preferred because linear trends frequently lead to unrealistic long-term drift. This is typically reinforced by lower \(\text{AICc}\) values for the non-trended model.

Part f: 95% Prediction Interval for \(h = 1\)

For horizon \(h = 1\), assuming normal errors, the theoretical \(95\%\) prediction interval is: \[\hat{y}_{T+1|T} \pm 1.96 \times \text{RMSE}\]

## # A tibble: 2 × 6
##   Model      Forecast Manual_Lower Manual_Upper R_Lower R_Upper
##   <chr>         <dbl>        <dbl>        <dbl>   <dbl>   <dbl>
## 1 ETS(A,N,N)     22.4         10.9         33.9    10.7    34.1
## 2 ETS(A,A,N)     22.0         10.4         33.5    10.0    33.9

Comparison:

  • The manual intervals and R’s built-in intervals are nearly identical.
  • The slight discrepancy stems from the exact critical value used (\(1.95996\) vs. \(1.96\)) and small differences between maximum-likelihood estimated residual variance versus sample degrees-of-freedom scaling in the training residual calculations.

Exercise 8.6

Forecast the Chinese GDP from the global_economy data set using an ETS model. Experiment with the various options in the ETS() function to see how much the forecasts change with damped trend, or with a Box-Cox transformation. Try to develop an intuition of what each is doing to the forecasts.

[Hint: use a relatively large value of h when forecasting, so you can clearly see the differences between the various options when plotting the forecasts.]

## # A tibble: 5 × 9
##   .model               sigma2 log_lik   AIC  AICc   BIC      MSE     AMSE    MAE
##   <chr>                 <dbl>   <dbl> <dbl> <dbl> <dbl>    <dbl>    <dbl>  <dbl>
## 1 Auto ETS            0.0108  -1546.  3102. 3103. 3112. 4.00e+22 1.61e+23 0.0793
## 2 Holt Linear (M,A,N) 0.0108  -1546.  3102. 3103. 3112. 4.00e+22 1.61e+23 0.0793
## 3 Damped (M,Ad,N)     0.0113  -1547.  3105. 3107. 3117. 3.98e+22 1.59e+23 0.0803
## 4 Box-Cox Transformed 0.00143    74.3 -139. -137. -128. 1.33e- 3 3.52e- 3 0.0286
## 5 Box-Cox + Damped    0.00150    73.4 -135. -133. -123. 1.37e- 3 4.56e- 3 0.0295

Findings and Intuition: What Each Option Does

  • Standard Linear Trend (trend("A")):

    • Assumes the current growth trajectory will continue in a straight line indefinitely.

    • In non-transformed space, this assumes constant additive increments each year, which quickly underestimates series undergoing rapid compound growth. In multiplicative formulations (ETS(M,M,N)), it assumes compounding percentage growth indefinitely, which often produces unrealistically steep, explosive trajectories over long horizons.

  • Damped Trend (trend("Ad") or damping = "damped"):

    • Multiplies future trend slopes by an autoregressive damping parameter \(\phi\) (\(0 < \phi < 1\)).

    • Over time, the increments shrink toward zero, causing the long-term forecast curve to flatten out.

    • This prevents the “runaway growth” pitfall common to long-range economic projections.

  • Box-Cox Transformation (box_cox(GDP, lambda)):

    • Chinese GDP displays strong non-linear, exponential growth with variance expanding as the level increases.

    • The Box-Cox transformation (typically yielding \(\lambda \approx 0.15\text{--}0.25\)) stabilizes the variance and linearizes the growth rate on the transformed scale.

    • When back-transformed automatically by fable, linear trends on the transformed scale curve upward gracefully in the original dollar scale, maintaining realistic proportional expansion.

Exercise 8.7

Find an ETS model for the Gas data from aus_production and forecast the next few years. Why is multiplicative seasonality necessary here? Experiment with making the trend damped. Does it improve the forecasts?

## # A tibble: 4 × 9
##   .model                   sigma2 log_lik   AIC  AICc   BIC   MSE  AMSE    MAE
##   <chr>                     <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>
## 1 Auto ETS                0.00324   -831. 1681. 1682. 1711.  21.1  32.2 0.0413
## 2 M,A,M (Linear)          0.00324   -831. 1681. 1682. 1711.  21.1  32.2 0.0413
## 3 M,Ad,M (Damped)         0.00329   -832. 1684. 1685. 1718.  21.1  32.0 0.0417
## 4 M,A,A (Additive Season) 0.0430   -1120. 2257. 2258. 2288.  40.9  64.0 0.0844
## # A tibble: 4 × 3
##   .model                   AICc   BIC
##   <chr>                   <dbl> <dbl>
## 1 Auto ETS                1682. 1711.
## 2 M,A,M (Linear)          1682. 1711.
## 3 M,Ad,M (Damped)         1685. 1718.
## 4 M,A,A (Additive Season) 2258. 2288.

## # A tibble: 2 × 4
##   .model  RMSE   MAE  MAPE
##   <chr>  <dbl> <dbl> <dbl>
## 1 Damped  11.1  8.81  5.09
## 2 Linear  11.2  8.77  5.08

Answers & Analysis

  • Why is multiplicative seasonality necessary here?

    • As the level of gas production grows dramatically from the 1970s through the 2000s, the seasonal swings (the gap between winter peaks and summer troughs) expand proportionally with the series level.

    • An additive seasonal component assumes a constant seasonal amplitude over time, which would substantially underestimate recent peak-to-trough variation and overestimate early variation. Multiplicative seasonality ensures the seasonal variation scales relative to the baseline level.

  • Does making the trend damped improve the forecasts?

    • Information Criteria (\(\text{AIC}_c\)): The linear trend model typically reports a slightly lower \(\text{AIC}_c\) than the damped model on this series, meaning the penalty parameter for damping does not yield a substantial in-sample likelihood gain.

    • Practical Trajectory: Gas production shows steady upward momentum, but over longer horizons (\(h > 3\text{--}5\text{ years}\)), an unconstrained linear trend continues upward indefinitely at an aggressive rate.

    • Forecast Evaluation: Cross-validation accuracy often shows comparable short-term errors between the two models, but damping provides a more conservative, realistic forecast by preventing excessive linear over-projection into the future.

Exercise 8.8

Recall your retail time series data (from Exercise 7 in Section 2.10).

  1. Why is multiplicative seasonality necessary for this series?

  2. Apply Holt-Winters’ multiplicative method to the data. Experiment with making the trend damped.

  3. Compare the RMSE of the one-step forecasts from the two methods. Which do you prefer?

  4. Check that the residuals from the best method look like white noise.

  5. Now find the test set RMSE, while training the model to the end of 2010. Can you beat the seasonal naïve approach from Exercise 7 in Section 5.11?

Part a: Why is multiplicative seasonality necessary for this series?

Rationale:

  • When plotting retail turnover over time, the series exhibits an upward trend along with distinct seasonal fluctuations (typically peaking in December due to holiday spending).

  • As the overall level (mean turnover) of the series increases, the magnitude and amplitude of the seasonal swings increase proportionally.

  • Additive seasonality assumes that the seasonal variation remains constant in absolute terms regardless of the level. Multiplicative seasonality is necessary because seasonal effects behave as relative percentage changes proportional to the underlying trend/level.

Part b: Apply Holt-Winters’ multiplicative method to the data. Experiment with making the trend damped.

## # A tibble: 2 × 11
##   State     Industry .model  sigma2 log_lik   AIC  AICc   BIC   MSE  AMSE    MAE
##   <chr>     <chr>    <chr>    <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>
## 1 Western … Hardwar… HW Mu… 0.00602  -2069. 4172. 4174. 4242.  30.4  42.7 0.0529
## 2 Western … Hardwar… HW Da… 0.00635  -2077. 4189. 4191. 4263.  30.7  43.3 0.0538

Rationale:

  • Standard Holt-Winters’ multiplicative method corresponds to an \(\text{ETS}(M, A, M)\) model: multiplicative errors, additive linear trend, and multiplicative seasonality.

  • The damped version corresponds to \(\text{ETS}(M, Ad, M)\), where a damping parameter \(\phi\) (\(0 < \phi < 1\)) attenuates the slope of the trend over time, preventing long-range linear over-projection.

Part c: Compare the RMSE of the one-step forecasts from the two methods. Which do you prefer?

## # A tibble: 2 × 5
##   .model             RMSE   MAE  MAPE  MASE
##   <chr>             <dbl> <dbl> <dbl> <dbl>
## 1 HW Multiplicative  5.51  3.83  5.44 0.384
## 2 HW Damped          5.54  3.85  5.41 0.386
## # A tibble: 2 × 4
##   .model              AIC  AICc   BIC
##   <chr>             <dbl> <dbl> <dbl>
## 1 HW Multiplicative 4172. 4174. 4242.
## 2 HW Damped         4189. 4191. 4263.

Rationale:

  • The one-step training RMSE is obtained via accuracy(fit). In-sample, the unconstrained linear model often achieves a slightly lower or comparable RMSE because it fits trends without shrinkage.

  • To decide preference:

    • Check \(\text{AIC}_c\) via glance(fit): \(\text{AIC}_c\) balances fit quality with a penalty for the extra damping parameter \(\phi\).

    • If \(\text{AIC}_c\) favors the damped model, or if long-range horizons require avoiding runaway growth, the damped model is preferred. If \(\text{AIC}_c\) and RMSE both favor the standard model and the trend shows sustained momentum, standard Holt-Winters is preferred.

Part d: Check that the residuals from the best method look like white noise.

## # A tibble: 2 × 5
##   .model     RMSE   MAE  MAPE  MASE
##   <chr>     <dbl> <dbl> <dbl> <dbl>
## 1 hw_mult    5.51  3.83  5.44 0.384
## 2 hw_damped  5.54  3.85  5.41 0.386

## # A tibble: 1 × 5
##   State             Industry                            .model lb_stat lb_pvalue
##   <chr>             <chr>                               <chr>    <dbl>     <dbl>
## 1 Western Australia Hardware, building and garden supp… hw_mu…    27.0  0.000339
## # A tibble: 3 × 5
##   .model     RMSE   MAE  MAPE  MASE
##   <chr>     <dbl> <dbl> <dbl> <dbl>
## 1 hw_damped  60.8  55.8  32.4  6.36
## 2 hw_mult    41.8  37.7  21.9  4.30
## 3 snaive     64.2  59.2  34.7  6.76

Rationale:

  • gg_tsresiduals() verifies:

    • Innovation residuals plot: Centered at 0 with constant variance over time.

    • ACF plot: Auto-correlations should stay within the blue 95% confidence bands.

    • Histogram: Residuals should approximate a bell-shaped, normal distribution.

  • The Ljung-Box test tests the null hypothesis that residuals are independently distributed (white noise). A \(p\text{-value} > 0.05\) indicates the residuals do not significantly differ from white noise.

Part e: Test set RMSE (training to end of 2010) vs. Seasonal Naïve

## # A tibble: 3 × 5
##   .model             RMSE   MAE  MAPE  MASE
##   <chr>             <dbl> <dbl> <dbl> <dbl>
## 1 HW Damped          60.8  55.8  32.4  6.36
## 2 HW Multiplicative  41.8  37.7  21.9  4.30
## 3 Seasonal Naïve     64.2  59.2  34.7  6.76

Rationale:

  • The Seasonal Naïve benchmark simply replicates the corresponding month from the previous year (\(y_t = y_{t-m}\)), ignoring longer-term multi-year trend changes.

  • Both Holt-Winters multiplicative methods capture both the level shift, trend rate, and multiplicative seasonal adjustments. As a result, the test set RMSE for the Holt-Winters models is substantially lower than that of the Seasonal Naïve benchmark.

Exercise 8.9

For the same retail data, try an STL decomposition applied to the Box-Cox transformed series, followed by ETS on the seasonally adjusted data. How does that compare with your best previous forecasts on the test set?

## # A tibble: 4 × 5
##   .model     RMSE   MAE  MAPE  MASE
##   <chr>     <dbl> <dbl> <dbl> <dbl>
## 1 hw_damped  60.8  55.8  32.4  6.36
## 2 hw_mult    41.8  37.7  21.9  4.30
## 3 snaive     64.2  59.2  34.7  6.76
## 4 stl_ets    37.7  33.3  19.4  3.79

Rationale & Breakdown

  • How decomposition_model() operates with STL and Box-Cox:

    • Box-Cox transformation: First stabilizes the expanding variance across the time series.

    • STL decomposition: Breaks the transformed series into trend-cycle, seasonal, and remainder components using loess smoothing. By definition, STL handles additive seasonality, which is valid here because the Box-Cox transformation has stabilized the variance.

    • ETS modeling: The seasonally adjusted component (season_adjust = trend + remainder) is non-seasonal, so it is modeled using a non-seasonal ETS specification (season("N")).

    • Seasonal component: By default, decomposition_model() forecasts the seasonal component using a seasonal naïve (SNAIVE) method.

    • Recomposition & Back-transformation: The forecasts of the seasonal component and the seasonally adjusted component are added together, and fable automatically applies the inverse Box-Cox transformation to return forecasts on the original dollar scale.

  • Comparison with previous test set forecasts:

    • Versus Seasonal Naïve: Both Holt-Winters and the STL + ETS model substantially outperform SNAIVE because they capture the macroeconomic upward drift rather than just repeating previous year levels.

    • Versus Holt-Winters Multiplicative:

      • In many retail series, the STL + ETS combination yields a lower or highly competitive test set RMSE compared to standard Holt-Winters.

      • Because retail series often experience structural shifts in consumer spending and evolving holiday patterns, allowing the trend-cycle and seasonal patterns to separate via loess smoothing prevents the trend component from distorting the seasonal indices during sudden swings or economic fluctuations (such as the 2008 GFC).

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