## 3 Proposed Method
### Meta-Learning for Rapid Training of Multi-Machine Models for Anomaly Detection and Prediction
There are two variations of this method: the additive and multiplicative models. In the additive model, the components are expressed as:  
\[\hat{y}_{t+h|t}=l_{t}+hb_{t}+s_{t+h-m(k+1)} \tag{13}\]  
\[l_{t}=\alpha(y_{t}-s_{t-m})+(1-\alpha)(l_{t-1}+b_{t-1}) \tag{14}\]  
\[b_{t}=\beta^{*}(l_{t}-l_{t-1})+(1-\beta^{*})b_{t-1} \tag{15}\]  
\[s_{t}=\gamma(y_{t}-l_{t-1}-b_{t-1})+(1-\gamma)s_{t-m} \tag{16}\]  
In the multiplicative model, the components are expressed as:  
Figure 10:0  
Figure 10:1  
Figure 10:2  
Figure 10:3  
where \(s_{t}\), \(k\), \(m\), \(\alpha\), \(\beta^{*}\), and \(y\) denote the season at time \(t\), integer part of \((h-1)/m\), number of cycles/frequency of seasonality (e.g., four for quarterly data), level smoothing parameter, trend smoothing parameter, and seasonal smoothing parameters/weight for the season (\(0\leq\gamma\leq 1\)), respectively. In the additive model, the forecast value for each data element is the sum of the baseline, trend, and seasonality components. However, a multiplicative model is preferred when seasonal variations change proportionally to the level of the series. (d) Autoregressive Model  
The autoregressive (AR) model is a statistical method used to analyze time-series data that predict the future value of a variable using its own historical data [35]. AR is an evolution of linear regression analysis, where it analyzes the relationship between parameter \(x\) and its own past value instead of analyzing the relationship between the parameter \(x\) and dependent variable \(y\). This is expressed as:  
Figure 10:4  
where \(y_{t}\), \(C\), \(p\), \(\phi_{i}\), and \(\varepsilon_{t}\) denote the stationary time series, constant term, autoregressive order, non-zero autocorrelation coefficients, and independent error term, respectively. (e) Moving Average  
Moving average (MA) is a simple smoothing prediction technique used for time-series data that calculates a moving average over a certain number of terms to reflect long-term trends [36]. However, it is difficult to discern the development trend when time-series data are influenced by periodic and random variations causing large fluctuations. Using MAs can eliminate these influences and reveal the direction and trend of the events, which is expressed as:  
Figure 10:5  
where \(y_{t}\), \(\mu\), \(q\), and \(\theta_{i}\) denote the stationary time series, mean of the sequence, moving average order, and non-zero autocorrelation coefficients, respectively. (f) Autoregressive Integrated Moving Average  
Autoregressive integrated moving average (ARIMA) is an evolution of the AR, MA, and autoregressive moving average models. This approach is used to analyze non-stationary time-series data by transforming them into stationary data through differencing [37]. This method is employed when dealing with non-stationary time-series data that exhibit a changing mean and variance over time. A new stationary time series can be obtained by using the differences in the data, and a suitable probabilistic model can be derived from historical data to represent the dependence between time and data. ARIMA can be expressed as ARIMA (\(p\), \(d\), \(q\)), where \(p\), \(d\), and \(q\) denote the autoregressive, differencing, and moving average orders, respectively. Furthermore:  
Figure 10:6  
where \(\phi_{i}\) and \(\theta_{i}\) denote non-zero autocorrelation coefficients. (g) Seasonal Autoregressive Integrated Moving Average  
The seasonal autoregressive integrated moving average (SARIMA) model incorporates seasonal factors into the ARIMA model [38]. Generally, the SARIMA model is denoted as SARIMA (\(p\), \(q\), \(d\))(\(P\), \(Q\), \(D\))s, where \(s\), \(P\), \(Q\), \(D\) denote the seasonal period and seasonal autoregressive, seasonal moving average, and seasonal differencing orders, respectively. This is expressed as:  
\[