## 3 Experimental Results and Analysis
### Testing for Damage Assessment (O2)
The stochastic output-only CC-AR representation relating the current (at time instant \(t\)) value of the modeled output signal to older ones has the following generic form:  
\[ \begin{array}{l}y[t]=\sum\limits_{i=1}^{L}\theta_{i}p_{i}[t]+e[t]\\ e[t]\sim NID\Big{(}0,\sigma_{e}^{2}\Big{)}\end{array}  \tag{1}\]  
with \(t\) designating the normalized discrete time and \(y[t]\), \(e[t]\) the output and one-step-ahead prediction error [or residual, assumed to be a zero-mean uncorrelated sequence with variance \(\sigma_{e}^{2}\) signals, respectively. The term \(NID(\mu,\sigma_{e}^{2})\) stands for Normally Independently Distributed (with the indicated mean \(\mu\) and variance \(\sigma_{e}^{2}\)). The terms \(p_{i}[t]\) are referred to as regressors and involve lagged output values ranging from \(y[t-1]\) up to \(y[t-ny]\), with \(ny\) designating the maximum value of lags admitted for the signal \(y[t]\) in (1) (model order). The \(i\)-th regressor coefficient is noted \(\theta_{i}\) and has constant value. For instance, a CC-AR representation designated as \(y[t]=0.3\cdot y[t-1]+0.7\cdot y[t-2]-0.1\cdot y[t-3]+e[t]\) has an order \(ny\) equal to 3 and is usually referred to as a CC-AR(3) model. Its regressor coefficients are \(\theta_{1}=0.3\), \(\theta_{2}=0.7\) and \(\theta_{3}=-0.1\). The objective is to identify the CC-AR representation, that is, to select the regressors \(p_{i}[t]\) and to estimate the associated coefficients \(\theta_{i}\) in order to most effectively represent the dynamics of the considered signal. The identification of CC-AR representation along with the estimation of coefficients \(\theta_{i}\) are trivial procedures extensively presented in, among others, [21]. The interested reader may refer to this work for details on determining the model order and estimating the coefficients. Briefly, a signal is effectively modeled by a CC-AR(\(ny\)) representation if one determines the lowest \(ny\) value for which the associated CC-AR(\(ny\)) model accurately describes the frequency content of the signal (by assigning poles at those frequencies) and the residual sequence \(e[t]\) is uncorrelated. The latter means that the values of \(e[t]\) for various \(t\) feature insignificant mutual dependencies, as checked by specific hypothesis tests [21]. At this point, note that identifying models for fault diagnosis applications is a somewhat different task than performing identification for purposes such as control, signal pre-whitening and so on. For fault diagnosis, the identified model should include poles with frequencies as close as possible to the peak locations (frequencies) of interest in the power spectrum plot. At the same time, the identified CC-AR(_ny_) model should have as few regressors as possible: Extra regressors (over-modeling) may lead to fewer correlated residuals \(e[t]\) but tend to assign poles even at frequency spikes corresponding to noise or other parasitic phenomena. Over-modeling should, thus, be avoided, as far as possible. In the current case, these peaks of interest were located at 40 and 50 KHz. Ideally, the user should determine the value range for _ny_, for which all associated identified CC-AR(_ny_) models include poles at the desired frequency peaks. In other words, one should plot CC-AR(_ny_) poles at each frequency as a function of the corresponding value for _ny_ (stabilization plot), and select a value range for _ny_ leading to obtaining poles at the desired frequency regions. At the same time, these CC-AR(_ny_) models should produce residuals as uncorrelated as possible. Ideally, the model structure selected should have the smallest _ny_ of those in the range, thus avoiding any over-modeling issues. In Figure 10 typical stabilization plot from a type-C1 test data and another from a type-C2 slabs are shown. The considered values for _ny_ ranged from 4 up to 180. With data from either -C1 or -C2 slabs, for _ny_\(\geq\) 80 the associated CC-AR(_ny_) models consistently assign poles at frequencies around 40 and 50 KHz.