Multiplicative Models

Chapter 1: Lesson 5

Learning Outcomes

Decompose time series into trends, seasonal variation, and residuals

Explain the differences between additive and multiplicative models
Implement multiplicative decomposition
Compute the estimators of seasonal variation for a multiplicative model
Calculate the random component for a multiplicative model
Compute a seasonally-adjusted time series based on a multiplicative model

Preparation

Review Sections 1.5.1-1.5.3

Learning Journal Exchange (10 min)

Review another student’s journal
What would you add to your learning journal after reading your partner’s?
What would you recommend your partner add to their learning journal?
Sign the Learning Journal review sheet for your peer

Packages

pacman::p_load(
  tidyverse,  # ggplot, mutate(), cleaning...
  tsibble,    # as_tsibble()
  fable,      # model(...), forecast(), tidy(), glance()...
  feasts,     # ACF(), PACF()
  ggtime,     # autoplot() for tsibbles
  rio,        # import()
  pander,     # pander() markdown tables
  patchwork   # + and / for ggplots
)

Class Activity: Comparing Models in the Textbook Versus R (5 min)

Both the textbook and R use the same model in the additive case:

\[ x_t = m_t + s_t + z_t \]

However, there is a discrepancy in the definitions for the mulitplicative models. The textbook defines the multiplicative model as

\[ x_t = m_t \cdot s_t + z_t \] but R defines the multiplicative model as \[ x_t = m_t \cdot s_t \cdot z_t \] You can investigate R’s definition by executing this command in RStudio.

?classical_decomposition

Class Activity: Exploring Simulated Time Series Data (15 min)

So far, you have learned how to estimate a trend using aggregated data (i.e., an annual average) or a moving average. We will compute the seasonal effect and use this to get the random component.

Additive Model

The code hidden below simulates 10 years of a monthly time series with a linear trend and seasonal variation based on an additive model. Because the data are simulated, we know exactly which functions were used to create it, and we can observe what happens when we decompose this function.

Table 1: Simulated Data (Additive Model)

Show the code

# Set random seed for reproducibility
set.seed(20) 

# Set parameters & initialize vectors
num_years <- 10
n <- 12 * num_years
sigma <- .75
a <- 0.05
b <- 1
c <- 0.5
trend <- seasonal <- x_t <- rep(0,n)
time_seq <- seq(1,n)

# Generate correlated error terms
w <- rnorm(n + 4, 0, 1)
z = w + lead(w,1) + lead(w,2) + lead(w,3) + lead(w,4)
z  = head(z, n) / 2

# Get date
year_seq <- lubridate::year(today()) - num_years  + (time_seq - 1) %/% 12
month_seq <- (time_seq - 1) %% 12 + 1
date_seq <- ymd(paste0(year_seq,"-",month_seq,"-01"))

# Get data
for (t in 1:n) {
  trend[t] <- a * t + 10
  seasonal[t] <- b * sin(t / 12 * 2 * pi * 1)  + c * cos(t / 12 * 2 * pi * 3)
  x_t[t] <- trend[t] + seasonal[t] + z[t]
}

x_df <- data.frame(x_t = x_t, trend = trend, seasonal = seasonal)

start_year <- lubridate::year(today()) - num_years
start_date <- lubridate::ymd(paste0(start_year,"-01-01"))

# start_date <- lubridate::ymd("1958-01-01")
date_seq <- seq(start_date,
    start_date + months(nrow(x_df)-1),
    by = "1 months")

x_df_ts <- x_df |>
  mutate(
    date = date_seq,
    month = tsibble::yearmonth(date)
  ) |>
  select(date, month, trend, seasonal, x_t) |> 
  as_tsibble(index = month)

Date	Month	Trend, $$m_t$$	Seasonal, $$s_t$$	Data, $$x_t$$
2016-01-01	2016 Jan	10.05	0.5	10.842
2016-02-01	2016 Feb	10.1	0.366	10.461
2016-03-01	2016 Mar	10.15	1	9.993
2016-04-01	2016 Apr	10.2	1.366	9.082
2016-05-01	2016 May	10.25	0.5	8.701
2016-06-01	2016 Jun	10.3	-0.5	7.697
⋮	⋮	⋮	⋮	⋮
2025-11-01	2025 Nov	15.95	-0.5	16.264
2025-12-01	2025 Dec	16	0.5	18.165

The code above has generated simulated data, where the trend is linear with equation

\[ m_t = \frac{t}{20} \]

and the seasonal effect follows the function

\[ s_t = \sin \left( \frac{t\pi}{6} \right) + \frac{1}{2}\cos\left(\frac{t \pi}{18} \right) \]

Letting $t$ represent the month number across 10 years, we simulate a time series. Click on the tabs below to compare the actual construction of the time series (using the components generated in the code above) to the decomposition in R.

Here is a plot of the components of the simulated data.

Show the code

trend_plot <- ggplot(x_df_ts, aes(x=month, y=trend)) + 
  geom_line() +
  labs(
    title="Plot of Trend", 
    x="Month", 
    y="Trend"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

seasonal_plot <- ggplot(x_df_ts, aes(x=month, y=seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Seasonal Effect", 
    x="Month", 
    y="Seasonal"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

error_plot <- ggplot(x_df_ts, aes(x = month, y = x_t - trend - seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Random Error Term", 
    x="Month", 
    y="Random"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- ggplot(x_df_ts, aes(x=month, y=x_t)) + 
  geom_line() +
  labs(
    title="Plot of Simulated Time Series", 
    x="Month", 
    y="x_t"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- x_plot  + labs(title = "True (Simulated) Values", x = NULL)
trend_plot <- trend_plot + labs(title = NULL, x = NULL)
seasonal_plot <- seasonal_plot + labs(title = NULL, x = NULL)
error_plot <- error_plot + labs(title = NULL)

x_plot / trend_plot / seasonal_plot / error_plot # uses patchwork

Now, we use R to decompose the time series $\{x_t\}$.

Show the code

x_decompose <- x_df_ts |>
    model(feasts::classical_decomposition(x_t,
        type = "add"))  |>
    components()

autoplot(x_decompose)

Check Your Understanding

How does the (estimated) decomposition compare to the theoretical values above?
- How well is the trend estimated?
- How well is the seasonal effect estimated?
- How well is the random effect estimated?
Make changes to the simulated data and observe the effect on the plots

Multiplicative Model

We now simulate data and apply R’s multiplicative model. This implies that the error term, $z_t$, has a mean of 1, rather than 0.

Table 2: Simulated Data (Multiplicative Model)

Show the code

# Set random seed for reproducibility
set.seed(123) 

# Set parameters & initialize vectors
num_years <- 10
n <- 12 * num_years
sigma <- .75
a <- 0.03
b <- 1
c <- 0.5 
trend <- seasonal <- x_t <- rep(0,n)
time_seq <- seq(1,n)

# Generate correlated error terms
w <- rnorm(n + 4, 0.2, 0.1) # Changed to a mean of 1 and sd of 0.03
z = w + lead(w,1) + lead(w,2) + lead(w,3) + lead(w,4)
z  = head(z, n)

# Get date
year_seq <- lubridate::year(today()) - num_years  + (time_seq - 1) %/% 12
month_seq <- (time_seq - 1) %% 12 + 1
date_seq <- ymd(paste0(year_seq,"-",month_seq,"-01"))

# Get data
for (t in 1:n) {
  trend[t] <- exp(a * t)
  seasonal[t] <- exp( b * sin(t / 12 * 2 * pi * 1)  + c * cos(t / 12 * 2 * pi * 3) + 1 )
  x_t[t] <- trend[t] * seasonal[t] * z[t] # Note R's definition of the mult. model
}

x_df <- data.frame(x_t = x_t, trend = trend, seasonal = seasonal)

start_year <- lubridate::year(today()) - num_years
start_date <- lubridate::ymd(paste0(start_year,"-01-01"))

# start_date <- lubridate::ymd("1958-01-01")
date_seq <- seq(start_date,
    start_date + months(nrow(x_df)-1),
    by = "1 months")

x_df_ts <- x_df |>
  mutate(
    date = date_seq,
    month = tsibble::yearmonth(date)
  ) |>
  select(date, month, trend, seasonal, x_t) |>
  as_tsibble(index = month)

Date	Month	Trend, $$m_t$$	Seasonal, $$s_t$$	Data, $$x_t$$
2016-01-01	2016 Jan	1.03	4.482	5.065
2016-02-01	2016 Feb	1.062	3.92	5.512
2016-03-01	2016 Mar	1.094	7.389	11.266
2016-04-01	2016 Apr	1.127	10.655	13.348
2016-05-01	2016 May	1.162	4.482	5.391
2016-06-01	2016 Jun	1.197	1.649	1.93
⋮	⋮	⋮	⋮	⋮
2025-11-01	2025 Nov	35.517	1.649	39.853
2025-12-01	2025 Dec	36.598	4.482	121.366

The code above simulated data, where the trend is exponential with equation

\[ m_t = e^{0.03 t} \]

and the seasonal effect follows the function

\[ s_t = \sin \left( \frac{t\pi}{6} \right) + \frac{1}{2}\cos\left(\frac{t \pi}{18} \right) + 1 \]

Letting $t$ represent the month number across 10 years, we simulate a time series with multiplicative effects. Click on the tabs below to compare the actual construction of the time series (using the components generated in the code above) to the decomposition in R.

Here is a plot of the components of the simulated data.

Show the code

trend_plot <- ggplot(x_df_ts, aes(x=month, y=trend)) + 
  geom_line() +
  labs(
    title="Plot of Trend", 
    x="Month", 
    y="Trend"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

seasonal_plot <- ggplot(x_df_ts, aes(x=month, y=seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Seasonal Effect", 
    x="Month", 
    y="Seasonal"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

error_plot <- ggplot(x_df_ts, aes(x = month, y = x_t / trend / seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Random Error Term", 
    x="Month", 
    y="Random"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- ggplot(x_df_ts, aes(x=month, y=x_t)) + 
  geom_line() +
  labs(
    title="Plot of Simulated Time Series", 
    x="Month", 
    y="x_t"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- x_plot  + labs(title = "True (Simulated) Values", x = NULL)
trend_plot <- trend_plot + labs(title = NULL, x = NULL)
seasonal_plot <- seasonal_plot + labs(title = NULL, x = NULL)
error_plot <- error_plot + labs(title = NULL)

x_plot / trend_plot / seasonal_plot / error_plot

Now, we use R to decompose the time series $\{x_t\}$.

Show the code

x_decompose <- x_df_ts |>
    model(feasts::classical_decomposition(x_t,
        type = "mult"))  |>
    components()

autoplot(x_decompose)

Check Your Understanding

How does the (estimated) decomposition compare to the theoretical values above?
- How well is the trend estimated?
- How well is the seasonal effect estimated?
- How well is the random effect estimated?
Make changes to the simulated data and observe the effect on the plots

Which Model Should I Use: Additive or Multiplicative?

Compare the following two time series.

Check Your Understanding

Complete a table like the following in your Learning Journal to compare characteristics of these two time series. Be sure to include a sketch of the respective time plots.

	Rexburg Temperature	S&P 500 Closing Price
Deterministic or stochastic trend?
Is there a seasonal effect?
Is there evidence of cycles?
Does the variation get bigger over time?
Additive or multiplicative model?

Is an additive or multiplicative model more appropriate? Why?

Small Group Activity: Apple’s Quarterly Revenue (45 min)

Please download this file chapter_1_5_handout.xlsx for this activity.

The code below imports and shows a few rows of the quarterly revenues for Apple Inc. (in billions of U.S. dollars).

Show the code

apple_ts <- rio::import("https://byuistats.github.io/timeseries/data/apple_revenue_2026.xlsx") |>
  mutate(
    dates = as.Date(date),      # <-- just coerce/assign directly, no mdy()
    year = lubridate::year(dates),
    quarter = lubridate::quarter(dates),
    value = revenue_billions
  ) |>
  dplyr::select(dates, year, quarter, period_covered, value)  |> 
  arrange(dates) |>
  mutate(index = tsibble::yearquarter(dates)) |>
  as_tsibble(index = index) |>
  rename(revenue = value)

apple_ts |>
  select(index, period_covered, revenue) |>
  display_partial_table(6,4)

index	period_covered	revenue
2005 Q1	Oct-Dec 2004	3.49
2005 Q2	Jan-Mar 2005	3.243
2005 Q3	Apr-Jun 2005	3.52
2005 Q4	Jul-Sep 2005	3.678
2006 Q1	Oct-Dec 2005	5.749
2006 Q2	Jan-Mar 2006	4.359
⋮	⋮	⋮
2025 Q4	Jul-Sep 2025	102.466
2026 Q1	Oct-Dec 2025	143.756
2026 Q2	Jan-Mar 2026	111.184
2026 Q3	Apr-Jun 2026	109.417

Here are some time plots of the Apple revenue data.

This time plot displays Apple’s quarterly revenue data from 2005 to 2026.

Show the code

# apple_ts |>
#   autoplot(.vars = revenue) +
#   expand_limits(y = 0) +
#   labs(
#     x = "Quarter",
#     y = "Apple Revenue, Billions $US",
#     title = "Apple's Quarterly Revenue, Billions of U.S. Dollars"
#   ) +
#   theme(plot.title = element_text(hjust = 0.5))

apple_ts |>
  autoplot(.vars = revenue) +
  aes(color = "Revenue (Billions $US)") +
  scale_color_manual(
    name = NULL,
    values = c("Revenue (Billions $US)" = "black")
  ) +
  expand_limits(y = 0) +
  labs(
    x = "Quarter",
    y = "Apple Revenue, Billions $US",
    title = "Apple's Quarterly Revenue, Billions of U.S. Dollars"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    legend.position = c(0.02, 0.98),
    legend.justification = c(0, 1),
    legend.background = element_rect(fill = "white", color = "grey70")
  )

Here is a plot illustrating the last few years’ of the Apple revenue data:

Show the code

# apple_ts |>
#   filter(year >= max(year) - 4) |>
#   autoplot(.vars = revenue) +
#   expand_limits(y = 0) +
#   labs(
#     x = "Quarter",
#     y = "Apple Revenue, Billions $US",
#     title = "Apple's Quarterly Revenue, Billions of U.S. Dollars"
#   ) +
#   theme(plot.title = element_text(hjust = 0.5))

apple_ts |>
  filter(year >= max(year) - 4) |>
  autoplot(.vars = revenue) +
  aes(color = "Revenue (Billions $US)") +
  scale_color_manual(
    name = NULL,
    values = c("Revenue (Billions $US)" = "black")
  ) +
  expand_limits(y = 0) +
  labs(
    x = "Quarter",
    y = "Apple Revenue, Billions $US",
    title = "Apple's Quarterly Revenue, Billions of U.S. Dollars"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    legend.position = c(0.02, 0.98),
    legend.justification = c(0, 1),
    legend.background = element_rect(fill = "white", color = "grey70")
  )

Check Your Understanding

What do you notice?
- Does it seem like there is a trend in the time series?
- Is there evidence of a seasonal effect? If so, during which quarter(s) are the revenues particularly high? particularly low?
- Can you attribute a reason for this behavior?
Is an additive or multiplicative model more appropriate? Why?

We want to find the seasonally adjusted series for a multiplicative model. This is a multi-step process.

Centered Moving Average

First, we compute the centered moving average, $\hat m_t$.

Check Your Understanding

Write a mutate statement that will compute the 4-quarter moving average for the variable “revenue” in the tsibble apple_ts. You can use the mutate statement from Chapter 1 Lesson 4 as a starting point.

# computes the 4-quarter centered moving average (m_hat)
apple_ts <- apple_ts |> 
  mutate(
    m_hat = 
      # Your code goes here
  )

To emphasize the computation of the centered moving average, the observed data values that were used to find $\hat m_t$ for the fourth quarter of 2005 are shown in green in the table below.

Estimated Quarterly Multiplicative Effect

Next, the centered moving average, $\hat m_t$, is used to compute the quarterly multiplictive effect, $\hat s_t$:

\[ \hat s_t = \dfrac{ x_t }{ \hat m_t } \]

Table 3: Computation of the Centered Moving Average, $\hat m_t$, and the Estimated Quarterly Multiplicative Effect, $\hat s_t$

quarter	Revenue $$x_t$$	$$ \hat m_t $$	$$ \hat s_t $$
2005 Q1	3.49	NA	NA
2005 Q2	3.243	NA	NA
2005 Q3	3.52	3.765	0.935
2005 Q4	3.678	4.187	0.878
2006 Q1	5.749	______	______
2006 Q2	4.359	______	______
2006 Q3	4.37	______	______
2006 Q4	4.837	______	______
2007 Q1	7.123	______	______
2007 Q2	5.286	______	______
2007 Q3	5.563	______	______
2007 Q4	6.606	______	______
2008 Q1	10.43	______	______
2008 Q2	7.98	______	______
⋮	⋮	⋮	⋮
2025 Q3	94.036	______	______
2025 Q4	102.466	______	______
2026 Q1	143.756	______	______
2026 Q2	111.184	______	______
2026 Q3	109.417	______	______

Check Your Understanding

Working with your assigned partner, fill in the missing values of $\hat m_t$ in the table above.
Then, find the missing values of $\hat s_t$.

Seasonally Adjusted Factors

Next, we need to compute the mean (across years) of $\hat s_t$ by quarter. To help us calculate this, it can be convenient to organize the values of $\hat s_t$ in a table, where the rows give the year and the columns give the quarter.

The overall mean of these means will be reasonably close to, but not exactly one. We adjust these values by dividing the quarterly means by the overall mean.

Table 4: Computation of the Seasonally Adjusted Factors, $\bar s_t$

Year	Q1	Q2	Q3	Q4
2005	NA	NA	0.935	0.878
2006	______	______	______	______
2007	______	______	______	______
2008	______	______	______	______
2009	______	______	______	______
2010	1.159	0.883	0.888	0.994
2011	1.139	0.946	0.968	0.837
2012	1.273	1.027	0.872	0.862
2013	1.288	1.025	0.819	0.856
2014	1.301	1.012	0.783	0.818
2015	1.367	1.013	0.847	0.891
2016	1.355	0.928	0.781	0.855
2017	1.412	0.935	0.776	0.864
2018	1.405	0.939	0.808	0.968
2019	1.303	0.894	0.816	0.956
2020	1.356	0.851	0.84	0.835
2021	1.326	1.005	0.875	0.872
2022	1.282	0.995	0.849	0.933
2023	1.219	0.989	0.851	0.933
2024	1.247	0.935	0.872	0.954
2025	1.229	0.925	0.883	0.924
2026	1.252	NA	NA	NA
Mean	1.284	______	______	______
$$ \bar s_t $$	1.286	$$~$$ ______	$$~$$ ______	$$~$$ ______

Check Your Understanding

The table above gives the values of $\hat s_t$. Fill in the missing values. (Note you already computed these.)
The second-to-last row (labeled “Mean”) gives the values of $\bar s_t$. Find the mean of the $\hat s_t$ values for each quarter. Call these means $\bar {\hat s_t}$.
Compute the mean of the $\bar {\hat s_t}$ values. Call this number $\bar {\bar {\hat s_t}}$ (This number should be somewhat close to 1.)
Divide each of the $\bar {\hat s_t}$ values by $\bar {\bar {\hat s_t}}$ to get $\bar s_t$, the seasonally adjusted factor for quarter $t$. (Note that the mean of the $\bar s_t$ values will be 1.)

\[ \bar s_t = \frac{ \left( \bar {\hat s_t} \right) }{ \left( \bar {\bar {\hat s_t}} \right) } \]

Random Component and the Seasonally Adjusted Time Series

Using R’s definition of the multiplicative model, \[ x_t = m_t \cdot s_t \cdot z_t \] we calculate the random component by dividing the values in the time series by the product of the trend and the seasonally adjusted factor:

\[ \text{random component} = \dfrac{ x_t }{ \hat m_t \cdot \bar s_t } \]

The seasonally adjusted series is computed by dividing the respective observed values by $\bar s_t$:

\[ \text{seasonally adjusted series} = \dfrac{ x_t }{ \bar s_t } \]

Use these equations to calculate the values missing from the table below. The adjusted seasonal effect $\bar s_t$ (s_bar) was computed in the last row of the previous table.

Table 5: Computation of the Random Component and the Seasonally Adjusted Time Series

Quarter	Revenue $$x_t$$	$$ \hat m_t $$	$$ \hat s_t $$	$$ \bar s_t $$	Random	Seasonally Adjusted $$x_t$$
2005 Q1	3.49	NA	NA	1.286	NA	2.713
2005 Q2	3.243	NA	NA	______	NA	3.428
2005 Q3	3.52	3.765	0.935	______	1.098	4.136
2005 Q4	3.678	4.187	0.878	______	0.958	4.013
2006 Q1	5.749	______	______	______	______	______
2006 Q2	4.359	______	______	______	______	______
2006 Q3	4.37	______	______	______	______	______
2006 Q4	4.837	______	______	______	______	______
2007 Q1	7.123	______	______	______	______	______
2007 Q2	5.286	______	______	______	______	______
⋮	⋮	⋮	⋮	⋮	⋮	⋮
2025 Q4	102.466	______	______	______	______	______
2026 Q1	143.756	______	______	______	______	______
2026 Q2	111.184	______	______	______	______	______
2026 Q3	109.417	______	______	______	______	______

Class Activity: Computing the Multiplicative Decomposition in R (5 min)

The R code below calculates the decomposition, including the seasonally adjusted time series, beginning with the tsibble apple_ts.

Table 6: First Few Rows of the Decomposition of the Apple Revenue Time Series

Show the code

apple_decompose <- apple_ts |>
  model(feasts::classical_decomposition(revenue,
          type = "mult"))  |>
  components()

apple_decompose |>
  head(8) |>
  display_table()

.model	index	revenue	trend	seasonal	random	season_adjust
feasts::classical_decomposition(revenue, type = "mult")	2005 Q1	3.490	NA	1.2863724	NA	2.713056
feasts::classical_decomposition(revenue, type = "mult")	2005 Q2	3.243	NA	0.9459724	NA	3.428218
feasts::classical_decomposition(revenue, type = "mult")	2005 Q3	3.520	3.765125	0.8511409	1.0984033	4.135626
feasts::classical_decomposition(revenue, type = "mult")	2005 Q4	3.678	4.187000	0.9165143	0.9584501	4.013031
feasts::classical_decomposition(revenue, type = "mult")	2006 Q1	5.749	4.432750	1.2863724	1.0082131	4.469157
feasts::classical_decomposition(revenue, type = "mult")	2006 Q2	4.359	4.683875	0.9459724	0.9837916	4.607957
feasts::classical_decomposition(revenue, type = "mult")	2006 Q3	4.370	5.000500	0.8511409	1.0267544	5.134285
feasts::classical_decomposition(revenue, type = "mult")	2006 Q4	4.837	5.288125	0.9165143	0.9980105	5.277604

Check Your Understanding

How do our results compare with the R output?

The figure below illustrates the original time series (in black), the centered moving average $\hat m_t$ (in blue), and the seasonally adjusted series (in red).

From 2005 to 2026:

Show the code

apple_decompose |>
  ggplot(aes(x = index)) +
  geom_line(aes(y = revenue, color = "Revenue")) +
  geom_line(aes(y = season_adjust, color = "Seasonally Adjusted")) +
  geom_line(aes(y = trend, color = "Trend")) +
  expand_limits(y = 0) +
  scale_color_manual(
    name = NULL,
    values = c(
      "Revenue" = "black",
      "Seasonally Adjusted" = "#D55E00",
      "Trend" = "#0072B2"
    )
  ) +
  labs(
    x = "Quarter",
    y = "Quarterly Revenue, Billions",
    title = "Apple Inc. Quarterly Revenue (in Billions of U.S. Dollars)"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    legend.position = c(0.02, 0.98),
    legend.justification = c(0, 1),
    legend.background = element_rect(fill = "white", color = "grey70")
  )

From 2022 to 2026:

Show the code

apple_decompose |>
  filter(year(index) >= max(year(index)) - 4) |>
  ggplot(aes(x = index)) +
  geom_line(aes(y = revenue, color = "Revenue")) +
  geom_line(aes(y = season_adjust, color = "Seasonally Adjusted")) +
  geom_line(aes(y = trend, color = "Trend")) +
  expand_limits(y = 0) +
  scale_color_manual(
    name = NULL,
    values = c(
      "Revenue" = "black",
      "Seasonally Adjusted" = "#D55E00",
      "Trend" = "#0072B2"
    )
  ) +
  labs(
    x = "Quarter",
    y = "Quarterly Revenue, Billions",
    title = "Apple Inc. Quarterly Revenue (in Billions of U.S. Dollars)"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    legend.position = c(0.02, 0.98),
    legend.justification = c(0, 1),
    legend.background = element_rect(fill = "white", color = "grey70")
  )

Check Your Understanding

How would you interpet the seasonally-adjusted series?

The following figures compare the decomposition obtained for the Apple revenue data based on the multiplicative and additive models.

Decomposition based on a multiplicative model:

Show the code

autoplot(apple_decompose)

Decomposition based on an additive model:

Show the code

apple_ts |>
  model(feasts::classical_decomposition(revenue,
          type = "add"))  |>
  components() |>
  autoplot()

Check Your Understanding

Is the multiplicative model appropriate for these data?

Homework Preview (5 min)

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Homework

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homework_1_5.qmd

Solutions

Moving Average

Handout

Multiplicative Models

Learning Outcomes

Preparation

Learning Journal Exchange (10 min)

Packages

Class Activity: Comparing Models in the Textbook Versus R (5 min)

Class Activity: Exploring Simulated Time Series Data (15 min)

Additive Model

Table 1: Simulated Data (Additive Model)

Multiplicative Model

Table 2: Simulated Data (Multiplicative Model)

Which Model Should I Use: Additive or Multiplicative?

Small Group Activity: Apple’s Quarterly Revenue (45 min)

Centered Moving Average

Estimated Quarterly Multiplicative Effect

Table 3: Computation of the Centered Moving Average, \(\hat m_t\), and the Estimated Quarterly Multiplicative Effect, \(\hat s_t\)

Seasonally Adjusted Factors

Table 4: Computation of the Seasonally Adjusted Factors, \(\bar s_t\)

Random Component and the Seasonally Adjusted Time Series

Table 5: Computation of the Random Component and the Seasonally Adjusted Time Series

Class Activity: Computing the Multiplicative Decomposition in R (5 min)

Table 6: First Few Rows of the Decomposition of the Apple Revenue Time Series

Homework Preview (5 min)

Homework

Solutions