Introduction

Holt (1957) and Winters (1960) extended Holt’s method to capture seasonality. The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level \(\ell_{t}\), one for the trend \(b_{t}\) , and one for the seasonal component \(s_{t}\) , with corresponding smoothing parameters \(\alpha\), \(\beta^*\) and \(\gamma\). We use m to denote the frequency of the seasonality, i.e., the number of seasons in a year. For example, for quarterly data m = 4 , and for monthly data m = 12 .

1. Holt-Winters’ Additive Method

The component form for the additive method is:

\[\begin{align*} \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} + s_{t+h-m(k+1)} \\ \ell_{t} &= \alpha(y_{t} - s_{t-m}) + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 - \beta^*)b_{t-1}\\ s_{t} &= \gamma (y_{t}-\ell_{t-1}-b_{t-1}) + (1-\gamma)s_{t-m}, \end{align*}\]

where k is the integer part of \((h-1)/m\), which ensures that the estimates of the seasonal indices used for forecasting come from the final year of the sample. The level equation shows a weighted average between the seasonally adjusted observation \((y_{t} - s_{t-m})\) and the non-seasonal forecast \((\ell_{t-1}+b_{t-1})\) for time t . The trend equation is identical to Holt’s linear method. The seasonal equation shows a weighted average between the current seasonal index, \((y_{t}-\ell_{t-1}-b_{t-1})\), and the seasonal index of the same season last year (i.e., m time periods ago).

2. Holt-Winters’ Multiplicative Method

The component form for the multiplicative method is:

\[\begin{align*} \hat{y}_{t+h|t} &= (\ell_{t} + hb_{t})s_{t+h-m(k+1)} \\ \ell_{t} &= \alpha \frac{y_{t}}{s_{t-m}} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ b_{t} &= \beta^*(\ell_{t}-\ell_{t-1}) + (1 - \beta^*)b_{t-1} \\ s_{t} &= \gamma \frac{y_{t}}{(\ell_{t-1} + b_{t-1})} + (1 - \gamma)s_{t-m} \end{align*}\]

3. An Application: International Tourist Visitors in Australia

We apply Holt-Winters’ method with both additive and multiplicative seasonality to forecast quarterly visitors in Australia spent by international tourists. Figure 1 shows the data from 1999 to 2013, and the forecasts for 2014 – 2015. Note that the data show an obvious seasonal pattern, with peaks observed in the March quarter of each year, corresponding to the Australian summer.

library(tidyverse)

#---------------------------
#  Some Ultility Funstions
#---------------------------

# A function for theme: 
my_theme <- function(...) {
  theme(
    axis.line = element_blank(),  
    axis.text.x = element_text(color = "white", lineheight = 0.9),  
    axis.text.y = element_text(color = "white", lineheight = 0.9),  
    axis.ticks = element_line(color = "white", size  =  0.2),  
    axis.title.x = element_text(color = "white", margin = margin(0, 10, 0, 0)),  
    axis.title.y = element_text(color = "white", angle = 90, margin = margin(0, 10, 0, 0)),  
    axis.ticks.length = unit(0.3, "lines"),   
    legend.background = element_rect(color = NA, fill = " gray10"),  
    legend.key = element_rect(color = "white",  fill = " gray10"),  
    legend.key.size = unit(1.2, "lines"),  
    legend.key.height = NULL,  
    legend.key.width = NULL,      
    legend.text = element_text(color = "white"),  
    legend.title = element_text(face = "bold", hjust = 0, color = "white"),  
    legend.text.align = NULL,  
    legend.title.align = NULL,  
    legend.direction = "vertical",  
    legend.box = NULL, 
    panel.background = element_rect(fill = "gray10", color  =  NA),  
    panel.border = element_blank(),
    panel.grid.major = element_line(color = "grey35"),  
    panel.grid.minor = element_line(color = "grey20"),  
    panel.spacing = unit(0.5, "lines"),   
    strip.background = element_rect(fill = "grey30", color = "grey10"),  
    strip.text.x = element_text(color = "white"),  
    strip.text.y = element_text(color = "white", angle = -90),  
    plot.background = element_rect(color = "gray10", fill = "gray10"),  
    plot.title = element_text(color = "white", hjust = 0, lineheight = 1.25,
                              margin = margin(2, 2, 2, 2)),  
    plot.subtitle = element_text(color = "white", hjust = 0, margin = margin(2, 2, 2, 2)),  
    plot.caption = element_text(color = "white", hjust = 0),  
    plot.margin = unit(rep(1, 4), "lines"))
  
}

# A function for calcualting MAE: 

mae <- function(actual, predicted) {
  y <- actual - predicted
  y %>% abs() %>% mean() %>% return()
}


#--------------------------------
# Holt-Winters’ seasonal methods
#--------------------------------

library(fpp2)
# Data for training model: 
aust <- austourists[1:60] %>% ts(start = 1999, frequency = 4)

# Inspect our data: 
autoplot(aust) + 
  geom_line(color = "cyan") + 
  geom_point(color = "cyan") + 
  my_theme() + 
  labs(x = NULL, y = NULL, 
       title = "Figure 1: Quarterly Visitors (in millions) to Australia: 1999-2013", 
       subtitle = "Data Source: Tourism Research Australia.")

## [1] 2.362556
## [1] 2.633565

Further reading

  • Two articles by Ev Gardner (Gardner, 1985, 2006) provide a great overview of the history of exponential smoothing, and its many variations.

  • A full book treatment of the subject providing the mathematical details is given by Hyndman et al. (2008).

References

  1. Gardner, E. S. (1985). Exponential smoothing: The state of the art. Journal of Forecasting, 4(1), 1–28. https://doi.org/10.1002/for.3980040103

  2. Gardner, E. S. (2006). Exponential smoothing: The state of the art — Part II. International Journal of Forecasting, 22, 637–666. https://doi.org/10.1016/j.ijforecast.2006.03.005

  3. Hyndman, R. J., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with exponential smoothing: The state space approach. Berlin: Springer-Verlag. http://www.exponentialsmoothing.net

---
title: "Holt-Winters’ Seasonal Methods for Forecasting Time Series" 
subtitle: "Self-training Project"
author: "Nguyen Chi Dung"
output:
  html_document: 
    code_download: true
    code_folding: hide
    highlight: pygments
    # number_sections: yes
    theme: "flatly"
    toc: TRUE
    toc_float: TRUE
---

```{r setup,include=FALSE}
knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE)
```

# Introduction

Holt (1957) and Winters (1960) extended Holt’s method to capture seasonality. The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level $\ell_{t}$, one for the trend $b_{t}$ , and one for the seasonal component $s_{t}$ , with corresponding smoothing parameters $\alpha$, $\beta^*$ and $\gamma$. We use m to denote the frequency of the seasonality, i.e., the number of seasons in a year. For example, for quarterly data m = 4 , and for monthly data m = 12 .

## 1. Holt-Winters’ Additive Method

The component form for the additive method is:

$$\begin{align*}
  \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} + s_{t+h-m(k+1)} \\
  \ell_{t} &= \alpha(y_{t} - s_{t-m}) + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\
  b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 - \beta^*)b_{t-1}\\
  s_{t} &= \gamma (y_{t}-\ell_{t-1}-b_{t-1}) + (1-\gamma)s_{t-m},
\end{align*}$$


where k is the integer part of $(h-1)/m$, which ensures that the estimates of the seasonal indices used for forecasting come from the final year of the sample. The level equation shows a weighted average between the seasonally adjusted observation $(y_{t} - s_{t-m})$ and the non-seasonal forecast $(\ell_{t-1}+b_{t-1})$ for time t . The trend equation is identical to Holt’s linear method. The seasonal equation shows a weighted average between the current seasonal index, $(y_{t}-\ell_{t-1}-b_{t-1})$, and the seasonal index of the same season last year (i.e., m time periods ago).

## 2. Holt-Winters’ Multiplicative Method

The component form for the multiplicative method is:

$$\begin{align*}
  \hat{y}_{t+h|t} &= (\ell_{t} + hb_{t})s_{t+h-m(k+1)} \\
  \ell_{t} &= \alpha \frac{y_{t}}{s_{t-m}} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\
  b_{t} &= \beta^*(\ell_{t}-\ell_{t-1}) + (1 - \beta^*)b_{t-1}                \\
  s_{t} &= \gamma \frac{y_{t}}{(\ell_{t-1} + b_{t-1})} + (1 - \gamma)s_{t-m}
\end{align*}$$



## 3. An Application: International Tourist Visitors in Australia

We apply Holt-Winters’ method with both additive and multiplicative seasonality to forecast quarterly visitors in Australia spent by international tourists. Figure 1 shows the data from 1999 to 2013, and the forecasts for 2014 – 2015. Note that the data show an obvious seasonal pattern, with peaks observed in the March quarter of each year, corresponding to the Australian summer.


```{r}
library(tidyverse)

#---------------------------
#  Some Ultility Funstions
#---------------------------

# A function for theme: 
my_theme <- function(...) {
  theme(
    axis.line = element_blank(),  
    axis.text.x = element_text(color = "white", lineheight = 0.9),  
    axis.text.y = element_text(color = "white", lineheight = 0.9),  
    axis.ticks = element_line(color = "white", size  =  0.2),  
    axis.title.x = element_text(color = "white", margin = margin(0, 10, 0, 0)),  
    axis.title.y = element_text(color = "white", angle = 90, margin = margin(0, 10, 0, 0)),  
    axis.ticks.length = unit(0.3, "lines"),   
    legend.background = element_rect(color = NA, fill = " gray10"),  
    legend.key = element_rect(color = "white",  fill = " gray10"),  
    legend.key.size = unit(1.2, "lines"),  
    legend.key.height = NULL,  
    legend.key.width = NULL,      
    legend.text = element_text(color = "white"),  
    legend.title = element_text(face = "bold", hjust = 0, color = "white"),  
    legend.text.align = NULL,  
    legend.title.align = NULL,  
    legend.direction = "vertical",  
    legend.box = NULL, 
    panel.background = element_rect(fill = "gray10", color  =  NA),  
    panel.border = element_blank(),
    panel.grid.major = element_line(color = "grey35"),  
    panel.grid.minor = element_line(color = "grey20"),  
    panel.spacing = unit(0.5, "lines"),   
    strip.background = element_rect(fill = "grey30", color = "grey10"),  
    strip.text.x = element_text(color = "white"),  
    strip.text.y = element_text(color = "white", angle = -90),  
    plot.background = element_rect(color = "gray10", fill = "gray10"),  
    plot.title = element_text(color = "white", hjust = 0, lineheight = 1.25,
                              margin = margin(2, 2, 2, 2)),  
    plot.subtitle = element_text(color = "white", hjust = 0, margin = margin(2, 2, 2, 2)),  
    plot.caption = element_text(color = "white", hjust = 0),  
    plot.margin = unit(rep(1, 4), "lines"))
  
}

# A function for calcualting MAE: 

mae <- function(actual, predicted) {
  y <- actual - predicted
  y %>% abs() %>% mean() %>% return()
}


#--------------------------------
# Holt-Winters’ seasonal methods
#--------------------------------

library(fpp2)
# Data for training model: 
aust <- austourists[1:60] %>% ts(start = 1999, frequency = 4)

# Inspect our data: 
autoplot(aust) + 
  geom_line(color = "cyan") + 
  geom_point(color = "cyan") + 
  my_theme() + 
  labs(x = NULL, y = NULL, 
       title = "Figure 1: Quarterly Visitors (in millions) to Australia: 1999-2013", 
       subtitle = "Data Source: Tourism Research Australia.")
  
# Use two methods for forecasting the next 8 observations: 
fit1 <- hw(aust, h = 8, seasonal = "additive")
fit2 <- hw(aust, h = 8, seasonal = "multiplicative")


# Compare actuals vs predcitions: 
my_df <- data_frame(Actual = austourists[61:68], 
                    Additive = fit1$mean %>% as.vector(), 
                    Multiplicative = fit2$mean %>% as.vector(), t = 1:8)

my_df %>% 
  gather(Series, b, -t) %>% 
  ggplot(aes(t, b, color = Series)) + 
  geom_line() + 
  geom_point() + 
  my_theme() + 
  scale_color_manual(values = c("cyan", "purple", "orange")) + 
  scale_x_continuous(breaks = seq(1:8)) + 
  labs(x = NULL, y = NULL, 
       title = "Figure 2: Actuals vs Additive and Multiplicate Predictions")

# Compare MAEs: 
mae(my_df$Actual, my_df$Additive)
mae(my_df$Actual, my_df$Multiplicative) 

```


# Further reading

- Two articles by Ev Gardner (Gardner, 1985, 2006) provide a great overview of the history of exponential smoothing, and its many variations.

- A full book treatment of the subject providing the mathematical details is given by Hyndman et al. (2008).

# References

1. Gardner, E. S. (1985). Exponential smoothing: The state of the art. Journal of Forecasting, 4(1), 1–28. https://doi.org/10.1002/for.3980040103

2. Gardner, E. S. (2006). Exponential smoothing: The state of the art — Part II. International Journal of Forecasting, 22, 637–666. https://doi.org/10.1016/j.ijforecast.2006.03.005

3. Hyndman, R. J., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with exponential smoothing: The state space approach. Berlin: Springer-Verlag. http://www.exponentialsmoothing.net



