Unsupervised Mode

In most practical implementations of SHM, only undamaged datasets are avail- able. Therefore, most SHM concepts focus on employing SHM methods which assess datasets based exclusively on information gained from the undamaged structure. In this section, the concept of detecting structural changes based on the locations of TF poles is presented in the context of unsupervised machine learning. More specifically, in the current thesis, TF pole migration is investigated in the context of the three-tier SHM framework, which was presented in section 2.4. [36].

Firstly, the appropriate model order is selected by comparing the MSE of a train- ing dataset to that of a validation dataset, as described in the previous section. In the training phase, datasets from different healthy system states are assigned to clus- ters depending on their EOCs (tier 1-ML). Data clusters can either be predefined by the user (manual clustering) or generated by automatic clustering methods. CPs are calculated for the data contained in each cluster (tier 2-CP). In HT, probabilistic models are set up for the CPs of each cluster obtained in ML based on the EOCs (tier 3-HT).

In the testing phase, new incoming datasets are assigned to the existing clusters (tier 1-ML) and their CPs are calculated (tier 2-CP). Relative CPs are calculated using information from the reference datasets defined during the training phase. In this work, the reference values of each cluster occur as the mean values of the corre- sponding quantities coming from all the datasets contained in each cluster. Finally, in HT, the extracted CPs are compared to cluster statistics and are evaluated within different probabilistic models for different confidence intervals (tier 3-HT).

At this point, a new CP, which makes use of the TF pole locations, is defined and assessed using the three-tier SHM framework. It was previously shown that TF poles are essentially system zeros, and since the location of system zeros is affected by the input location, data clustering is necessary. Hence, in the training phase, datasets are grouped in clusters either by manual clustering or by automatic clustering algorithms in order to compensate the effect of prevailing EOCs. In tier 2, output-only ARX models are built for datasets obtained from different healthy states, i.e., datasets from the healthy structure acquired under different EOCs. The TF poles of all datasets are calculated and superimposed on the z-plane for each cluster obtained during tier 1. Subsequently, for each clustering case of tier 1, TF poles of the healthy states are clustered upon the z-plane using a clustering algorithm, such as k-means clustering or AP. Thus, several TF pole clusters emerge. Each TF pole cluster is described by the mean value µ and the 3σ level curves of a bivariate normal distribution. Alter- natively, the level curves can represent other multiples of the standard deviation σ or the 100th_{percentile of the bivarite distribution. The two variables of the normal} distribution are, in this case, the real part and the imaginary part of the TF poles.

In HT, decision boundaries can be defined for a series of confidence regions. As shown in section 2.3.1.2, the Mahalanobis squared distances (MSDs) of the points of a bivariate normal distribution follow a χ2_{-distribution. Consequently, the decision}

boundaries of the new CP correspond to contours of constant probability (1 − a), which are obtained as the percentiles of the χ2_{-distribution (see Eq. 2.71). This se-} lection of decision boundaries results in accurate detections, assuming that the TF poles of each cluster are normally distributed. If this assumption of Gaussianity does not hold, other decision boundaries have to be defined. To be more specific, the MSD between each point in a TF pole cluster and the distribution of the TF pole cluster is calculated. The percentiles of the MSDs of the TF poles contained in a cluster can be used as thresholds for classifying the TF poles of new datasets based on their MSD value.

In testing, new datasets are evaluated one at a time. First, the datasets are assigned to the existing clusters of tier 1, depending on the EOCs they correspond to. Subsequently, the output-only ARX model is built for each test dataset and the TF poles are calculated (tier 2). The TF poles of the new datasets are assigned to their closest TF pole clusters and the MSDs between the TF poles and the corre- sponding clusters are calculated. In HT, the datasets are assessed for two cases: (i) for the assumption that TF poles within a cluster are normally distributed and (ii) for the case that the TF pole distribution deviates from Gaussianity. In the first case, the MSDs of the current datasets are assessed with respect to the percentiles of the χ2_{-distribution. In the latter case, the MSDs of the current dataset are assessed} with respect to the percentiles of the MSDs within the clusters obtained in training. Hence, the MSD can be conceived as a new CP, which indicates whether the location of the TF poles has changed significantly. Figure 3.6 depicts the concept of evaluating the TF pole of a dataset by calculating the MSD to its closest cluster. The current dataset can be assessed for a series of confidence regions, which are equivalent to confidence intervals for univariate distributions.

This procedure is followed for all TF poles of a dataset, i.e., for ncl TF poles, where ncl is the number of pole clusters, and yields ncl damage indices, which can either be conceived as seperate CPs or can be used to build one CP. In the latter case, the CP hypothesis is (i) the majority vote of the decisions derived from HT of the individual pole clusters and (ii) H1, i.e., a positive detection is returned, if at least one of the TF pole clusters indicates changes of the TF pole location.

Several output-only ARX models can be built, when several structural responses are available. Each of these uses different inputs and outputs and, therefore, expresses another relation between input and output. As a result, the TF poles of each model are different. The number of TF poles, on the other hand, depend on the order of the ARX model. Hence, the number of TF poles are the same regardless of the clustering case used in tier 1 of the SHM framework. The clustering case, however, affects the cluster characteristics, i.e., the cluster mean and standard deviation.

μ

σ

σ

x

θ

cluster of healthy state

pole of current dataset

Figure 3.6: Ellipse describing the TF poles of the healthy state, and TF pole of the current

dataset, which is evaluated with respect to the distribution of the healthy state. The figure depicts: the cluster means (µh), the standard deviations in the principle axis direction (σ1h) and in the perpendicular direction (σ2h) and the angle of the ellipse principle direction with respect to the horizontal axis (θh)