26 M. Horner and S.N. Pakzad In Eqs. (4.6) through (4.12), the expectations are now included as the data may be incomplete. Given a current iteration’s guess at the two parameter vectors, the kth missing observation is estimated for the output y(k) and input u(k), respectively, with Eq. (4.10) below: y.k/ D'T 1 .k/ 1 u.k/ D'T 2 .k/ 2 (4.13) The above equations define the EM algorithm for the ARX model. The process involves defining initial unknown parameter guesses, completing the datasets based on those guesses, computing the maximum likelihood parameter values based on those completed datasets, and iterating until convergence. The differentiations in Eqs. (4.6), (4.7), and (4.9) through (4.12) mean the algorithm may converge to a local maximum rather than global, a characteristic of the EM algorithm in general. Therefore, the closer the initial guesses to the actual parameter values, the more likely the convergence to the global maximum parameters. As of this analysis, several randomly-generated initial guesses were used in multiple runs of the algorithm with the intention of selecting the converged parameter estimates that appear most often. 4.3 Damage-Sensitive Features The success of a damage detection methodology depends on the sensitivity of the selected dataset features to changes in structural behavior due to damage. These features must be able to distinguish changes due to damage from environmental changes typical in real-world applications. A missing data analysis for structural identification, like that presented in [16] could in theory use the identified frequencies, mode shapes, or damping ratios as damage-sensitive features. However, there is conflicting evidence in the literature on the effectiveness of these features for damage detection. Fundamental frequency shifts have been used for damage identification in [18], but these shifts are not able to localize damage, and may not be as sensitive as other features, such as damping. Damping, however, is well-known as harder to accurately predict than other structural characteristics, and often is associated with larger error [19]. Finally, mode shapes have some benefit in local damage sensitivity, but are as of yet not experimentally validated for this use [13]. Damage-sensitive features associated with regression model parameters have not yet been used in mobile sensing applications, but their computational and damage identification benefits over modal property shifts are discussed in [20]. Autoregressive (AR), ARX, and autoregressive-moving-average (ARMA) models are mentioned. A sensitivity analysis associated with these regression model parameters is provided in [21]. AR models are not considered in this paper, but for a similar derivation of the EM algorithm for their estimation, the reader is referred to [22]. Along with collinear regression (CR), and single-variate regression (SVR) models, AR and ARX models are used for damage detection with complete fixed sensor network data taken from the same laboratory frame as in this paper in [23]. One such feature will be considered in this paper, deemed the Angle Coefficient in [23], which measures the angle between regressed lines from two different system states. It is computed as in Eq. (4.14), where v and v0 are vectors for undamaged and unknown system states, respectively, and are of the formh 1 a 1 an b1 bn i T for the ARX model presented in Sect. 4.2. Dcos 1 ˇ ˇ ˇ ˇ v v’ kvkkv’k ˇ ˇ ˇ ˇ (4.14) Once damage-sensitive features are extracted, it is important to distinguish whether or not a change indicated by these features is due to random variations of the measurements over time, or if a damaging event has occurred to alter the measurements. In [23], several healthy and damaged test runs are conducted from which damage-sensitive features are extracted, then change point analysis is conducted with a normalized likelihood ratio test (NLRT) [24] or two-sample t-test [25] to detect the test number where damage was first simulated. In this paper, angle coefficient means are determined for a series of undamaged structural scenarios, and for a set of damaged structural scenarios. The shifts in these means are explored for different locations on the structure, elaborated upon in Sect. 4.5.
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