Assignment 5 D624
Jose Fuentes
2025-03-08
Homework 5
Do exercises 8.1, 8.5, 8.6, 8.7, 8.8, 8.9 in Hyndman. Please submit both the link to your Rpubs and the .pdf file with your run code
8.1)
#Consider the the number of pigs slaughtered in Victoria, available in the aus_livestock dataset.
library(dplyr)## Warning: package 'dplyr' was built under R version 4.4.2##
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## intersect, setdiff, unionlibrary(fpp3)## Warning: package 'fpp3' was built under R version 4.4.2## ── Attaching packages ──────────────────────────────────────────── fpp3 1.0.1 ──## ✔ tibble 3.2.1 ✔ ggplot2 3.5.1
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## ✖ dplyr::lag() masks stats::lag()
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## ✖ tsibble::union() masks base::union()# Load the dataset
data("aus_livestock")
data <- aus_livestock
# Filter the data for pigs slaughtered in Victoria
fit <- data %>%
filter(Animal == "Pigs", State == "Victoria") %>%
model(
SES = ETS(Count ~ error("A") + trend("N") + season("N"))
)
# Get the optimal parameter estimates
optimal_params <- tidy(fit) %>%
select(term, estimate) %>%
mutate(estimate = round(estimate, 4))
print(optimal_params)## # A tibble: 2 × 2
## term estimate
## <chr> <dbl>
## 1 alpha 0.322
## 2 l[0] 100647.# Generate the forecast for the next four months
fc <- fit %>%
forecast(h = 4)
# Select forecasted values
forecasted_values <- fc %>%
select(Month, Count, .mean)
print(forecasted_values)## # A fable: 4 x 3 [1M]
## Month Count
## <mth> <dist>
## 1 2019 Jan N(95187, 8.7e+07)
## 2 2019 Feb N(95187, 9.7e+07)
## 3 2019 Mar N(95187, 1.1e+08)
## 4 2019 Apr N(95187, 1.1e+08)
## # ℹ 1 more variable: .mean <dbl># Extract the first forecast value
y <- fc %>%
slice(1) %>%
pull(.mean)
# Compute the standard deviation of the residuals
s <- augment(fit) %>%
pull(.resid) %>%
sd()
# Compute and print the 95% prediction interval for the first forecast
cat("The 95% prediction interval for the first forecast is between", y - 1.96 * s, "and", y + 1.96 * s, "\n")## The 95% prediction interval for the first forecast is between 76871.01 and 113502.1# Compare with the interval produced by R's `hilo` function
interval <- fc %>%
slice(1) %>%
hilo(95) %>%
pull()
print(interval)## <hilo[1]>
## [1] [76854.79, 113518.3]958.5)
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. Use an ETS(A,N,N) model to forecast the series, and plot the forecasts. Compute the RMSE values for the training data. 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. Compare the forecasts from both methods. Which do you think is best? 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.
library(fpp3)
# Load the dataset
data("global_economy")
data <- global_economy
# Select the data for one country, e.g., Australia
australia_exports <- data %>%
filter(Country == "Australia") %>%
select(Year, Exports)
# Plot the Exports series
autoplot(australia_exports, Exports) +
ggtitle("Exports for Australia") +
xlab("Year") +
ylab("Exports (in millions)")
# Fit an ETS(A,N,N) model
fit_ANN <- australia_exports %>%
model(ETS(Exports ~ error("A") + trend("N") + season("N")))
# Generate the forecast for the next four years using ETS(A,N,N)
fc_ANN <- fit_ANN %>%
forecast(h = "4 years")
# Plot the forecasts from ETS(A,N,N)
fc_ANN %>%
autoplot(australia_exports) +
ggtitle("Exports Forecast for Australia using ETS(A,N,N)") +
xlab("Year") +
ylab("Exports (in millions)")
# Compute the RMSE for the ETS(A,N,N) model
rmse_ANN <- fit_ANN %>%
accuracy() %>%
filter(.model == "ETS(Exports ~ error(\"A\") + trend(\"N\") + season(\"N\"))") %>%
pull(RMSE)
# Fit an ETS(A,A,N) model
fit_AAN <- australia_exports %>%
model(ETS(Exports ~ error("A") + trend("A") + season("N")))
# Generate the forecast for the next four years using ETS(A,A,N)
fc_AAN <- fit_AAN %>%
forecast(h = "4 years")
# Plot the forecasts from ETS(A,A,N)
fc_AAN %>%
autoplot(australia_exports) +
ggtitle("Exports Forecast for Australia using ETS(A,A,N)") +
xlab("Year") +
ylab("Exports (in millions)")
# Compute the RMSE for the ETS(A,A,N) model
rmse_AAN <- fit_AAN %>%
accuracy() %>%
filter(.model == "ETS(Exports ~ error(\"A\") + trend(\"A\") + season(\"N\"))") %>%
pull(RMSE)
# Compute the 95% prediction interval for the first forecast using ETS(A,N,N)
fc1_ANN <- fc_ANN %>%
slice(1) %>%
pull(.mean)
s_ANN <- rmse_ANN
lower_bound_ANN <- fc1_ANN - 1.96 * s_ANN
upper_bound_ANN <- fc1_ANN + 1.96 * s_ANN
cat("The 95% prediction interval for the first forecast using ETS(A,N,N) is between", lower_bound_ANN, "and", upper_bound_ANN, "\n")## The 95% prediction interval for the first forecast using ETS(A,N,N) is between 18.35944 and 22.85488# Compute the 95% prediction interval for the first forecast using ETS(A,A,N)
fc1_AAN <- fc_AAN %>%
slice(1) %>%
pull(.mean)
s_AAN <- rmse_AAN
lower_bound_AAN <- fc1_AAN - 1.96 * s_AAN
upper_bound_AAN <- fc1_AAN + 1.96 * s_AAN
cat("The 95% prediction interval for the first forecast using ETS(A,A,N) is between", lower_bound_AAN, "and", upper_bound_AAN, "\n")## The 95% prediction interval for the first forecast using ETS(A,A,N) is between 18.64986 and 23.02743# Compare with the intervals produced by R's `hilo` function
interval_ANN <- fc_ANN %>%
slice(1) %>%
hilo(95) %>%
pull()
print(interval_ANN)## <hilo[1]>
## [1] [18.3197, 22.89462]95interval_AAN <- fc_AAN %>%
slice(1) %>%
hilo(95) %>%
pull()
print(interval_AAN)## <hilo[1]>
## [1] [18.57028, 23.107]95This analysis examined the 95% prediction intervals generated by two Exponential Smoothing (ETS) models: ETS(A,N,N) and ETS(A,A,N). The goal was to compare intervals calculated manually using Root Mean Squared Error (RMSE) with those produced by R’s hilo function. For both models, the manually derived intervals closely matched the output of the hilo function, demonstrating consistency and validating the reliability of both calculation methods.
The ETS(A,A,N) model, which incorporates a trend component, produced slightly wider prediction intervals than the simpler ETS(A,N,N) model. This difference is expected, as the trend component introduces additional variability into the forecasts. The selection between these models hinges on the nature of the data: ETS(A,N,N) is suitable for data without a discernible trend, while ETS(A,A,N) is preferred when a trend is present. Ultimately, both models offer dependable prediction intervals, and the optimal choice depends on the specific characteristics of the data and the forecasting requirements.
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.]
library(fpp3)
# Load the dataset
data("global_economy")
data <- global_economy
# Select the data for China
china_gdp <- data %>%
filter(Country == "China") %>%
select(Year, GDP)
# Plot the GDP series
autoplot(china_gdp, GDP) +
ggtitle("GDP for China") +
xlab("Year") +
ylab("GDP (in billions)")
# Fit ETS(A,N,N) model
fit_ANN <- china_gdp %>%
model(ETS(GDP ~ error("A") + trend("N") + season("N")))
# Generate the forecast for the next 20 years
fc_ANN <- fit_ANN %>%
forecast(h = "20 years")
# Plot the forecasts from ETS(A,N,N)
fc_ANN %>%
autoplot(china_gdp) +
ggtitle("GDP Forecast for China using ETS(A,N,N)") +
xlab("Year") +
ylab("GDP (in billions)")
# Fit ETS(A,A,N) model with damped trend
fit_AAN_damped <- china_gdp %>%
model(ETS(GDP ~ error("A") + trend("Ad") + season("N")))
# Generate the forecast for the next 20 years
fc_AAN_damped <- fit_AAN_damped %>%
forecast(h = "20 years")
# Plot the forecasts from ETS(A,A,N) with damped trend
fc_AAN_damped %>%
autoplot(china_gdp) +
ggtitle("GDP Forecast for China using ETS(A,A,N) with Damped Trend") +
xlab("Year") +
ylab("GDP (in billions)")
# Fit ETS model with Box-Cox transformation
fit_boxcox <- china_gdp %>%
model(ETS(GDP ~ error("A") + trend("A") + season("N"), lambda = 0.3))## Warning: 1 error encountered for ETS(GDP ~ error("A") + trend("A") + season("N"), lambda = 0.3)
## [1] unused argument (lambda = 0.3)# Generate the forecast for the next 20 years
fc_boxcox <- fit_boxcox %>%
forecast(h = "20 years")
# Plot the forecasts from ETS model with Box-Cox transformation
fc_boxcox %>%
autoplot(china_gdp) +
ggtitle("GDP Forecast for China using ETS model with Box-Cox Transformation") +
xlab("Year") +
ylab("GDP (in billions)")## Warning in max(ids, na.rm = TRUE): no non-missing arguments to max; returning
## -Inf## Warning in max(ids, na.rm = TRUE): no non-missing arguments to max; returning
## -Inf## Warning: Removed 20 rows containing missing values or values outside the scale range
## (`geom_line()`).
# Compute accuracy metrics for each model
accuracy_ANN <- fit_ANN %>%
accuracy()
accuracy_AAN_damped <- fit_AAN_damped %>%
accuracy()
accuracy_boxcox <- fit_boxcox %>%
accuracy()
print(accuracy_ANN)## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(GDP ~ error(… Trai… 2.10e11 4.16e11 2.13e11 8.14 11.0 0.983 0.991 0.789print(accuracy_AAN_damped)## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(GDP ~ err… Trai… 2.95e10 1.90e11 9.49e10 1.62 7.62 0.438 0.454 -0.00187print(accuracy_boxcox)## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(GDP ~ error(\"A\")… Trai… NaN NaN NaN NaN NaN NaN NaN NAThis analysis compared two Exponential Smoothing (ETS) models for GDP forecasting: Model 1 (ETS(A,N,N)), with an additive error and no trend or seasonality, and Model 2 (ETS(A,Ad,N)), incorporating a damped additive trend. Model 2 demonstrably outperformed Model 1, as evidenced by significantly lower error metrics. Specifically, Model 2 had a Mean Error (ME) of approximately 29.5 billion compared to Model 1’s 209.8 billion. The Root Mean Square Error (RMSE) was reduced from 415.7 billion in Model 1 to 190.2 billion in Model 2, and the Mean Absolute Error (MAE) dropped from 213.2 billion to 94.9 billion. These substantial reductions indicate that the inclusion of a damped trend in Model 2 significantly improved the model’s accuracy in capturing the underlying patterns in the GDP data.
Further analysis revealed that Model 1 had a high positive autocorrelation of errors (ACF1) of 0.789, suggesting that its forecast errors were not independent and that the model could be further refined. Model 1 also exhibited a Mean Percentage Error (MPE) of 8.14 and a Mean Absolute Percentage Error (MAPE) of 10.99, indicating relatively high error percentages. The Mean Absolute Scaled Error (MASE) and Root Mean Square Scaled Error (RMSSE) for Model 1 were both close to 1, specifically 0.983 and 0.991 respectively, indicating that the model’s forecasts were roughly as accurate as a naive model. The absence of MASE and RMSSE values for Model 2 prevents a full comparative assessment of scaled errors. However, the substantial reduction in ME, RMSE, and MAE strongly supports the conclusion that Model 2, with its damped trend, provided a more accurate and reliable forecast of GDP compared to the simpler Model 1.
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?
library(fpp3)
# Load the dataset
data("aus_production")
gas_data <- aus_production %>%
select(Quarter, Gas)
# Plot the Gas series
autoplot(gas_data, Gas) +
ggtitle("Gas Production in Australia") +
xlab("Year") +
ylab("Gas Production")
# Fit an ETS model with multiplicative seasonality
fit_multiplicative <- gas_data %>%
model(ETS(Gas ~ error("A") + trend("A") + season("M")))
# Generate the forecast for the next few years
fc_multiplicative <- fit_multiplicative %>%
forecast(h = "5 years")
# Plot the forecasts from the ETS model with multiplicative seasonality
fc_multiplicative %>%
autoplot(gas_data) +
ggtitle("Gas Production Forecast using ETS Model with Multiplicative Seasonality") +
xlab("Year") +
ylab("Gas Production")
# Fit an ETS model with multiplicative seasonality and damped trend
fit_multiplicative_damped <- gas_data %>%
model(ETS(Gas ~ error("A") + trend("Ad") + season("M")))
# Generate the forecast for the next few years with damped trend
fc_multiplicative_damped <- fit_multiplicative_damped %>%
forecast(h = "5 years")
# Plot the forecasts from the ETS model with multiplicative seasonality and damped trend
fc_multiplicative_damped %>%
autoplot(gas_data) +
ggtitle("Gas Production Forecast using ETS Model with Multiplicative Seasonality and Damped Trend") +
xlab("Year") +
ylab("Gas Production")
# Compute accuracy metrics for each model
accuracy_multiplicative <- fit_multiplicative %>%
accuracy()
accuracy_multiplicative_damped <- fit_multiplicative_damped %>%
accuracy()
print(accuracy_multiplicative)## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(Gas ~ error(\"A\… Trai… 0.218 4.19 2.84 -0.920 5.03 0.510 0.553 0.0405print(accuracy_multiplicative_damped)## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(Gas ~ error(\"A\"… Trai… 0.548 4.22 2.81 1.32 4.11 0.505 0.556 0.0265fit_multiplicative %>%
accuracy()## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(Gas ~ error(\"A\… Trai… 0.218 4.19 2.84 -0.920 5.03 0.510 0.553 0.0405fit_multiplicative_damped %>%
accuracy()## # A tibble: 1 × 10
## .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 "ETS(Gas ~ error(\"A\"… Trai… 0.548 4.22 2.81 1.32 4.11 0.505 0.556 0.0265This analysis compares two ETS models for gas consumption forecasting: Model 1 (ETS(A,A,M)) with an additive error, additive trend, and multiplicative seasonality, and Model 2 (ETS(A,Ad,M)) which incorporates a damped additive trend. Model 1 demonstrates a lower Mean Error (ME) of 0.218 compared to Model 2’s 0.548, suggesting that Model 1’s forecasts are, on average, closer to the actual values. Furthermore, Model 1 exhibits a slightly lower Root Mean Square Error (RMSE) of 4.19 compared to Model 2’s 4.22, indicating a marginally better overall accuracy. However, Model 2 showcases a slightly lower Mean Absolute Error (MAE) of 2.81 compared to Model 1’s 2.84, suggesting a better fit in terms of absolute error magnitudes.
Examining the Mean Percentage Error (MPE), Model 1’s negative value of -0.92 indicates a tendency to underestimate gas consumption, while Model 2’s positive 1.32 suggests overestimation. The analysis highlights that trend damping in Model 2 slightly increased RMSE but decreased MAE. Therefore, the choice between models depends on the analysis’s priorities: Model 1 is preferable for minimizing RMSE, while Model 2 is better for minimizing absolute errors. Both models effectively capture the data’s multiplicative seasonality, which is crucial for representing the increasing seasonal variations. If the trend is expected to slow over time, Model 2 with its damped trend might offer better long-term forecasting.
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. 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. 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?
# Load necessary libraries
library(fpp3)
library(tsibble)
library(dplyr)
# Set seed for reproducibility
set.seed(1234567)
# Sample a series from aus_retail data
myseries <- aus_retail |>
filter(`Series ID` == sample(aus_retail$`Series ID`, 1))
# Convert the dataset to a tsibble and fill gaps
myseries <- myseries %>%
as_tsibble(index = Month) %>%
fill_gaps()
# Create a training dataset consisting of observations before 2011
myseries_train <- myseries |>
filter(year(Month) < 2011)
# Plot the training data
autoplot(myseries_train, Turnover) +
ggtitle("Retail Turnover") +
xlab("Month") +
ylab("Turnover")
# Fit Holt-Winters' multiplicative method and damped trend method
fit <- myseries_train |>
model(
MAM = ETS(Turnover ~ error("M") + trend("A") + season("M")),
MAdM = ETS(Turnover ~ error("M") + trend("Ad") + season("M"))
)
# Produce forecasts for the test data
fc <- fit |>
forecast(h = 24)
# Plot the forecasts
autoplot(fc, myseries_train, level = NULL) +
ggtitle("Forecasts using Holt-Winters' Multiplicative Method") +
xlab("Month") +
ylab("Turnover")
# Check the accuracy of the fitted models
fit |>
accuracy()## # A tibble: 2 × 12
## State Industry .model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Victo… Cafes, … MAM Trai… 1.34 12.7 8.84 0.0736 3.08 0.320 0.339 0.222
## 2 Victo… Cafes, … MAdM Trai… 1.31 12.4 8.57 0.384 3.01 0.310 0.329 0.0640# Check that the residuals from the best method look like white noise
fit |>
components() |>
filter(.model == "MAdM") |>
select(remainder) |>
autoplot(.vars = remainder, na.rm = TRUE) +
ggtitle("Residuals of Holt-Winters' Method with Damped Trend")
# Produce forecasts for the full dataset
fc <- fit |>
forecast(new_data = anti_join(myseries, myseries_train))## Joining with `by = join_by(State, Industry, `Series ID`, Month, Turnover)`# Plot the forecasts for the full dataset
fc |>
autoplot(myseries, level = NULL) +
ggtitle("Forecasts using Holt-Winters' Method") +
xlab("Month") +
ylab("Turnover")
# Check the accuracy of the forecasts
fc |>
accuracy(myseries)## # A tibble: 2 × 12
## .model State Industry .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 MAM Victoria Cafes, … Test 15.4 67.1 55.9 1.03 7.05 2.02 1.78 0.920
## 2 MAdM Victoria Cafes, … Test 10.2 74.8 62.4 0.207 7.92 2.26 1.99 0.931This analysis compared two ETS models, MAM and MAdM, focusing on forecast accuracy. The MAM model demonstrated superior performance with lower Root Mean Squared Error (RMSE) of 67.05 and Mean Absolute Error (MAE) of 55.95, compared to MAdM’s RMSE of 74.81 and MAE of 62.39. While MAdM exhibited a lower Mean Error (ME) of 10.18 compared to MAM’s 15.45, indicating less biased forecasts on average, the higher RMSE and MAE for MAdM suggest that MAM provides more reliable predictions overall.
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?
library(fpp3)
library(tsibble)
library(dplyr)
# Set seed for reproducibility
set.seed(1234567)
# Sample a series from aus_retail data
myseries <- aus_retail |>
filter(`Series ID` == sample(aus_retail$`Series ID`, 1))
# Create a training dataset consisting of observations before 2011
myseries_train <- myseries |>
filter(year(Month) < 2011)
# Apply Guerrero method to find optimal lambda for Box-Cox transformation
lambda <- myseries_train |>
features(Turnover, features = guerrero) |>
pull(lambda_guerrero)
# Apply STL decomposition with Box-Cox transformation to training data
stl <- myseries_train |>
model(
STL(box_cox(Turnover, lambda))
)
# Apply STL decomposition with Box-Cox transformation to the full data
stl2 <- myseries |>
model(
STL(box_cox(Turnover, lambda))
)
# Fit ETS models to the seasonally adjusted component
fit <- stl |>
components() |>
model(
MAM = ETS(season_adjust ~ error("M") + trend("A") + season("M")),
MAdM = ETS(season_adjust ~ error("M") + trend("Ad") + season("M"))
)
# Produce forecasts for the test data
fc <- fit |>
select(-.model) |>
forecast(new_data = stl2 |> components() |> select(-.model) |> filter(year(Month) >= 2011))
# Check the accuracy of the forecasts
accuracy_stl_ets <- fc |>
accuracy(stl2 |> components() |> select(-.model))
print(accuracy_stl_ets)## # A tibble: 2 × 12
## .model State Industry .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 MAM Vict… Cafes, … Test -0.542 0.558 0.542 -3.98 3.98 1.96 1.63 0.811
## 2 MAdM Vict… Cafes, … Test -0.0928 0.342 0.300 -0.756 2.21 1.08 1.00 0.948This analysis compared two Holt-Winters’ Multiplicative models, MAM and MAdM, for retail time series forecasting. The MAdM model, incorporating a damped trend, outperformed the MAM model. MAdM showed a lower Mean Error (ME) of -0.09 compared to MAM’s -0.54, indicating less biased forecasts. Crucially, MAdM also exhibited a lower Root Mean Squared Error (RMSE) of 0.34 compared to MAM’s 0.56, demonstrating superior overall predictive accuracy.
These outputs suggest that the damped trend in MAdM effectively improved forecast accuracy by achieving a better balance between trend and seasonality. Thus, MAdM is the preferred model for this retail dataset due to its reduced bias and enhanced prediction performance.
Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.