## 3 Proposed Method
### Meta-Learning for Rapid Training of Multi-Machine Models for Anomaly Detection and Prediction
The proposed anomaly detection models trained on the anomaly detection dataset could only detect current and past data anomalies. Therefore, an additional anomaly prediction model was required for future anomaly states. The historical data of the AI degradation level index obtained from anomaly detection were compared with various commonly used prediction models. Finally, the Holt-Winters algorithm was applied to the anomaly prediction model [7]. Various anomaly prediction algorithms are introduced below and the overview is in Table 3. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Algorithm** & **Overview (or Main Features)** \\ \hline Simple exponential smoothing (SES) & By placing a large weight on the most recent data, only two values and constants are needed to predict the next period of value \\ \hline Holt (Holt’s linear trend) & Holt’s linear trend, which predicts trends in data, consists of a prediction equation and two smoothing equations for sequences that have a linear trend and no seasonality \\ \hline Holt–Winters (Holt–Winters seasonal method) & The Holt–Winters method forecasts time series with both trends and seasonality \\ \hline Autoregressive (AR) & The historical data of the variable itself are used to predict its own data, and the autoregression must meet the requirements of stationarity \\ \hline Moving Average (MA) & A simple smoothing forecasting technique that calculates the sequence average of a certain number of items in turn according to the time-series data and the passage of time items to reflect the long-term trend \\ \hline Autoregressive integrated moving average (ARIMA) & In the case analysis of non-stationary time series, the originally non-stationary time series becomes a stationary time series after many differences \\ \hline Seasonal ARIMA (SARIMA) & ARIMA (differentially integrated moving average autoregressive) time-series-forecasting method with seasonal periodicity \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overview of time-series algorithms. Figure 10: Structure of the autoencoder. (a) Simple Exponential Smoothing (SES)  
This algorithm is used when there is no clear trend or seasonal pattern in the predictive data [33; 34]. The prediction is calculated using weighted averages, meaning that the largest and smallest weights are associated with the most recently and least recently observed values, respectively. This is expressed as:  
\[\hat{y}_{T+1|T}=\alpha y_{T}+\alpha(1-\alpha)y_{T-1}+\alpha(1-\alpha)^{2}y_{T-2}+\ldots \tag{9}\]  
where \(\hat{y}_{T+1|T}\) denotes the one-step-ahead forecast for time \(T+1\), \(y_{T}\) denotes the most recent observation, and \(0\leq\alpha\leq 1\) denotes the smoothing parameter. (b) Holt (Double Exponential Smoothing Method)  
The Holt double exponential smoothing method is an extension of the simple exponential smoothing method that predicts trends in data [9]. This method is suitable for linear trending sequences without seasonal patterns and consists of one prediction and two smoothing equations that represent the level and trend components (\(l_{t},b_{t}\)), which are respectively expressed as:  
\[\hat{y}_{t+h|t}=l_{t}+hb_{t} \tag{10}\]  
\[l_{t}=\alpha y_{t}+(1-\alpha)(l_{t-1}+b_{t-1}) \tag{11}\]  
\[b_{t}=\beta^{*}(l_{t}-l_{t-1})+(1-\beta^{*})b_{t-1} \tag{12}\]  
where \(y_{t}\) and \(l_{t}\) denote the observed value and level at time \(t\), respectively. \(b_{t}\), \(h\), \(\alpha\), and \(\beta^{*}\) denote the trend at time \(t\), weight for the level (\(0\leq\alpha\leq 1\)), and weight for the trend (\(0\leq\beta^{*}\leq 1\)), respectively. (c) Holt-Winters Forecasting (Triple Exponential Smoothing)  
Holt-Winters forecasting, also known as triple exponential smoothing, is a method used to predict the behavior of time-series data that includes trends and seasonality. This algorithm considers three factors: the level \(l_{t}\), trend \(b_{t}\), and seasonal component \(s_{t}\). It is effective for forecasting time-series data with seasonality patterns.