Data_624: Homework 5
Arutam Antunish
2026-03-09
CHAPTER 8 Exercises
8.1 Consider the the number of pigs slaughtered in Victoria, available in the aus_livestock dataset.
- Use the ETS() function to estimate the equivalent model for simple exponential smoothing. Find the optimal values of α and ℓ 0, and generate forecasts for the next four months.
We use the ETS() function to specify a Simple Exponential Smoothing (SES) model. By setting error(“A”), trend(“N”), and season(“N”), we tell R to treat the series as having no trend and no seasonality, which is exactly what SES does.
library(fpp3)## Warning: package 'fpp3' was built under R version 4.5.2## Registered S3 method overwritten by 'tsibble':
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## ✖ tsibble::union() masks base::union()# 1. Filter the data for Pigs in Victoria
pigs_vic <- aus_livestock %>%
filter(Animal == "Pigs", State == "Victoria")
# 2. Estimate the SES model
fit <- pigs_vic %>%
model(SES = ETS(Count ~ error("A") + trend("N") + season("N")))
# 3. Extract optimal parameters
report(fit)## 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# 4. Generate forecasts for the next 4 months
fc <- fit %>%
forecast(h = 4)
View(fc)-Alpha (\(\alpha = 0.322\)): The smoothing parameter for the level. Since it’s around 0.3, the model is balanced—it values recent data but doesn’t overreact to every small monthly fluctuation in the pig count.
-Initial State (\(\ell_0 = 100,646.6\)): The estimated starting level of the series at the beginning of the dataset.
- Compute a 95% prediction interval for the first forecast using ^ y ± 1.96 s where s is the standard deviation of the residuals. Compare your interval with the interval produced by R.
Here we compare the mathematical approximation of the 95% interval against the internal computation performed by the fable package.
# Get the standard deviation of residuals (sigma)
s <- tidy(fit) %>%
filter(term == "sigma") %>%
pull(estimate)
# Calculate the manual 95% interval for the first forecast
y_hat <- fc$.mean[1]
lower_manual <- y_hat - 1.96 * s
upper_manual <- y_hat + 1.96 * s
# Display results
cat("Manual Interval: [", lower_manual, ",", upper_manual, "]\n")## Manual Interval: [ , ]print(fc[1,])## # A fable: 1 x 6 [1M]
## # Key: Animal, State, .model [1]
## Animal State .model Month
## <fct> <fct> <chr> <mth>
## 1 Pigs Victoria SES 2019 Jan
## # ℹ 2 more variables: Count <dist>, .mean <dbl>8.5 Data set global_economy contains the annual Exports from many countries. Select one country to analyse.
- Plot the Exports series and discuss the main features of the data.
# Filter data for Canada
can_exports <- global_economy %>% filter(Country == "Canada")
# Plot the series
can_exports %>% autoplot(Exports) +
labs(title = "Annual Exports: Canada", y = "% of GDP")
Discussion: The series shows significant volatility and a clear upward trend from the 1960s until roughly 2000, followed by a decline and stabilization. This suggests that a model capable of capturing trend (like ETS(A,A,N)) might outperform simple smoothing.
- Use an ETS(A,N,N) model to forecast the series, and plot the forecasts.
fit_nn <- can_exports %>% model(ETS(Exports ~ error("A") + trend("N") + season("N")))
fc_nn <- fit_nn %>% forecast(h = 5)
fc_nn %>% autoplot(can_exports) + labs(title = "ETS(A,N,N) Forecast")
Analysis: SES (A,N,N) produces a flat forecast line, assuming the future will simply reflect the most recent average level, ignoring the long-term historical trend.
- Compute the RMSE values for the training data.
accuracy(fit_nn) %>% select(RMSE)The ETS(A,A,N) model (Holt’s Linear Method) allows for a slope component. If your resulting RMSE is lower than the previous 1.617713, it confirms that the trend component successfully captured more variance in the Canadian export data. However, since the data exhibits a structural break (the decline after 2000), a simple linear trend may still struggle to forecast the post-2000 period accurately.
- 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.
fit_an <- can_exports %>% model(ETS(Exports ~ error("A") + trend("A") + season("N")))
accuracy(fit_an) %>% select(RMSE)Analysis: ETS(A,N,N) vs. ETS(A,A,N) Performance Comparison: The ETS(A,N,N) model achieved an RMSE of 1.6177 while the ETS(A,A,N) model resulted in 1.6256.
Model Selection: Adding a trend component increased the RMSE. This indicates that the trend parameter did not improve predictive accuracy for this specific dataset.
Conclusion: Per the principle of parsimony, the simpler ETS(A,N,N) model is preferred, as the extra complexity of the trended model provided no benefit in error reduction.
- Compare the forecasts from both methods. Which do you think is best?
# Plot both forecasts
fit_both <- can_exports %>% model(ANN = ETS(Exports ~ error("A") + trend("N") + season("N")), AAN = ETS(Exports ~ error("A") + trend("A") + season("N")))
fit_both %>% forecast(h = 5) %>% autoplot(can_exports)
- Comparison of Forecasts Visual Interpretation: The ETS(A,N,N) (ANN) model produces a flat, horizontal forecast based on the most recent data level. In contrast, the ETS(A,A,N) (AAN) model attempts to project the most recent linear trend.
Method Selection: The ANN model is the preferred choice for this dataset. The AAN model’s trend projection is misleading because it incorrectly assumes the recent historical decline will continue linearly into the future.
Conclusion: Given that the AAN model had a higher RMSE (1.6256 vs 1.6177) and fails to capture the stabilization observed in the series, the simpler ANN model is superior for these data.
- 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.
# Extract RMSE
rmse_nn <- accuracy(fit_nn)$RMSE
y_hat <- fc_nn$.mean[1]
# Manual 95% Interval (using RMSE as a proxy for sigma)
lower_manual <- y_hat - 1.96 * rmse_nn
upper_manual <- y_hat + 1.96 * rmse_nn8.6 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.]
# 1. Filter data for China
china_gdp <- global_economy %>% filter(Country == "China")
# 2. Define models
fit_china <- china_gdp %>%
model(
Standard = ETS(GDP ~ error("A") + trend("A") + season("N")),
Damped = ETS(GDP ~ error("A") + trend("Ad") + season("N")),
BoxCox = ETS(box_cox(GDP, 0.3) ~ error("A") + trend("A") + season("N"))
)
# 3. Forecast with a large horizon (h = 20)
fc_china <- fit_china %>% forecast(h = 20)
# 4. Plot
fc_china %>% autoplot(china_gdp) + labs(title = "China GDP: ETS Model Comparison")
Analysis: China GDP Forecast Comparison Standard Trend (A,A,N): Projects historical growth linearly. It is often too aggressive for long-term horizons as it assumes growth rates remain constant.
Damped Trend (A,Ad,N): Acts as a “stabilizer.” By forcing the trend to flatten over time, it produces a more realistic, conservative forecast that avoids explosive, unrealistic growth.
Box-Cox Transformation: Adjusts for rapid acceleration in the GDP data. It stabilizes the variance, preventing the forecast intervals from becoming artificially narrow or wide due to the exponential nature of the series.
Intuition: Use Damped trends to avoid over-optimistic projections and Box-Cox transformations when the absolute size of the data changes the underlying volatility of the series.
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?
# Filter data for Gas
gas_data <- aus_production %>% filter(!is.na(Gas))
# Fit models: Additive vs. Multiplicative seasonality and Damped trend
fit_gas <- gas_data %>%
model(
Multiplicative = ETS(Gas ~ error("M") + trend("A") + season("M")),
Damped = ETS(Gas ~ error("M") + trend("Ad") + season("M"))
)
# Forecast
fc_gas <- fit_gas %>% forecast(h = "5 years")
# Plot
fc_gas %>% autoplot(gas_data) + labs(title = "Gas Production Forecast")
Analysis: Gas Production Forecast
Seasonality: The use of multiplicative seasonality (season(“M”)) is necessary because the magnitude of the seasonal fluctuations in gas production increases proportionally as the overall production level grows over time. Damped Trend Experiment: Applying a damped trend (trend(“Ad”)) moderates the long-term growth projection. While a standard trend assumes growth continues at a constant rate, the damped model assumes the growth rate will eventually plateau.
Conclusion: Whether the damped trend improves the forecast depends on your expectations for the industry; if you anticipate that the recent historical growth will eventually slow down, the damped model provides a more realistic and conservative forecast compared to the linear trend.
8.8 Recall your retail time series data (from Exercise 7 in Section 2.10).
Why is multiplicative seasonality necessary for this series? Apply Holt-Winters’ multiplicative method to the data. Experiment with making the trend damped.
Multiplicative seasonality is necessary because the seasonal peaks (Christmas/holiday spikes) grow in proportion to the total sales volume. An additive model would inaccurately keep the seasonal variance constant.
library(fpp3)
# Filter the series
retail_data <- aus_retail %>%
filter(State == "Victoria", Industry == "Clothing retailing")
# Fit models: Holt-Winters vs. Damped
fit <- retail_data %>%
model(
HW = ETS(Turnover ~ error("M") + trend("A") + season("M")),
Damped = ETS(Turnover ~ error("M") + trend("Ad") + season("M"))
)Compare the RMSE of the one-step forecasts from the two methods. Which do you prefer? Check that the residuals from the best method look like white noise.
# Compare RMSE
accuracy(fit) %>% select(.model, RMSE)# Residual check for the best model
fit %>% select(HW) %>% gg_tsresiduals()## Warning: `gg_tsresiduals()` was deprecated in feasts 0.4.2.
## ℹ Please use `ggtime::gg_tsresiduals()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
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?
# Train to 2010
train <- retail_data %>% filter(Month <= yearmonth("2010 Dec"))
# Fit models and Seasonal Naïve
fit_test <- train %>%
model(
HW = ETS(Turnover ~ error("M") + trend("A") + season("M")),
SNAIVE = SNAIVE(Turnover)
)
# Forecast and compare to test data
fc <- fit_test %>% forecast(h = "5 years")
fc %>% accuracy(retail_data)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?
# Filter your retail series
retail_data <- aus_retail %>%
filter(State == "Victoria", Industry == "Clothing retailing")
# Apply STL decomposition + ETS on seasonally adjusted data
# We use decomposition_model() to handle the re-seasonalization
fit_stl <- retail_data %>%
model(
stl_ets = decomposition_model(
STL(box_cox(Turnover, 0.3) ~ trend(window = 21) + season(window = "periodic")),
ETS(season_adjust ~ error("A") + trend("A") + season("N"))
)
)
# Forecast and compare to test data
train <- retail_data %>% filter(Month <= yearmonth("2010 Dec"))
fit_test_stl <- train %>%
model(stl_ets = decomposition_model(
STL(box_cox(Turnover, 0.3) ~ trend(window = 21) + season(window = "periodic")),
ETS(season_adjust ~ error("A") + trend("A") + season("N"))
))
fc_stl <- fit_test_stl %>% forecast(h = "5 years")
fc_stl %>% accuracy(retail_data)Analysis: STL Decomposition and ETS Methodology: By applying a Box-Cox transformation followed by STL decomposition, we effectively stabilized the variance and separated the seasonal component from the trend. This allows the ETS model to focus exclusively on the seasonally adjusted series .
Performance: With an RMSE of 72.27, we can compare this model directly against the previous Holt-Winters and Seasonal Naïve approaches.
Evaluation: The relatively high ACF1 value (0.775) indicates that significant information remains in the residuals . This suggests that while the STL+ETS approach is robust, it may not have outperformed your best previous model in this specific case. Further optimization of the trend window or seasonality parameters may be required to reduce the residual autocorrelation.