18.6 Shipments of Household Appliances.

The time plot in Figure 18.9 shows the series of quarterly shipments (in million dollars) of US household appliances between 1985–1989 (data are available in ApplianceShipments.csv, data courtesy of Ken Black).

a.

Which of the following methods would be suitable for forecasting this series if applied to the raw data? 1. Moving average 2. Simple exponential smoothing 3. Double exponential smoothing 4. Holt–Winter’s exponential smoothing

# Load libraries
library(tidyverse) # for data manipulation
library(forecast) # for time series analysis
library(lubridate) # for date manipulation
library(mlba) # for load data
library(zoo) # for moving average
library(knitr) # for tables
# Load the data
data("ApplianceShipments") # Load the data to ApplianceShipments

# Create a time series object
appliance.ts <- ts(ApplianceShipments$Shipments, 
                               start = c(1985, 1), 
                               # end = c(1989, 4),
                               frequency = 4)

# Decompose the time series
appliance.decomp <- decompose(appliance.ts)

# Plot the decomposed time series
autoplot(appliance.decomp) +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Decomposition of Appliance Shipments") +
  theme_minimal()

b.

Apply a moving average with window span w = 4 to the data. Use all but the last year as the training set. Create a time plot of the moving average series.

  1. What does the MA(4) chart reveal?
  2. Use the MA(4) model to forecast appliance sales in Q1-1990.
  3. Use the MA(4) model to forecast appliance sales in Q1-1991.
  4. Is the forecast for Q1-1990 most likely to under-estimate, over-estimate or accurately estimate the actual sales on Q1-1990? Explain.
  5. Management feels most comfortable with moving averages. The analyst therefore plans to use this method for forecasting future quarters. What else should be considered before using the MA(4) to forecast future quarterly shipments of household appliances?
# Split the data into training and test set
nValid <- 4
nTrain <- length(appliance.ts) - nValid
train.ts <- window(appliance.ts, start = c(1985, 1), end = c(1985, nTrain + 1))
valid.ts <- window(appliance.ts, start = c(1985, nTrain + 1), 
                   end = c(1985, nTrain + nValid))

# Plot train and test set
autoplot(train.ts, series = "Training") +
  autolayer(valid.ts, series = "Validation") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Training and Validation Set of Appliance Shipments") +
  theme_minimal()

# Create a moving average model
appliance.ma <- rollmean(train.ts, k=4, align = "right")

# Create a time plot of the moving average series
autoplot(appliance.ma, series = "Moving Average") +
  autolayer(train.ts, series = "Actual") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Moving Average (w = 4) of Appliance Shipments") +
  theme_minimal()

# Full moving average model
appliance.fullma <- rollmean(appliance.ts, k=4, align = "right")

# Forecast appliance sales in Q1-1990 and Q1-1991
forecast(appliance.ma, h = 9) |> kable()
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
1989 Q2 4429.759 4342.672 4516.845 4296.571 4562.946
1989 Q3 4429.759 4306.606 4552.912 4241.413 4618.105
1989 Q4 4429.759 4278.930 4580.587 4199.086 4660.431
1990 Q1 4429.759 4255.598 4603.919 4163.404 4696.114
1990 Q2 4429.759 4235.043 4624.475 4131.966 4727.551
1990 Q3 4429.759 4216.459 4643.059 4103.544 4755.973
1990 Q4 4429.759 4199.369 4660.149 4077.408 4782.110
1991 Q1 4429.759 4183.462 4676.056 4053.080 4806.437
1991 Q2 4429.759 4168.522 4690.996 4030.232 4829.286 
forecast(appliance.fullma) |> kable()
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
1990 Q1 4565.743 4482.992 4648.494 4439.186 4692.300
1990 Q2 4565.743 4448.721 4682.765 4386.773 4744.713
1990 Q3 4565.743 4422.423 4709.063 4346.554 4784.932
1990 Q4 4565.743 4400.253 4731.233 4312.648 4818.838
1991 Q1 4565.743 4380.720 4750.766 4282.775 4848.711
1991 Q2 4565.743 4363.062 4768.425 4255.768 4875.718
1991 Q3 4565.743 4346.823 4784.664 4230.933 4900.553
1991 Q4 4565.743 4331.708 4799.779 4207.817 4923.669 
# Plot the forecasts
autoplot(forecast(appliance.ma, h = 9)) +
  autolayer(appliance.ts, series = "Actual") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Forecast of Appliance Shipments using MA(4)") +
  theme_minimal()

autoplot(forecast(appliance.fullma)) +
  autolayer(appliance.ts, series = "Actual") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Forecast of Appliance Shipments using MA(4)") +
  theme_minimal()

  1. The last trend is stable -> the forecast for Q1-1990 is most likely to accurately estimate the actual sales on Q1-1990.
  2. The analyst should consider the seasonality of the data before using the MA(4) to forecast future quarterly shipments of household appliances.

c.

We now focus on forecasting beyond 1989. In the following, continue to use all but the last year as the training set, and the last four quarters as the validation set. First, fit a regression model to sales with a linear trend and quarterly seasonality to the training data. Next, apply Holt–Winter’s exponential smoothing (with the default smoothing values) to the training data. Choose an adequate “season length.”

  1. Compute the MAPE for the validation data using the regression model.
# Time series regression model
appliance.lm <- tslm(train.ts ~ trend + season, lambda = 1)

# Forecast appliance sales using the regression model
forecast.lm <- forecast(appliance.lm, h = nValid)

# Compute the MAPE for the validation data using the regression model
accuracy(forecast.lm, valid.ts) |> kable()
ME RMSE MAE MPE MAPE MASE ACF1 Theil’s U
Training set 0.00000 151.56085 128.10941 -0.1191684 2.908784 0.5804888 0.2760747 NA
Test set -18.96667 99.01135 81.73333 -0.4136366 1.782656 0.3703497 -0.6158021 0.5425461
  1. Compute the MAPE for the validation data using Holt–Winter’s exponential smoothing.
# Holt-Winter's exponential smoothing
appliance.ets <- ets(train.ts, model = "MAA")

# Forecast appliance sales using Holt-Winter's exponential smoothing
forecast.ets <- forecast(appliance.ets, h = nValid)

# Compute the MAPE for the validation data using Holt-Winter's exponential smoothing
accuracy(forecast.ets, valid.ts) |> kable()
ME RMSE MAE MPE MAPE MASE ACF1 Theil’s U
Training set -37.94926 145.2715 108.5604 -0.8678832 2.444005 0.4919082 -0.1638919 NA
Test set 473.25984 509.2908 473.2598 10.2195003 10.219500 2.1444329 -0.1940605 2.675792
  1. Which model would you prefer to use for forecasting Q1-1990? Give three reasons.
# Plot the forecasts
autoplot(forecast.lm, series = "Regression") +
  # training data
  autolayer(train.ts, series = "Training") +
  # validation data
  autolayer(valid.ts, series = "Validation") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Forecast of Appliance Shipments using Regression Model") +
  theme_minimal()

autoplot(forecast.ets, series = "Holt-Winter's") +
  # training data
  autolayer(train.ts, series = "Training") +
  # validation data
  autolayer(valid.ts, series = "Validation") +
  xlab("Year") +
  ylab("Shipments (in million dollars)") +
  ggtitle("Forecast of Appliance Shipments using Holt-Winter's Exponential Smoothing") +
  theme_minimal()

  1. If we optimize the smoothing parameters in the Holt–Winter’s method, is it likely to get values that are close to zero? Why or why not?