# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 2 Proposed Method
### Modal Parameter Estimation  
The first step is calculating pole candidates for all model orders \(p_{i}=1,2,\ldots,p_{max}\) up to and including the maximum model order \(p_{max}\). In the present research, this is done using the PolyMAX algorithm [9], but a variety of other methods are also applicable (see Section 1). The found pole candidates are then classified as certainly spurious and potentially physical by applying four criteria: The relative difference in frequency of a pole of order \(p_{i}\) to any pole of model order \(p_{i-1}\) must be smaller than \(k_{\Delta f}\) (a), the relative difference in damping of a pole of order \(p_{i}\) to any pole of model order \(p_{i-1}\) must be smaller than \(k_{\Delta D}\) (b), and the pole's damping must be larger than \(k_{Dmin}\) (c) and smaller than \(k_{Dmax}\) (d). If all conditions (a) to (d) for a given pole are met, it is classified as potentially physical, otherwise as certainly spurious. Figure 2 exemplarily shows the frequency of all potentially physical poles of a given frequency response function (FRF).  
Spurious poles are discarded and thus excluded from further analysis. Given that many model orders were considered, there are generally several remaining poles for any mode shape being identified. Agglomerative clustering is used to group poles into modes. Given that this is a hierarchical clustering algorithm, the algorithm determines the number of clusters. However, a threshold has to be set, in this case the maximum frequency distance of poles within a cluster \(d_{max}\). If the clustering is applied to the poles in Figure 2, six possible modes are found, as illustrated in Figure 3.  
Two more rules were applied in order to avoid clusters having spurious poles only. First, it was assumed that clusters reflecting actual modes contain at least a certain percentage of the model order \(p_{i}\) of poles, in this case \(q_{min}\). This rule directly corresponds to the original assumption that physical poles show up at many model orders [14]. Second, the variance in frequency of poles of spurious-only clusters is assumed to be higher than the one from a cluster reflecting an actual mode. Thus, all clusters with an average Euclidean distance larger than \(k_{eukl}\cdot d_{max}\) are discarded, where \(d_{max}\) is the maximum frequency distance of poles within a cluster from the clustering stage and \(k_{eukl}\) is an euclidean frequency distance factor.  
Figure 1: Flowchart showing the AutoEMA method; the modal parameter estimation is based on a set of hyperparameters explained in Table 1. Their value is determined by a Bayesian optimization strategy such that the resulting model optimally replicates the original input data.  
Figure 2: Exemplarily FRF and potentially physical poles identified after applying the PolyMAX algorithm and criteria (a) to (d) (see Section 2.1) for the given hyperparameters \(k_{Af}\), \(k_{AD}\), \(k_{Dim}\), and \(k_{Dmax}\).  
Finally, one pole per cluster can be selected by choosing the pole closest to the median pole location of a cluster, which is indicated by crosses in Figure 3. The pole determines both eigenfrequency and modal damping of the mode corresponding to the cluster. After all poles have been selected, the mode shapes are calculated using the least squares frequency-domain method [9].  
Table 1 summarizes the hyperparameters involved in the modal parameter estimation. The code used in this work is based on the Python package pyEMA[17].