Exponetial Smoothing JH

Holt-Winters’ Seasonal Method

An extension of Holt’s method to capture seasonality.

Comprised of forecast Equation and three smoothing equations: - One for the level - One for the trend - One for the seasonal component with corresponding smoothing parameters: \[ \alpha, \beta*, and \gamma \]

There are two variations to this method that differ in the nature of the seasonal component.

First, the additive method is preferred when the seasonal variations are roughly constant through the series,

Holt Seasonal Additive

With this method, the seasonal component is expressed in absolute terms in the scale of the observed series, and in the level equation the series is seasonally adjusted by subtracting the seasonal component. Within each year, the seasonal component will add up to approximately zero.

Second, the multiplicative method is preferred when the seasonal variations are changing proportional to the level of the series.

Holt Seasonal Multiplicative

With the multiplicative method, the seasonal component is expressed in relative terms (percentages), and the series is seasonally adjusted by dividing through by the seasonal component. Within each year, the seasonal component will sum up to \(\m\) approximately .

aust <- window(austourists,start=2005)
fit1 <- hw(aust,seasonal="additive")
fit2 <- hw(aust,seasonal="multiplicative")
autoplot(aust) +
autolayer(fit1, series="HW additive forecasts", PI=FALSE) +
autolayer(fit2, series="HW multiplicative forecasts", PI=FALSE) +
xlab("Year") +
ylab("Visitor nights (millions)") +
ggtitle("International visitors nights in Australia") +
guides(colour=guide_legend(title="Forecast"))

summary(fit1)

## 
## Forecast method: Holt-Winters' additive method
## 
## Model Information:
## Holt-Winters' additive method 
## 
## Call:
##  hw(y = aust, seasonal = "additive") 
## 
##   Smoothing parameters:
##     alpha = 0.3063 
##     beta  = 1e-04 
##     gamma = 0.4263 
## 
##   Initial states:
##     l = 32.2597 
##     b = 0.7014 
##     s = 1.3106 -1.6935 -9.3132 9.6962
## 
##   sigma:  1.9494
## 
##      AIC     AICc      BIC 
## 234.4171 239.7112 250.4748 
## 
## Error measures:
##                       ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 0.008115785 1.763305 1.374062 -0.2860248 2.973922 0.4502579
##                     ACF1
## Training set -0.06272507
## 
## Forecasts:
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2016 Q1       76.09837 73.60011 78.59664 72.27761 79.91914
## 2016 Q2       51.60333 48.99039 54.21626 47.60718 55.59947
## 2016 Q3       63.96867 61.24582 66.69153 59.80443 68.13292
## 2016 Q4       68.37170 65.54313 71.20027 64.04578 72.69762
## 2017 Q1       78.90404 75.53440 82.27369 73.75061 84.05747
## 2017 Q2       54.40899 50.95325 57.86473 49.12389 59.69409
## 2017 Q3       66.77434 63.23454 70.31414 61.36069 72.18799
## 2017 Q4       71.17737 67.55541 74.79933 65.63806 76.71667

summary(fit2)

## 
## Forecast method: Holt-Winters' multiplicative method
## 
## Model Information:
## Holt-Winters' multiplicative method 
## 
## Call:
##  hw(y = aust, seasonal = "multiplicative") 
## 
##   Smoothing parameters:
##     alpha = 0.4406 
##     beta  = 0.0134 
##     gamma = 0.0023 
## 
##   Initial states:
##     l = 32.4875 
##     b = 0.6974 
##     s = 1.0237 0.9618 0.7704 1.2442
## 
##   sigma:  0.0367
## 
##      AIC     AICc      BIC 
## 221.1313 226.4254 237.1890 
## 
## Error measures:
##                      ME     RMSE     MAE           MPE    MAPE      MASE
## Training set 0.09206228 1.575631 1.25496 -0.0006505533 2.70539 0.4112302
##                     ACF1
## Training set -0.07955726
## 
## Forecasts:
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2016 Q1       80.08894 76.31865 83.85922 74.32278 85.85509
## 2016 Q2       50.15482 47.56655 52.74309 46.19640 54.11324
## 2016 Q3       63.34322 59.80143 66.88502 57.92652 68.75993
## 2016 Q4       68.17810 64.08399 72.27221 61.91670 74.43950
## 2017 Q1       83.80112 78.43079 89.17146 75.58790 92.01434
## 2017 Q2       52.45291 48.88795 56.01787 47.00077 57.90504
## 2017 Q3       66.21274 61.46194 70.96353 58.94702 73.47845
## 2017 Q4       71.23205 65.85721 76.60690 63.01194 79.45217

fitted(fit1)

##          Qtr1     Qtr2     Qtr3     Qtr4
## 2005 42.65723 24.21081 32.66618 36.37206
## 2006 45.53781 27.51872 36.21481 40.33713
## 2007 49.17134 32.18209 39.31365 43.50887
## 2008 49.89746 32.85434 39.71474 43.48280
## 2009 53.65576 35.82730 43.37632 45.34947
## 2010 56.83932 37.31366 45.42533 47.91400
## 2011 60.42249 37.71212 47.59100 49.91681
## 2012 63.19432 40.58637 49.32571 53.47546
## 2013 65.75739 44.06492 54.19500 55.53369
## 2014 68.24659 43.48545 54.81621 58.70628
## 2015 69.05311 47.59377 59.24376 64.22408

fitted(fit2)

##          Qtr1     Qtr2     Qtr3     Qtr4
## 2005 41.28689 26.36042 32.62011 35.43591
## 2006 44.91855 28.43989 36.70598 39.63640
## 2007 50.24528 31.40864 39.84476 42.20533
## 2008 50.30134 31.91456 40.44309 44.16408
## 2009 53.76967 34.34469 43.00433 46.12632
## 2010 56.35302 36.28628 45.13587 48.57174
## 2011 58.97855 37.27806 47.78102 51.14259
## 2012 62.76659 39.02701 49.48525 54.86171
## 2013 67.26005 42.02854 52.46503 56.18995
## 2014 69.83611 42.43956 53.92965 58.43127
## 2015 72.59030 45.62125 58.77277 64.38368

Accuracy measures for the Holt Seasonal Additive:

round(accuracy(fit1),2)

##                ME RMSE  MAE   MPE MAPE MASE  ACF1
## Training set 0.01 1.76 1.37 -0.29 2.97 0.45 -0.06

Accuracy measures for the Holt Seasonal Multiplicative:

round(accuracy(fit2),2)

##                ME RMSE  MAE MPE MAPE MASE  ACF1
## Training set 0.09 1.58 1.25   0 2.71 0.41 -0.08

Additive_States <- fit1$model$states[,1:3]
Multiplicative_States <- fit2$model$states[,1:3]

colnames(Additive_States) <- c("level", "slope", "Season")

colnames(Multiplicative_States) <- c("level", "slope", "Season")

Additive_States_plot <- autoplot(Additive_States, facets=TRUE, color="blue") + labs(title="Additive states") 
Multiplicative_States_plot <- autoplot(Multiplicative_States,facets=TRUE, color="red") + labs(title="Multiplicative States")
grid.arrange(Additive_States_plot, Multiplicative_States_plot, nrow=1)

The small value of \(\gamma\) for the multiplicative model means that the seasonal component hardly changes over time. The small value of \(\beta\) for the additive model means the slope component hardly changes over time (check the vertical scale). The increasing size of the seasonal component for the additive model suggests that the model is less appropriate than the multiplicative model.

Holt-Winter’s Damped Method

As stated previously, Gardner & McKenzie (1985) introduced a parameter that “dampens” the trend to a flat line some time in the future. A method that often provides accurate and robust forecasts for seasonal data is the Holt-Winters method with a damped trend and multiplicative seasonality.

Holt Seasonal Damped Method

fc <- hw(subset(hyndsight,end=length(hyndsight)-35),
         damped = TRUE, seasonal="multiplicative", h=35)
autoplot(hyndsight) +
  autolayer(fc, series="HW multi damped", PI=FALSE)+
  guides(colour=guide_legend(title="Daily forecasts"))