4 J. Kullaa Eqs. (10) or (11). A covariance matrix of the time-shifted training data is estimated. The covariance matrix is subjected to a whitening transformation for data normalization. When new data (ACFs) arrive, the same whitening transformation is applied to these data. If the dynamic characteristics of the structure have changed, the new data are assumed to appear in the noise space and thus outside the hypersphere (for Gaussian data). This can be detected by applying principal component analysis to all data. The first principal component is now dictated by the data points outside the hypersphere. The first principal component scores are only retained, reducing the data dimensionality to one. Extreme value statistics (EVS) distributions [16] are identified for the scores of the training data by dividing the data into equal sized subgroups. The extreme values of each subgroup are plotted on a control chart with control limits determined from the probability of a false alarm [17]. If the data points of the test data are outside the control limits, an alarm is raised. Numerical Experiment The structure being monitored was a bridge deck with a concrete slab and steel stiffeners (Fig. 2). A detailed description of the model can be found in [18]. The first 7 modes were active, having natural frequencies of 3.95 Hz, 5.35 Hz, 13.7 Hz, 15.4 Hz, 17.9 Hz, 24.1 Hz, and 24.8 Hz. The corresponding damping ratios were 0.01, 0.01, 0.02, 0.02, 0.03, 0.03, and 0.03. The sampling interval was ∆t =0.01 s and the measurement period was almost 44 min including 218 =262144 samples. A long measurement period is necessary to provide accurate ACF estimates [19]. Fig. 2 Finite element model of the bridge deck with labeled sensor positions. Damage location is close to sensor 11 Because environmental influences were not considered in this study, the excitation was the primary cause of varying the ACFs (operational variability). Some variability came from the measurement noise, which was also different in each measurement. The number of independent excitations varied randomly between 1 and 3. Furthermore, the excitation points were randomly selected from all nodes at the joints between the steel webs and flanges. The load histories were generated in the frequency domain between 0 and 25 Hz. Each measurement had different load functions with random amplitudes and phases at each frequency pin. In addition, it was randomly determined if a loading had zero contribution at a randomly located 5 Hz interval. This simulated measurements in which some modes were not excited. The FE analysis was performed in the frequency domain [20]. Measurement noise was simulated by adding Gaussian noise to each sensor. The average signal-to-noise ratio (SNR) was 30 dB. The ACFs were estimated from the long response time histories at lags τ between 0 and 3 s, resulting in a length of 300 data points in each measurement (Fig. 4a). In the data analysis for damage detection, the ACFs were concatenated to form a long data matrix (Fig. 4b). To remove discontinuities between measurements, mdata points had to be removed from each data set. The first 100 data sets were from the undamaged structure and the next 36 data sets from the damaged structure with an increasing crack length (Fig. 3). The number of different crack lengths was 6, and each damage level was monitored with 6 simulations. The first 70 data sets were used as training data.
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