PSI - Issue 84

Fulvio Busatta et al. / Procedia Structural Integrity 84 (2026) 797–804

803

5. Anomaly detection using different control charts The natural frequency tracking shown in Fig. 5a suggests that the modal parameters of the Torrente Reggia bridge are subjected to EOV effects. Therefore, EOV effects must be filtered out before any anomaly detection. In this regard, a well-assessed procedure is based on the PCA algorithm (Sharma, 1995), that is a multivariate technique capable of finding out common features in the variability of the identified natural frequencies. Once trained, within a sufficiently long period covering a wide range of EOV, the PCA regression model reconstructs the natural frequencies. Thus, the residuals between identified and reconstructed natural frequences should reflect structural changes only. In the following, the PCA residuals are used to build both the control charts based on multi-variate Mahalanobis distance (Mahalanobis, 1936) and T 2 -Hotelling (Hotelling, 1947). Unlike the PCA-based control charts, the Cointegration technique (Cross et al., 2011) has been recently adopted for novelty detection in Civil Engineering. Given M non stationary time-series Y k  N  M (e.g. N observations of M natural frequencies), they are said to be cointegrated if a stationary linear combination  k  N of Y k exists (Eq. 1):

(1)

The unknown coefficients  j ( j = 1, …, M ) in Eq. (1) can be found using the Johansen procedure (Johansen, 1988) during the training period in which the structure is supposed to be in normal operating condition under usual EOVs. Johansen procedure evaluates a Vector Error Correction Model of order p from Y k , as follows:

(2)

where the matrices  N  M and B j  N  M describe, respectively, the long-run equilibrium and short-run adjustment of the time-series Y k in order to maintain them in equilibrium. A very-refined maximum-likelihood estimation aims to find the coefficients  j (Eq. 1) that are in the first row of the matrix  (Eq. 2). a)

5.2%

b)

5.2%

c)

42.9%

Fig. 7. Comparison between (a) Cointegration procedure with (b) Mahalanobis distance and (c) T 2 -Hotelling control charts obtained from the residuals of frequencies (PCA) by using a training period of 9 months (UCL and LCL are set on the 95 th percentile).