The Exponential Smoothing Methods (Double-Triple) and Its Applications On Time Series Data

- The research studied double and triple exponential smoothing methods and its applications on forecasting time series data. Due to the original of plot data (pattern) increasing seasonal, the exponential smoothing method and mean absolute percentage error (MAPE) are then used. Here, Zaitun and Minitab software are used to compute the result of the forecasting. The result showed that the smoothing parameters significant influenced the significant result of the forecasting with small MAPE around 0.09 and 0.1, in double smoothing exponential (DES) and triple smoothing exponential (TES), respectively.

To measure the significant result of the forecasting for p period ahead, there are three measurements of error, namely (1) mean absolute deviation (MAD), (2) mean absolute percentage error (MAPE), and (3) mean squared error (MSE) (see Najmudin, [11]). Due to the values of MAPE is smaller than MAD and MSE, we then used it. Note that the formula of MAPE is given 11 where Y t is an actual data at period t, ˆt Y is a forecasting data at period t, and n is a number of data.
Following Goh and Law [5], and Andini [2], the criteria of MAPE is noted as: (1) 0 < x < 10 is very good, (2) 10 x < 20 is good, (3) 20 x < 50 is enough, (4) 50 x  is bad. We note here that the small value of MAPE is then chosen as a best indicator in determining the significant forecast.
There are several steps to produce the significant result of the forecasting on p period ahead: (1) plot and identify the trend of the data (time series data), (2) find the suitable method, (3) give a simulation data, (4) used the formula and software to find the result of the forecasting, and (5) check the significant result using MAPE.
In this paper, the introduction is given in Section 1. The double exponential smoothing (DES) and its simulation are presented in Section 2. The triple exponential smoothing (TES) and its simulation are obtained in Section 3. Then, Section 4 described the conclusion of the research.

II. A SIMULATION STUDY AND RESULT 2.1. The Double Exponential Smoothing Method
Following Makridakis [7] and Makridakis, et al., [8], the DES had multiple smoothing weighted, namely , , and .  [7]. To make clear the concept of DES and TES, we refer to the first theory of smoothing exponential methods, that is the SES. Here, the formula of SES is given as where Y t is an actual data at period t, ˆt Y is a forecasting data period t, is a number of data, 1t Y  is the forecasting data period t+1, and is the parameter smoothing (0 < α <1). Furthermore, following Najmudin [11], the DES's formula is then presented in term of the Brown is given as a is a differences values of smoothing at t period, t b is a additional factor at period of t, p is a number of period ahead to forecast. To get the ˆt p Y  , we must follow some steps: (1) compute the smoothing exponential using Here, t L is an estimation value of single smoothing and * t L is an estimation value of double smoothing. Similarly, Following Najmudin [11], Holt method follows three steps, that are: (1) determine level of estimation using we also determine trend of estimation using , and furthermore, we written the Holt formula as where ˆt p Y  is a forecasting data on p period of t, t a is a differences values of smoothing at t period, t b is a additional factor at period of t, p is a number of period ahead to forecast, t L is a level estimation, and t T is a trend estimation. Note that 0 L and 0 T are intercept of linear estimation and slope, respectively (Montgomery [10]). A simulation study is given for forecasting data of a number of motorcycle in Figure 1. Figure 1 showed that the trend of the plot of the data increases as the month increases (Year of data: 2017-2018). It is clear that the data (Figure 1.)

   
for Holt. The forecasting data is then given in Table 1 and Table 2 for both methods (Brown and Holt).

The Triple Exponential Smoothing Method
Similarly with the Section 2, we then studied the TES in this section. Following Makridakis [7] and Makridakis, et al., [8], the triple exponential smoothing (TES) had three smoothing weighted, namely , and    . These parameters are chosen based on the smallest mean absolute percentage error (MAPE) on several trials. Detail TES is found in Makridakis [7]. Following Makridakis et al. [8] and Najmudin [11], the general formula of the TES is given as Level : where t L is value of level,  ,  and  are and smoothing weighted, t T is an estimation trend,  is smoothing constant of trend estimation, t S is an estimation of seasonal,  is smoothing constant of trend seasonal, s is length of seasonal, ˆt p Y  is a forecasting data on p period ahead, and p is period of forecasting. A simulation study is given to the data of the executive class train ticket as below. The scatter plot is presented in Figure 2. ( Niqatani [13] and Pratikno et al. [17]), we then chose the TES as a suitable method for this plot.     Figure 3. It is clear that the green line (forecasting) is in line with the trend of the actual data (red line). They both lie at 95% its confidence interval, with the accuracy measurement of the MAPE is 0.1 (small). The actual error is given in Table 3.  Table 3. we see that the error of the forecasting is around 0.006 -0. 2 (in average is 0.07). It means that the error is relative small. Thus, we conclude that it is an enough error for getting the good forecasting.

III. CONCLUSION
The research studied the double and triple smoothing exponential method in forecasting time series data. Both methods are suitable due to the trend of the plot increases (DES methods) and seasonal increases (TES). The MAPE is used to obtain the eligible forecasting. Zaitun and Minitab software are used to compute the result. The result showed that the forecasting data are really influenced by smoothing parameters, with the smallest MAPE are 0.09 (DES) and 0.1 (TES), respectively. For the DES, Brown method is better than Holt method.