294 L. A. Bull et al. contributed to training-set. The MSD is useful in this application, as it considers both the mean and the variance of the regression. Therefore, the averaged, univariate MSD is defined for the 1000-point random-sample from each FRF in the test-set, 500 samples from both the real and imaginary parts, MSDi = 1 V[y∗] (E[y∗]− ˜yi) 2 (32.12) MSDFRF = 1 1000 1000 i=1 MSDi (32.13) The expected values are defined by the predictive equations of the GP in (32.5), and the tilde is used to denote the experimental or simulated output, sampled from the test data. In order to define a detection threshold, which flags an FRF in as either inlyingor outlying(i.e. normal or novel), bootstrapsampling is used [12, 16]. This defines the threshold by randomly sampling 1000 points from the normal-condition data used to train the form. The MSDFRF is then calculated according to (32.12). These steps are repeated for a large number of trials, and the resulting MSDs are sorted in order of magnitude. The critical value is the threshold which contains 95.45% (two-sigma) of the MSDvalues beneath it. 32.3.3 Results Results for novelty detection across the population via. the form are shown in Figs. 32.5 and 32.6; these plots can be interpreted as a control chart, each sub-figure representing individual member. Figure 32.5 presents the MSD values (corresponding to test FRFs) for members {S1, . . . ,S10}; these members contributed (a separate set of) normal-condition data used to train the form, shown in Fig. 32.4. The MSD values for members {S11, . . . ,S20} are presented in Fig. 32.6, including the experimental rig, S20; importantly, these systems were in the hold-out group, that did not contribute data to train the form. For the normal-condition test data, relating to the hold-out and training systems, the MSD discordancy measure generally falls below the detection threshold, for all members in the population. This is expected, as variation in these data (compared to the form) should mostly relate to measurement noise, as the parameters of the population, i, remain unchanged for each member. There are some false positives present, corresponding to the normal condition FRFs; for example, S 4, S 19 andS 20.Aswell as noise effects, these false-positives are likely to be due to more ‘extreme’ parameter sets being drawn from the underlying distribution p( i) of the homogeneous population. Notably, for the experimental member S 20, a normal condition FRFs is flagged as an outlier, while the rest are close to the threshold. This is unsurprising, however, considering that this member did not contribute any training data to learn the form; additionally, errors in the estimated parameters from Table 32.1 will add to the discordancy. Observing the MSD values from the damaged-condition FRFs, intuitively, the number of true positives increases as the severity of damage increases. Generally speaking, the form fails to highlight 7% damage, with increased sensitivity to 14% damage, and successfully flagging all 24% damage observations as outlying. False negatives for 7% damage likely occur because, at low levels of damage, the variation across the population defined byp( ) is similar to (or more severe than) the variations due to damage. As a result, with the current model of the form, variations within the normal-condition training data are masking the variations due to low-level damage. To expose low-level damage, another definition of the form is required; this can be done by defining an alternative feature-space, an alternative model, or both. 32.3.4 Discussion This case study has demonstrated that the form can be used as a general representation of a strongly homogeneous population. Given training data from a subset of members, the form is able to model missing information from the hold-out group, to aid diagnostic decisions. The success of this initial approach, however, depends greatly onp( i), which, in turn, depends on the type of population. If the underlying density p( i) across members is expected to be too dispersed and/or multi-modal (unlike the Gaussian
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