data(airpass)
summary(airpass)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   104.0   180.0   265.5   280.3   360.5   622.0
data(airpass)
plot(airpass)

As you can see by the plot this series shows a definite positive trend between 1960 and 1971 and also some seasonality. Based on the above chart, it seems the count of international airline passengers spikes during the summertime through the holiday season, and then dips back down near the end of each year.

fsmoothing=ses(airpass, h=1)
plot(fsmoothing)

summary(fsmoothing)
## 
## Forecast method: Simple exponential smoothing
## 
## Model Information:
## Simple exponential smoothing 
## 
## Call:
##  ses(y = airpass, h = 1) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
## 
##   Initial states:
##     l = 111.9892 
## 
##   sigma:  33.8299
## 
##      AIC     AICc      BIC 
## 1733.787 1733.958 1742.696 
## 
## Error measures:
##                   ME     RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 2.22249 33.59418 25.68134 0.3758224 8.957112 0.8017825 0.3028609
## 
## Forecasts:
##          Point Forecast   Lo 80    Hi 80    Lo 95    Hi 95
## Jan 1972       431.9958 388.641 475.3506 365.6904 498.3012
round(accuracy(fsmoothing))
##              ME RMSE MAE MPE MAPE MASE ACF1
## Training set  2   34  26   0    9    1    0
fc1=holt(airpass, h=1)
plot(fc1)

summary(fc1)
## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = airpass, h = 1) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 119.7517 
##     b = 1.5963 
## 
##   sigma:  34.0125
## 
##      AIC     AICc      BIC 
## 1737.295 1737.729 1752.144 
## 
## Error measures:
##                     ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 0.5677345 33.53682 25.61104 -0.3624936 8.996892 0.7995878
##                   ACF1
## Training set 0.3024925
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80   Lo 95    Hi 95
## Jan 1972       433.6004 390.0116 477.1892 366.937 500.2638
round(accuracy(fc1))
##              ME RMSE MAE MPE MAPE MASE ACF1
## Training set  1   34  26   0    9    1    0
fc2=hw(airpass, seasonal="additive", h=1)
fc3=hw(airpass, seasonal="multiplicative", h=1)
plot(fc2)

plot(fc3)

summary(fc2)
## 
## Forecast method: Holt-Winters' additive method
## 
## Model Information:
## Holt-Winters' additive method 
## 
## Call:
##  hw(y = airpass, h = 1, seasonal = "additive") 
## 
##   Smoothing parameters:
##     alpha = 0.9935 
##     beta  = 2e-04 
##     gamma = 6e-04 
## 
##   Initial states:
##     l = 120.9608 
##     b = 1.3934 
##     s = -29.1816 -54.3842 -20.7169 15.0727 65.1554 66.1846
##            33.5822 -4.232 -8.0946 -3.8205 -34.3364 -25.2288
## 
##   sigma:  18.0471
## 
##      AIC     AICc      BIC 
## 1565.872 1570.729 1616.359 
## 
## Error measures:
##                     ME     RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 0.9638247 17.01495 12.81203 0.3539818 5.224715 0.3999971 0.1875311
## 
## Forecasts:
##          Point Forecast   Lo 80    Hi 80    Lo 95    Hi 95
## Jan 1972       437.2863 414.158 460.4145 401.9146 472.6579
summary(fc3)
## 
## Forecast method: Holt-Winters' multiplicative method
## 
## Model Information:
## Holt-Winters' multiplicative method 
## 
## Call:
##  hw(y = airpass, h = 1, seasonal = "multiplicative") 
## 
##   Smoothing parameters:
##     alpha = 0.3146 
##     beta  = 0.0071 
##     gamma = 0.5977 
## 
##   Initial states:
##     l = 120.3796 
##     b = 1.7757 
##     s = 0.9298 0.7946 0.9024 1.0451 1.1338 1.1388
##            1.0529 0.9638 1.0349 1.0807 0.9854 0.9378
## 
##   sigma:  0.0407
## 
##      AIC     AICc      BIC 
## 1405.654 1410.511 1456.141 
## 
## Error measures:
##                    ME     RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 1.256973 10.63256 7.790649 0.2182707 2.914411 0.2432275 0.2135914
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95   Hi 95
## Jan 1972       445.8901 422.6577 469.1225 410.3592 481.421

Using simple exponential smoothing method the alpha R chose was a value near 1 so it is implying more recent history is being used. the RMSE is 33.59418

Using Hotls method the alpha R chose was a value near 1 so it is implying more recent history is being used. The beta is close to zero indicating the trend is weak. The RMSE is 33.53682

Using Holt-Winter additive method the alpha R chose was a value near 1 so it is implying more recent history is being used. The beta is close to zero indicating the trend is weak. The gamma captures the strength of the seasonailty so with this method there is no seasonality. The RMSE is 17.01495

Using Holt-Winter multiplicative method the alpha R chose is closer to zero implying R went further back into the history. The beta is close to zero indicating the trend is weak. The gamma captures the strength of the seasonailty so with this method there is some seasonality. The RMSE is 10.63256

In conclusion based on the RMSE we will go with the Holt-Winter multiplicative method as is has the lowest RMSE.