Data collection

The prevalence and mortality time-series data of COVID-19 used in this study corresponded to the period between 27 February 2020 and 30 June 2020, all these data were collected from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (https://github.com/CSSEGISandData/COVID-19) and the WHO website (https://www.who.int/emergencies/diseases/novel-coronavirus-2019), and Microsoft Office Excel 2007 was utilised to collate database. Typically, to obtain a robust and effective model in practice, at least 50 observations and preferably 100 observations or more should be used [Reference Bartholomew, Box and Jenkins31]. Consequently, during the model-development process, the data samples from 27 February 2020 to 5 June 2020 (100 observations) were used for the training set, and the remaining 25 samples were taken for the testing set.

The study protocol was approved by the research institutional review board of the Xinxiang Medical University (No: XYLL-2019072), and it was exempt from the institutional review board assessment since all the data analysed in this study were obtained in an anonymous format and any non-essential identifying information were not accessed. In addition, this study meets all the guidelines in the Declaration of Helsinki.

ARIMA model construction

Time-series prediction aims to use a statistical technique to nowcast and forecast future unknown series by identifying the internal rules between the past and current series. Often, time series displays correlations between successive observations. The ARIMA model is designed to make forecasts by taking correlations existing in the time-series data into consideration [Reference Chen32]. ARIMA model types are listed using the standard notation of ARIMA (p, d, q), herein, p signifies the order of autoregression (AR), d refers to the order of integration and q represents the order of moving average (MA). The ARIMA method is defined for stationary time series. Hence, the stationary conditions of the targeted time series should first be judged by inspecting the time-series plots and by performing the Augmented Dickey–Fuller (ADF) statistic, if the time series displays a trend in change over time and the ADF statistic shows no statistical difference, indicating a non-stationary time series [Reference Wang33]. In this case, the time series requires to be differenced until a stationary series is obtained [Reference Wang34]. Afterward, the values of p and q can roughly be identified by examining the autocorrelation function (ACF) and partial ACF (PACF) graphs of the differenced time series [Reference Wang34]. The ‘Expert Modeler’ in SPSS software and ‘auto.arima()’ in R software have an ability to automatically identify the best-performing ARIMA model by considering the goodness of fit measures such as a larger value of R-squared (R ²), a lower value of normalised Bayesian information criterion (NBIC) and the appropriate ACF and PACF graphs of the errors [Reference Zeng35, Reference Wang36]. In this study, both these two tools were used to determine the best model. After that, statistical tests were conducted for the resulting best ARIMA model. The estimated key parameters of the AR and MA should be statistically significant under the t test [Reference Li37, Reference Liu38]. A Ljung-Box Q test was then applied to the residual series produced by the best-fitting ARIMA model, if this statistic provided a P-value greater than 0.05, suggesting that the residuals behaved like a white-noise series [Reference Li37, Reference Liu38]. At this time, the best ARIMA model passed all the required checking, and then it can be used to implement out-of-sample forecasting. The general forms of the AR process, the MA process and the ARIMA process are given in Eqs. (1), (2), and (3), respectively, below

(1)$$Y_t = \phi _1Y_{t-1} + \phi _2Y_{t-2} + \cdots + \phi _pY_{t-p} + \varepsilon _t$$

(2)$$Y_t = \varepsilon {}_t-\theta _1\varepsilon _{t-1}-\theta _2\varepsilon _{t-2}-\cdots -\theta _q\varepsilon _{t-q}$$

(3)$$Y_t = \phi _1Y_{t-1} + \phi _2Y_{t-2} + \cdots + \phi _pY_{t-p} + \varepsilon _t-\theta _1\varepsilon {}_{t-1}-\theta _2\varepsilon _{t-2}-\cdots -\theta _q\varepsilon _{t-q}$$

where ϕ and θ represent the key parameters of AR and MA, respectively, p and q correspond to the orders of AR and MA, respectively,Y _t refers to the observation at time t andɛ_tis the residual series that is assumed to be uncorrelated in the final ARIMA model.

α-Sutte Indicator

α-Sutte Indicator is a novel technical analysis based on the Sutte Indicator method that was initially proposed in 2017 by Ahmar [Reference Ahmar, Rahman and Mulbar26]. As known, the stock movements failed to be always consistent, sometimes suddenly descended and sometimes suddenly ascended, which caused less accuracy in the domain of stock forecasting using the common technical indicators such as simple moving average (SMA), moving average convergence/divergence (MACD), relative strength index (RSI), stochastic and Bolinger Band [Reference Ahmar23, Reference Ahmar, Rahman and Mulbar26, Reference Kurniasih39]. Hence, the development of α-Sutte Indicator is expected to overcome the weakness of the mentioned methods in predicting the stock movements by considering five elements of the stock movements, namely price at the time of opening, closing, highest and lowest, along with the volume of transactions [Reference Ahmar24]. Paralleling the advance in methodology, the α-Sutte Indicator is found to be not only limited to forecast the stock movements but can to perform prediction for the time series of data as it can track the dynamic dependence of certain data reasonably well [Reference Ahmar, Rahman and Mulbar26, Reference Kurniasih39, Reference Ahmar40]. The formula of α-Sutte Indicator can be in the form as below

(4)$$\alpha _t = \displaystyle{{\alpha {[ {\Delta x/\lpar \lpar \alpha + \delta \rpar /2\rpar } } ] + \beta {[ {\Delta y/\lpar \lpar \beta + \alpha \rpar /2\rpar } } ] + \gamma {[ {\Delta z/\lpar \lpar \gamma + \beta \rpar /2\rpar } } ] } \over 3}$$

where

(5)$$\delta = \alpha _{t-4}$$

(6)$$\alpha = \alpha _{t-3}$$

(7)$$\beta = \alpha _{t-2}$$

(8)$$\gamma = \alpha _{t-1}$$

(9)$$\Delta x = \alpha -\delta = \alpha _{t-3}-\alpha _{t-4}$$

(10)$$\Delta y = \beta -\alpha = \alpha _{t-2}-\alpha _{t-3}$$

(11)$$\Delta z = \gamma -\beta = \alpha _{t-1}-\alpha _{t-2}$$

where α _t denotes the observed value at t time and α _t−k signifies the observed value at (t−k) time.

Measuring for the prediction reliability level

The forecasting reliability level between two models was assessed by different measurement metrics, including the scale-dependent measurement metrics (e.g. root mean square error (RMSE) and mean absolute error (MAE)) and the measurement metrics based on percentage errors (e.g. mean error rate (MER), mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE)). A smaller value of these measurement metrics corresponded to the best model and this optimal model was then constructed to produce the out-of-sample forecasting. In the process of predicting data, the 95% confidence limits (CL) of the out-of-sample forecasts were generated by forecasting the in-sample counterparts.

(12)$${\rm RMSE} = \sqrt {\displaystyle{1 \over N}\sum\limits_{i = 1}^N {{\lpar Y_i-{\hat{Y}}_i\rpar }^ 2} } $$

(13)$${\rm MAE} = \displaystyle{1 \over N}\sum\limits_{i = 1}^N {\vert {Y_i-{\hat{Y}}_i} \vert } $$

(14)$${\rm MER} = \displaystyle{{\lpar 1/N\rpar \sum\nolimits_{i = 1}^N {\vert {Y_i-{\hat{Y}}_i} \vert } } \over {{\bar{Y}}_i}}$$

(15)$${\rm MAPE} = \displaystyle{1 \over N}\sum\limits_{i = 1}^N {\displaystyle{{\vert {Y_i} - {{\hat{Y}}_i} \vert } \over {Y_i}} \times 100} $$

(16)$${\rm RMSPE} = \sqrt {\displaystyle{1 \over N}\sum\limits_{i = 1}^N {{\left({\displaystyle{{Y_i-{\bar{Y}}_i} \over {Y_i}}} \right)}^2} } $$

where Y _i refers to the original observations, $\hat{Y}_i$ is the prediction values, denotes the average of the original observations and N signifies the number of original observations. In this research study, we used SPSS software (version 17.0, IBM Corp, Armonk, NY) and R software (version 4.0.0, R Development Core Team, Vienna, Austria) to construct the ARIMA model and the α-Sutte Indicator. The statistical significance level was set at P < 0.05.