## 1 Introduction  
Time series is a sequence of observations, measured typically at successive points in time spaced at uniform intervals. This exists in many situations, including the daily closing value of the stock market, manufacturing process, health status of patients, and economic indicators [1]. Using these time series data, forecasting future events has been of considerable interest in various fields, including control charts [2; 3; 4; 5], health care surveillance [6; 7; 8], inventory controls [9], stock market prediction [10; 11; 12], pandemic occurrence prediction [13], and electricity demand forecasting [14]. In such industrial areas, the accurate forecasting of future time series values helps establish more effective decision or management policy. To achieve satisfactory forecasting, a number of time series forecasting methods have been proposed, including exponentially weighted moving average (EWMA; [15; 16; 17; 18]), autoregressive integrated moving average (ARIMA; [19]), generalized autoregressive conditionally heteroskedastic (GARCH; [20]), and vector autoregression (VAR; [21; 22]) methods. In recent years, neural network-based approaches [23; 24] have been widely applied for the time series forecasting tasks [25; 26; 27]. Among these time series forecasting methods, in this study, we focus on the EWMA model. The EWMA model assigns different weights for individual observations collected over time, and more recent observations are weighted more heavily than remote observations, and hence,dynamics of time series can be effectively reflected into the forecasting models. Furthermore, such an EWMA model has a desirable property with a recursive form, which leads to computational efficiency. Finally, the EWMA models do not need any distributional assumption and prior knowledge on the time series [1; 28]. Because of these advantages, a number of EWMA models have been proposed, including single exponential smoothing (single ES; [15; 16]) model, double exponential smoothing (double ES; [17]) model, triple exponential smoothing (triple ES; [18]) model, and two-stage exponentially weighted moving average (two-stage EWMA; [29; 30]) model. We presented a more detailed description on these existing EWMA models in Section 2.1. To ensure the satisfactory forecasting performance of the EWMA models, historical time series observations should be sufficiently accumulated. However, lots of data analysists have suffered from the insufficiency of historical observations in many real situations [31]. For example, the demand prediction of newly launched products is not an easy task because there are no sales records of these products. In addition, the prognosis of a manufacturing system status is also difficult immediately after the preventive maintenance because meaningful process data are not sufficiently accumulated after these operations. Accordingly, the insufficiency of historical observations has been regarded as one of the challenging problems for time series data analysis [28]. To address the insufficiency of initial time series data, self-starting approaches [32; 33; 34] can be incorporated into the EWMA model. In the self-starting-based EWMA modeling process, an initial EMWA model is constructed only using small initial time series observations, and the EWMA model is continuously updated as a new observation is sequentially recorded. By doing so, forecasting the performance of EWMA models can be improved over time because historical records are continuously accumulated in the forecasting process. Besides, the self-starting forecasting process can effectively accommodate dynamically changing time-varying patterns in that the forecasting models are updated by adding the most recently recorded observation to the initial time series data. In Section 2.2, we also presented more details on the self-starting forecasting process. The main purpose of this study was to examine the characteristics of EWMA models as base-forecasting models for the self-starting forecasting process.