Lecture 7: Exponential Smoothing With Trend and Seasonality

Load packages

library(tidyverse)
library(astsa)
library(fpp3)
library(epidatasets)

Internet useage

www = as_tsibble(WWWusage)

# Fit models
www_fit = www |>
  model(SES = ETS(value ~ error("A") + trend("N") + season("N")),
        Holt = ETS(value ~ error("A") + trend("A") + season("N")),
        Damped = ETS(value ~ error("A") + trend("Ad") + season("N")))
  
# Make forecasts
www_fit |>
  forecast(h = 10) |>
  mutate(.model = factor(.model, levels = c("SES", "Holt", "Damped"))) |>
  autoplot(www) + 
  labs(x = "Minute", y = "Number of users",
       title = "Internet usage per minute") +
  facet_grid(vars(.model)) + theme_bw() +
  theme(legend.position = "bottom", legend.title = element_blank())

# Inspect fitted coefficients
www_fit |> select(SES) |> coef()

www_fit |> select(Holt) |> coef()

www_fit |> select(Damped) |> coef()

Australian holiday travel

holiday = tourism |>
  filter(Purpose == "Holiday") |>
  summarize(Trips = sum(Trips)/1e3)

# Fit Holt-Winters model
holiday_fit = holiday |>
  model(HoltWinters = ETS(Trips ~ error("A") + trend("A") + 
                            season("A", period = 4)))

# Make forecasts
holiday_fit |> forecast(h = "3 years") |>
  autoplot(holiday) +
  labs(y = "Overnight trips (millions)",
       title =" Australian holiday travel") + theme_bw()

ETS decomposition

# Fit additive and multiplicative Holt-Winters  
holiday_fit = holiday |>
    model(
      HWAdd = ETS(Trips ~ error("A") + trend("A") + 
                    season("A", period = 4)),
      HWMult = ETS(Trips ~ error("A") + trend("A") + 
                     season("M", period = 4)))

# Inspect fitted coefficients---note that gamma is really tiny in both models,
# which means that the seasonal pattern doesn't change much over time
holiday_fit |> select(HWAdd) |> coef()

holiday_fit |> select(HWMult) |> coef()

# An example of how we would pull out the components
holiday_fit |> select(HWAdd) |> components() |> head(10)

# Plot the decomposition according to the components
library(gridExtra)

## 
## Attaching package: 'gridExtra'

## The following object is masked from 'package:dplyr':
## 
##     combine

g1 = holiday_fit |> select(HWAdd) |> components() |> autoplot() +
  labs(title = "Holt-Winters: additive seasonality", subtitle = NULL) + 
  theme_bw()
g2 = holiday_fit |> select(HWMult) |> components() |> autoplot() +
  labs(title = "Holt-Winters: multiplicative seasonality", subtitle = NULL) + 
  theme_bw()
grid.arrange(g1, g2, ncol = 2)  

## Warning: Removed 4 rows containing missing values (`geom_line()`).
## Removed 4 rows containing missing values (`geom_line()`).