Holt-Winters Method (Multiplicative Models)

Chapter 3: Lesson 5

Learning Outcomes

Implement the Holt-Winter method to forecast time series

Explain the Holt-Winters method equations for multiplicative decomposition models
Explain the purpose of the paramters $\alpha$, $\beta$, and $\gamma$
Interpret the coefficient estimates $a_t$, $b_t$, and $s_t$ of the Holt-Winters smoothing algorithm
Explain the Holt-Winters forecasting equation for multiplicative decomposition models, Equation (3.23)
Use HoltWinters() to forecast multiplicative model time series
Plot the Holt-Winters decomposition of a TS (see Fig 3.10)
Plot the Holt-Winters fitted values versus the original time series (see Fig 3.11)
Superimpose plots of the Holt-Winters predictions with the time series realizations (see Fig 3.13)

Preparation

Read Sections 3.4.2-3.4.3, 3.5

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Review another student’s journal
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Packages

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

Class Discussion: Multiplicative Seasonality (10 min)

We can assume either additive or multiplicative seasonality. In the previous two lessons, we explored additive seasonality. In this lesson, we consider the case where the seasonality is multiplicative.

Additive seasonality is appropriate when the variation in the time series is roughly constant for any level. We assume multiplicative seasonality when the variation gets larger as the level increases.

Forecasting

Here are the forecasting equations we use, based on the model that is appropriate for the time series.

	Additive Seasonality	Multiplicative Seasonality
Additive Slope	\[ \hat x_{n+k \mid n} = \left( a_n + k \cdot b_n \right) + s_{n+k-p} \]	\[ \hat x_{n+k \mid n} = \left( a_n + k \cdot b_n \right) \cdot s_{n+k-p} \]

Check Your Understanding

With your partner, for the forecasting equations above, identify where the additive or multiplicative terms are represented for both the slope and the seasonality.
- How is an additive slope represented in the forecasting equation?
- How is additive seasonality represented in the forecasting equation?
- How is multiplicative seasonality represented in the forecasting equation?

Update Equations (Multiplicative Seasonals)

The update equations for the seasonals are:

\[\begin{align*} a_t &= \alpha \left( \frac{x_t}{s_{t-p}} \right) + (1-\alpha) \left( a_{t-1} + b_{t-1} \right) && \text{Level} \\ b_t &= \beta \left( a_t - a_{t-1} \right) + (1-\beta) b_{t-1} && \text{Slope} \\ s_t &= \gamma \left( \frac{x_t}{a_{t}} \right) + (1-\gamma) s_{t-p} && \text{Seasonal} \end{align*}\]

Note that when the seasonal effect is additive, we subtract it from the time series to remove its effect. If the seasonal effect is multiplicative, we divide.

Check Your Understanding

Work with your partner to answer the following questions about the update equations.

\[ a_t = \alpha \cdot \underbrace{ \left( \frac{x_t}{s_{t-p}} \right) }_{A} + (1-\alpha) \cdot \underbrace{ \left( a_{t-1} + b_{t-1} \right) }_{B} ~~~~~~~~~~~~~~~~~~~~ \text{Level} \]

Interpret the term $A = \dfrac{x_t}{s_{t-p}}$.
Interpret the term $B = a_{t-1} - b_{t-1}$.
Explain why this expression for $a_t$ estimates the level of the time series at time $t$.

\[ b_t = \beta \cdot \underbrace{ \left( a_t - a_{t-1} \right) }_{C} + (1-\beta) \cdot \underbrace{ b_{t-1} }_{D} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{Slope} \]

Interpret the term $C = a_t - a_{t-1}$.
Interpret the term $D = b_{t-1}$.
Explain why this expression for $b_t$ estimates the slope of the time series at time $t$.

\[ s_t = \gamma \cdot \underbrace{ \left( \frac{x_t}{a_{t}} \right) }_{E} + (1-\gamma) \cdot \underbrace{ s_{t-p} }_{F} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{Seasonal} \]

Interpret the term $E = \dfrac{x_t}{a_{t}}$.
Interpret the term $F = s_{t-p}$.
Explain why this expression for $s_t$ estimates the seasonal component of the time series at time $t$.
When the seasonal component appears on the right-hand side of the update equations, it always given as $s_{t-p}$. Why do we use the estimate of the seasonal effect $p$ periods ago? Why not apply a more recent value?
What do the following sets of terms have in common?
- $\{A, C, E \}$
- $\{B, D, F \}$
Explain why the Holt-Winters method for multiplicative seasonals works.

Small Group Activity: Holt-Winters Model for BYU-Idaho Enrollment Data (25 min)

We will now apply Holt-Winters filtering to the BYU-Idaho Enrollment data. First, we examine the time plot in Figure 1.

Tables-Handout-Excel

Show the code

# read in the data from a csv and make the tsibble
byui_enrollment_ts <- rio::import("https://byuistats.github.io/timeseries/data/byui_enrollment_2012.csv") |>
  rename(
    semester = "TermCode",
    year = "Year",
    enrollment = "On Campus Enrollment (Campus HC)"
  ) |>
  mutate(
    term = 
      case_when(
        left(semester, 2) == "WI" ~ 1,
        left(semester, 2) == "SP" ~ 2,
        left(semester, 2) == "FA" ~ 3,
        TRUE ~ NA
      )
  ) |>
  filter(!is.na(term)) |>
  mutate(dates = yearmonth( ym( paste(year, term * 4 - 3) ) ) ) |>
  mutate(enrollment_1000 = enrollment / 1000) |>
  dplyr::select(semester, dates, enrollment_1000) |>
  as_tsibble(index = dates) 

byui_enrollment_ts_expanded <- byui_enrollment_ts |>
  as_tibble() |>
  hw_additive_slope_multiplicative_seasonal_rounded("dates", "enrollment_1000", p = 3, predict_periods = 9) |>
  mutate(xhat_t = ifelse(t %in% c(1:36), round(a_t * s_t, 1), xhat_t)) |>
  select(-dates, -enrollment_1000)

# This is hard-coded for this data set, because I am in a hurry to get done.
# This revises the semester codes
byui_enrollment_ts_expanded$semester[1:3] <- c("SP11", "FA11", "WI12")
byui_enrollment_ts_expanded$semester[40:48] <- c("SP24", "FA24", "WI25",
                                                 "SP25", "FA25", "WI26",
                                                 "SP26", "FA26", "WI27")

byui_enrollment_ts_expanded |>
  tail(nrow(byui_enrollment_ts_expanded) - 3) |>
  ggplot(aes(x = date, y = x_t)) +
    geom_line(color = "black", linewidth = 1) +
    coord_cartesian(ylim = c(0, 22.5)) +
    labs(
      x = "Date",
      y = "Enrollment (in Thousands)",
      title = "BYU-Idaho On-Campus Enrollments (in Thousands)",
      color = "Components"
    ) +
    theme_minimal() +
    theme(legend.position = "none") +
    theme(plot.title = element_text(hjust = 0.5))

Figure 1: Time plot of BYU-Idaho campus enrollments

Check Your Understanding

Which of the Holt-Winter models described above is appropriate for this situation? Justify your answer.

Table 1 summarizes the intermediate values for Holt-Winters filtering with multiplicative seasonals.

Table 1: Holt-Winters smoothing for BYU-Idaho campus enrollments

$$Semester$$	$$t$$	$$x_t$$	$$a_t$$	$$b_t$$	$$s_t$$	$$\hat x_t$$
SP11	-2	—	—	—		—
FA11	-1	—	—	—		—
WI12	0	—	—	—		—
SP12	1	13.7
FA12	2	16.2	14.2	0.082	1.03	14.6
WI13	3	15.5	14.5	0.126	1.01	14.6
SP13	4	14	14.5	0.101	0.99	14.4
FA13	5	15.6	14.7	0.121	1.04	15.3
WI14	6	15.6
SP14	7	12.9
FA14	8	16.2	14.8	0.08	1.05	15.5
WI15	9	16.7	15.2	0.144	1.04	15.8
SP15	10	13.7	15.1	0.095	0.96	14.5
FA15	11	17.6	15.5	0.156	1.07	16.6
⋮	⋮	⋮	⋮	⋮	⋮	⋮
FA21	29	20.4	18.5	0.086	1.12	20.7
WI22	30	18.4	18.3	0.029	1.05	19.2
SP22	31	14.2	17.9	-0.057	0.87	15.6
FA22	32	19.4	17.7	-0.086	1.12	19.8
WI23	33	17.7	17.5	-0.109	1.04	18.2
SP23	34	12.8	16.9	-0.207	0.85	14.4
FA23	35	18.7	16.7	-0.206	1.12	18.7
WI24	36	17.6	16.6	-0.185	1.04	17.3
SP24	37	—	—	—
FA24	38	—	—	—
WI25	39	—	—	—
SP25	40	—	—	—
FA25	41	—	—	—	1.12	17.6
WI26	42	—	—	—	1.04	16.1

Check Your Understanding

Apply Holt-Winters filtering to these data.

\[ \alpha = 0.2, \beta = 0.2, \gamma = 0.2 \]

Identify the value of $p$
Let $a_1 = x_1$
Compute \[ b_1 = \frac{ \left( \dfrac{x_{p+1} - x_{1}}{p} + \dfrac{x_{p+2} - x_{2}}{p} + \cdots + \dfrac{x_{2p} - x_{p}}{p} \right) }{p} \]
Let $s_{1-p} = s_{2-p} = \cdots = s_{3-p} = 1$, and set $s_1 = s_{1-p}$.
Compute the values of $a_t$, $b_t$, and $s_t$ for all rows with observed time series data.
Find $\hat x_t = a_t \cdot s_t$ for all rows with data.
Compute the prediction $\hat x_{n+k|n} = \left( a_t + k \cdot b_n \right) \cdot s_{n+k-p}$ for the future values.
Superimpose a sketch of your Holt-Winters filter and the associated forecast on Figure 1.

Small Group Activity: Applying Holt-Winters in R to the Apple Quarterly Revenue Data (10 min)

Recall the Apple, Inc., revenue values reported by Bloomberg:

Show the code

apple_ts <- rio::import("https://byuistats.github.io/timeseries/data/apple_revenue.csv") |>
  mutate(
    dates = mdy(date),
    year = lubridate::year(dates),
    quarter = lubridate::quarter(dates),
    value = revenue_billions
  ) |>
  dplyr::select(dates, year, quarter, value)  |> 
  arrange(dates) |>
  mutate(index = tsibble::yearquarter(dates)) |>
  as_tsibble(index = index) |>
  dplyr::select(index, dates, year, quarter, value) |>
  rename(revenue = value) # rename value to emphasize data context

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

Here are a few rows of the summarized data.

index	dates	year	quarter	revenue
2005 Q1	2005-01-25	2005	1	1.24
2005 Q2	2005-04-25	2005	2	0.48
2005 Q3	2005-07-25	2005	3	1.24
2005 Q4	2005-10-30	2005	4	1.71
2006 Q1	2006-01-25	2006	1	2.42
2006 Q2	2006-04-25	2006	2	1.71
⋮	⋮	⋮	⋮	⋮
2023 Q2	2023-05-04	2023	2	94.836
2023 Q3	2023-08-03	2023	3	81.797
2023 Q4	2023-11-02	2023	4	89.498

Check Your Understanding

What do you notice about this time plot?
- Describe the trend
- Is there evidence of seasonality?
- Is the additive or multiplicative model appropriate?

We apply Holt-Winters filtering to the quarterly Apple revenue data with a multiplicative model:

Show the code

apple_hw <- apple_ts |>
  # tsibble::fill_gaps() |>
  model(Additive = ETS(revenue ~
        trend("A") +
        error("A") +
        season("M"),
        opt_crit = "amse", nmse = 1))
report(apple_hw)

Series: revenue 
Model: ETS(A,A,M) 
  Smoothing parameters:
    alpha = 0.8465418 
    beta  = 0.0142208 
    gamma = 0.0001000036 

  Initial states:
       l[0]       b[0]     s[0]     s[-1]    s[-2]     s[-3]
 0.05455891 -0.0543255 1.003384 0.9450199 1.100051 0.9515453

  sigma^2:  32.8701

     AIC     AICc      BIC 
604.1173 606.8446 625.0939

We can compute some values to assess the fit of the model:

Show the code

# SS of random terms
sum(components(apple_hw)$remainder^2, na.rm = T)

# RMSE
forecast::accuracy(apple_hw)$RMSE

# Standard devation of the quarterly revenues
sd(apple_ts$revenue)

The sum of the square of the random terms is: 2235.1651229.
The root mean square error (RMSE) is: 5.423105.
The standard deviation of the number of incidents each month is 33.0205.

Figure 2 illustrates the Holt-Winters decomposition of the Apple revenue data.

Show the code

autoplot(components(apple_hw))

Figure 2: Apple, Inc., Quarterly Revenue (in Billions)

In Figure 3, we can observe the relationship between the Holt-Winters filter and the Apple revenue time series.

Show the code

augment(apple_hw) |>
  ggplot(aes(x = index, y = revenue)) +
    geom_line() +
    geom_line(aes(y = .fitted, color = "Fitted")) +
    labs(color = "")

Figure 3: Superimposed plots of the Apple revenue and the Holt-Winters filter

Figure 4 contains the information from Figure 3, with the addition of an additional four years of forecasted values. The light blue bands give a 95% prediction bands for the forecast.

Show the code

apple_forecast <- apple_hw |>
  forecast(h = "4 years") 

apple_forecast |>
  autoplot(apple_ts, level = 95) +
  geom_line(aes(y = .fitted, color = "Fitted"),
    data = augment(apple_hw)) +
  scale_color_discrete(name = "")

Figure 4: Superimposed plots of Apple’s quarterly revenue and the Holt-Winters filter, with four additional years forecasted

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homework_3_5.qmd

BYU-Idaho Enrollment

$$Semester$$	$$t$$	$$x_t$$	$$a_t$$	$$b_t$$	$$s_t$$	$$\hat x_t$$
SP11	-2	—	—	—	1	—
FA11	-1	—	—	—	1	—
WI12	0	—	—	—	1	—
SP12	1	13.7	13.7	-0.022	1	13.7
FA12	2	16.2	14.2	0.082	1.03	14.6
WI13	3	15.5	14.5	0.126	1.01	14.6
SP13	4	14	14.5	0.101	0.99	14.4
FA13	5	15.6	14.7	0.121	1.04	15.3
WI14	6	15.6	14.9	0.137	1.02	15.2
SP14	7	12.9	14.6	0.05	0.97	14.2
FA14	8	16.2	14.8	0.08	1.05	15.5
WI15	9	16.7	15.2	0.144	1.04	15.8
SP15	10	13.7	15.1	0.095	0.96	14.5
FA15	11	17.6	15.5	0.156	1.07	16.6
⋮	⋮	⋮	⋮	⋮	⋮	⋮
FA22	32	19.4	17.7	-0.086	1.12	19.8
WI23	33	17.7	17.5	-0.109	1.04	18.2
SP23	34	12.8	16.9	-0.207	0.85	14.4
FA23	35	18.7	16.7	-0.206	1.12	18.7
WI24	36	17.6	16.6	-0.185	1.04	17.3
SP24	37	—	—	—	0.85	14
FA24	38	—	—	—	1.12	18.2
WI25	39	—	—	—	1.04	16.7
SP25	40	—	—	—	0.85	13.5
FA25	41	—	—	—	1.12	17.6
WI26	42	—	—	—	1.04	16.1
SP26	43	—	—	—	0.85	13
FA26	44	—	—	—	1.12	16.9
WI27	45	—	—	—	1.04	15.5

References

C. C. Holt (1957) Forecasting seasonals and trends by exponentially weighted moving averages, ONR Research Memorandum, Carnegie Institute of Technology 52. (Reprint at https://doi.org/10.1016/j.ijforecast.2003.09.015).
P. R. Winters (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6, 324–342. (Reprint at https://doi.org/10.1287/mnsc.6.3.324.)