Chapter10 Damage Detection Using Large-Scale Covariance Matrix Luciana Balsamo, Raimondo Betti, and Homayoon Beigi Abstract Statistical pattern recognition based structural damage detection is often developed exploiting the methods of outlier analysis. In this context, damage occurrence is assessed by analyzing whether a set of features extracted from the response of the system under unknown conditions departs from the population of features extracted from the response of the healthy system. The metric dominantly used for this purpose is the Mahalanobis Squared Distance (MSD). Evaluation of MSD of a point from a population requires the use of the inverse of the population’s covariance matrix. It is known that when the feature dimensions are comparable or larger than the number of observations, the covariance matrix is ill-conditioned and numerically problematic to invert in the former case, while singular and not even invertible in the latter. To overcome this difficulty, three alternatives to the canonical damage detection procedure are investigated: data compression through Discrete Cosine Transform, use of pseudo-inverse of the covariance matrix, and use of shrinkage estimate of the covariance matrix. The performance of the three methods is compared using the experimental data recorded on a four story steel frame excited at the base by means of the shaking table available at the Carleton Laboratory at Columbia University. Keywords Damage detection • Large-scale covariance matrix • Discrete cosine transform • Pseudo inverse • Shrinkage covariance matrix 10.1 Introduction Statistical pattern recognition based structural damage detection is the task of assessing damage occurrence using information extracted from the structural response. It is developed by first learning the patterns drawn by such information when extracted from the system under healthy conditions, and by then comparing the learnt patterns with the patterns drawn by the same information extracted from the response of the system under unknown conditions: if the new patterns depart from the learnt ones more than a prescribed threshold, the structure is declared damaged. The information extracted from the structural response are known as damage sensitive features (dsf). Damage sensitive features need to be sensitive to structural changes due to damage, while remain insensitive to structural changes due to external effects, like environmental or operational conditions. The process of learning the patterns drawn by the damage sensitive features extracted from the response of the healthy system is known as training, while the process of comparing the trained features with those extracted from the structure under unknown conditions is named testing. One approach to measure the departure of the two populations of features is to evaluate the squared Mahalanobis distance of the testing features from the trained ones [1, 2]. Let us denote as x a p-dimensional point representing the testing feature vector, and by and † the mean vector and covariance matrix of the trained feature population, the Mahalanobis squared distance of x from the trained model is defined as: D.x/ D.x /T† 1.x /: (10.1) L. Balsamo ( ) • R. Betti Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA e-mail: lb2591@columbia.edu H. Beigi Recognition Technologies, Inc. White Plains, NY 10601, USA A. Wicks (ed.), Structural Health Monitoring, Volume 5: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04570-2__10, © The Society for Experimental Mechanics, Inc. 2014 89
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