# An Optimized Fed-Batch Culture Strategy Based on Multidimensional Time Series Aggregation
## 3 Results and Discussion  
The evaluation of the proposed profit function prediction method is performed with the data of industrial 2-KGA fermentation. In Equation (13), the predicted profit function \(\hat{f}_{pre}\) is related to the choice of parameter \(k\). When the time series of the current online batch is compared for similarity distance to all other batches in the historical database, the nearest \(k\) batches are utilized for the calculation of the predicted profit function using the proposed methods in Equations (11)-(14). To find out the effect of the \(k\) value on the prediction results, a total of 600 historical batches were split into two parts: the 100 batches with the closest dates were used for model validation, whereas the remaining batches were used for the retrieval of similar batches. The model's prediction error (root mean square error, RMSE) with different values of \(k\) (from 1 to 10) are compared. Figure 5 presents the mean and standard deviation of prediction error. When \(k=6\), the minimum mean of prediction error is achieved. It is noteworthy that despite the marginally elevated standard deviation observed with \(k=6\), this value is still deemed optimal, as the foremost objective of the feeding strategy is to optimize the predictive accuracy of the profit function. Therefore, the application of \(k=6\) is chosen to minimize the prediction error, and then the predicted profit function \(\hat{f}_{pre}\) of an online batch can be obtained before the feeding operation. For model testing, the proposed method is applied to 90 industrial fermentation batches, and the process data are collected. The prediction error distribution with different \(k\) values for these batches is compared in Figure 6. With the choice of \(k=6\), the mean and standard deviation of the prediction error for these batches are 1.91% and 2.06%, respectively. Figure 5: Prediction error of the profit function with different values of \(k\). To validate the proposed approach's efficacy, other feasible methods have also been applied to predict the profit function, with the same limitation that only the process data before the feeding operation can be utilized. The Artificial Neural Network (ANN) [36; 37], Support Vector Machine (SVM) [38; 39], Random Forest (RF) [40; 41], and eXtreme Gradient Boosting (XGBoost) [42; 43; 44] have been shown to achieve state-of-the-art results in many applications with sequential data or time series. The moving data window technique is utilized to generate input-output data pairs for the models [45]. The data pair corresponding to the time stamp of feeding operation \(\mathit{Tv}\) is presented with Equations (17)-(19). \[X=\left[T_{\mathit{Tv}}x_{\mathit{Tv}-\mathit{\tau}}x_{\mathit{Tv}-\mathit{ \tau}}x_{\mathit{Tv}-\mathit{Tv}_{\mathit{D}}}\right]^{T} \tag{17}\]  
\[x_{\mathit{Tv}}=\left[P(T_{\mathit{v}})\mathit{Alk}(T_{\mathit{v}})\right]^{T} \tag{18}\]  
\[Y=J\left(t_{i,f}\right) \tag{19}\]  
where \(\mathit{\tau}\) is the time interval of the variables for discretization,\(t_{i,f}\) and \(J\left(t_{i,f}\right)\) are the termination time of the \(i\)th batch and the profit function at \(t_{i,f}\), respectively. The input variable consists of product formation (\(P\)) and alkali consumption (\(\mathit{Alk}\)). Since the feeding time varies between different batches, \(T_{\mathit{D}}\) is set to 20, so that the width of the input data window for each batch is consistent. The data pairs generated from the 600 historical batches are utilized for the training and validation of the other mentioned models. The testing of these models is performed with the same 90 industrial batches with the proposed multidimensional time series aggregation method (MTSA, \(k=6\)). Specifically, a classical three-layer back propagation network is employed in the ANN model, which encompasses three neurons in the hidden layer. In the SVM model, a three-degree polynomial kernel is utilized, and the regularization parameter C is set to 1.0.