Data description

The data set used for this analysis is the ausbeer data set. It represents the total quarterly beer production in Australia (in megalitres). The data set contains 211 observations and 2 variables. Summary output of the data set as well as its structure follows. the two variables in the data set are called QD and nd0. QD is a date variable that starts from 1956 to 2008. nd0 is an int, which counts the beer production in Australia for each observation. Each observation represents a quarter of a year. Its min, mean, and max are, respectively, 213, 415, 599.0. There seems to be no missing values.

'data.frame':   211 obs. of  2 variables:
 $ QD : int  1956 1956 1956 1956 1957 1957 1957 1957 1958 1958 ...
 $ nd0: int  284 213 227 308 262 228 236 320 272 233 ...
       QD            nd0       
 Min.   :1956   Min.   :213.0  
 1st Qu.:1969   1st Qu.:378.5  
 Median :1982   Median :423.0  
 Mean   :1982   Mean   :415.0  
 3rd Qu.:1995   3rd Qu.:465.5  
 Max.   :2008   Max.   :599.0

Training and Testing Data

Next the data was split into test and training data sets. The training data set had a size n = 111 of The most recent observations minus the 11 periods kept for the test data set. Tabular output of these amounts follow.

DataSplit
AMNTs
Train Set 111
Test Set 11
ORG Set 211

Time Series Object

A time series objects was constructed using the ts() function. The frequency was set to 4. Output of the ts object as well as a plot the ts object follows. Visual inspection of the graph may yield that the time series is additive. Frankly, there seems to be no amplification of the data over time. The time series begins in 1978 of the 2nd quarter and ends in 2005 of the 4th quarter.

     Qtr1 Qtr2 Qtr3 Qtr4
1978       464  431  588
1979  503  443  448  555
1980  513  427  473  526
1981  548  440  469  575
1982  493  433  480  576
1983  475  405  435  535
1984  453  430  417  552
1985  464  417  423  554
1986  459  428  429  534
1987  481  416  440  538
1988  474  440  447  598
1989  467  439  446  567
1990  485  441  429  599
1991  464  424  436  574
1992  443  410  420  532
1993  433  421  410  512
1994  449  381  423  531
1995  426  408  416  520
1996  409  398  398  507
1997  432  398  406  526
1998  428  397  403  517
1999  435  383  424  521
2000  421  402  414  500
2001  451  380  416  492
2002  428  408  406  506
2003  435  380  421  490
2004  435  390  412  454
2005  416  403  408  482

ETS

Next ETS objects were created using different smoothing methods. Those methods which correspond to the ETS objects 1-5 respectively are the Simple Exponential Smoothing, Holt additive and additive-damped, and Holt-Winters additive and additive-damped. Accuracy metrics for each models smoothing follows.

ETS Accuracy Metrics

Here Accuracy metrics for each of the training data sets were computed and output in a table. These accuracy metrics pertain to the smoothing conducted be each of the aforementioned smoothing methods. Along the top the metrics are named. along the left side are the ETS methods those metrics correspond to. Utilizing RMSE and MASE, it may be clear that the Holt-winters additive method is minimized the most among the other methods.

Accuracy Metrics for ETS METHOD
ME RMSE MAE MPE MAPE MASE ACF1
SES -7.3506 52.6575 45.9144 -2.8091 9.9331 2.7339 -0.1007
Holt Linear 0.2376 50.6016 41.8327 -1.0766 8.8954 2.4909 -0.1050
Holt Add. Damped -5.8908 51.8899 44.6026 -2.4480 9.6200 2.6558 -0.0850
HW Add. -0.1390 18.0274 14.2027 -0.1101 3.0653 0.8457 -0.2406
HW Add. Damp -1.6154 18.3315 14.3708 -0.4302 3.1040 0.8557 -0.2145

Plots

Next plots of the forecast from each ETS method follows. In the upper right hand corner of each plot is a legend. The legend identifies which smoothing method applies to which series. The series in black and orange represent the training and testing data respectively. Forecast produced by each smoothing method is overlayed on the testing series.

Forecast Accuracy Metrics

Next Accuracy metrics for each of the training data sets were computed and output in a table. The MAPE and MSE depicted along the top. The observation size for each training set is written across the left side. The MAPE is represented as percentages. Utilizing MSE and MAPE, it may be clear that the Holt-winters additive damped method is minimized the most among the other methods.

The accuracy measures of various exponential smoothing models based on the testing data
MSE MAPE
SES 1382.88490 7.564721
Holt.Add 1136.94146 6.428992
Holt.Add.Damp 1268.47990 7.168382
HW.Add 67.85332 1.587917
HW.Add.Damp 58.83956 1.478948

Discussion

An analysis was conducted to explore different exponential smoothing methods. Those methods were Simple, Holt’s, and Holt-Winter’s. The data set used was the Counts of beer production in Australia. Each observation represented a measure taken quarterly from 1956 to 2008. There were 211 observations, 2 varaibles: a date and an amount, the highest and lowest beer produced in this time period, respectively, was 599.0ML and 213.0ML. The data set appeared to contain both trend and seasonality and there seemed to be no amplification of the data over time.The data was split into test and training data sets, n = 111 of The most recent observations minus the 11 periods kept for the test data set. Then a time series objects was constructed using the ts() function, frequency=4, start= Q2:1978 end= Q4:2005.Next ETS objects were created using different smoothing methods. Those methods were Simple Exponential Smoothing, Holt additive and additive-damped, and Holt-Winters additive and additive-damped. Accuracy metrics for each models smoothing and forecasting were output. Utilizing RMSE and MASE for the smoothing and MSE and MAPE for the forecast, it may be clear that the Holt-winters additive method is minimized the most among the other methods for the smoothing and Holt-winters additive damped method is minimized the most among the other methods for the forecast. This may stem from the model possessing both trend and seasonality. Next plots of the forecast from each ETS method followed with explanations of each.

Conclussion

An optimal smoothing model could not be found. However, One might choose between Holt-Winter’s additive and A.Damped based of the accuracy metrics reported. However the time series used in this analysis possessed trend and seasonality, which none of the other smoothing methods may have been equipped to handle.