## 6 Results  
### Regression Using Artificial Neural Networks  
Typically, the measured EIS spectra are used to fit the parameters of an equivalent circuit that describes the chemical processes of the cell. Due to some components of the equivalent circuit having nonlinear behavior, minimizing the squared error does not result in a system of equations that allows a direct solution using linear regression. Instead, a Newton-Raphson-like algorithm must be used to iteratively approach the minimum of the error function. This step is necessary for each evaluation of a newly measured spectrum and therefore takes a lot of computational resources. To avoid an under determined system of equations, it is also necessary to measure at least as many frequencies as parameters [41]. This may also increase the time needed. Following the identification of the parameters of the electrical equivalent circuit, it is then still necessary to relate the model parameters to the desired SoC, for example with a quadratic approximation. An analysis of the EIS data based on the electrochemical model does provide insights into the chemical processes of the cell. However, insofar as the goal is only to determine the SoC, this comes at the expense of speed and the necessary computing power for each evaluation. ANN can offer an advantage here. While training the network still takes its time, evaluation in the field is fast and easy, not least as the SoC can be modeled directly. Thus, in a short experiment on a desktop PC, a speedup of about 3 orders of magnitude could be achieved. An evaluation of the equivalent circuit, for example, takes on average 82.23 ms, while the evaluation of the trained neural network took only 51.86 us. Typically, BMS use microcontrollers with much less computing power, making the evaluation there require correspondingly more computing time. Moreover, ANN have already been used successfully for the analysis of EIS spectra [42; 43]. As at this point it is intended to investigate whether the models also generalize to an unseen DC current component (=working point), the training data consist of all measurements which have taken place at the working points 1 A and 3 A, while the spectra which have been measured at a DC current of 2 A form the test data set. As the measurements have been performed alternating, the ratio of training to test data is 2:3. In a first approach, the feature vector, as described in Equation (23), is obtained from the DC working point and from the measured impedance data. For the regression of the SoC, multilayer perceptron networks are used, which have a hidden layer in addition to the input and output layer. As the neural network is sensitive to feature scaling, for each feature, the mean is removed and scaled to a variance of \(\sigma^{2}=1\). Impedance values at similar frequencies correlate strongly with each other. Therefore, it is additionally analyzed whether a reduction in the feature vector by means of a principal component transformation leads to better results or to a better generalization. Here, the principal component transformation is only applied to the measurement data of the spectrum, as the DC operating point is very important for the generalization, but has only a small variance due to the two different values in the training dataset. The quality of the resulting model significantly depends, among other factors, on the hyperparameters used. However, the use of a principal component transformation also affects the optimal values of the hyperparameters. Therefore, for each number of principal components, optimized values for the number of neurons and the L2 regularization term \(\alpha\) are found using the Grid Search procedure in combination with cross-validation. In Grid Search, a grid is spanned containing all defined combinations of parameter values. These are used sequentially by cross validation to train a model and test on unseen data. In this process, the training data is divided into five groups.