# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 2 Proposed Method
### Hyperparameter Optimization  
The modes identified and the corresponding modal parameters strongly depend on the hyperparameters used in the modal parameter estimation step (see Table 1). The key idea behind this publication is to find optimal values for these parameters, whereas the recent literature has dealt with deriving them based on the input data set [12; 15; 16]. On the one hand, the presented approach keeps the solution space as widely open as possible. On the other hand, it ensures a high concordance between the original input data and the resulting modal model. A variety of optimization algorithms exist in the literature. The decision on which depends, amongst others, on the amount of knowledge on the system being optimized, the availability of efficient gradient calculations, and the shape as well as the (computation) costs of evaluating the objective function. The specific objective \begin{table}
\begin{tabular}{l l l} \hline \hline
**Step** & **Symbol** & **Description** \\ \hline Pole calculation & \(p_{max}\) & Maximum model order \\ \hline Classification & \(k_{Af}\) & Maximum relative frequency difference of two poles \\  & \(k_{AD}\) & Maximum relative damping difference of two poles \\  & \(k_{Dmin}\) & Minimum damping of a mode \\  & \(k_{Dmax}\) & Maximum damping of a mode \\ \hline Clustering & \(d_{max}\) & Maximum frequency distance within a cluster \\  & \(q_{min}\) & Minimum percentage of the model order of poles per cluster \\  & \(k_{tail}\) & Maximum euclidean frequency distance factor between poles \\ \hline Pole selection & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of hyperparameters used in the modal parameter estimation; the classification parameters are defined similar to [PERSON] et al. [12] and [PERSON] et al. [15]. Figure 3: Identified modes by agglomerative clustering with a maximum frequency distance \(d_{max}\) of poles within a cluster; different clusters, that is, distinct modes, are indicated by different colors. The finally selected poles are marked with a cross. function used will be described below and will involve running the modal parameter estimation (see Section 2.1). As the relationship between the hyperparameters and the resulting modal parameters is rather complex, no gradients are available since \(p_{max}\) is a discrete number, the objective function is generally non-convex, and the costs of evaluating the objective function can be high (depending on the chosen maximum model order \(p_{max}\) and the size of the input \(\mathtt{FRF}\) data set), the Bayesian optimization approach was chosen here [18]. In the context of this method, the objective function is generally treated as random and approximated with a Gaussian process regression model. Previous objective function evaluations are treated as prior knowledge and are considered as a prior. New function evaluations increase the knowledge about the objective function, which is then used to update the prior, forming the so-called posterior. The posterior, in turn, can then be used in several ways to determine the location of further new, promising function evaluations. Those locations are determined with the help of an acquisition function. In this case, the so-called upper confidence bound optimization policy was applied, allowing the balance between exploitation and exploration to be easily set. A comprehensive guide on Bayesian optimization can be found in [PERSON] [18]. The AutoEMA method tries to find values for the hyperparameters (see Table 1) such that the identified modal parameters and the corresponding modal model best replicate the original input data, which is given in the form of FRFs.